Introduction to Cobordism and Complex Oriented Cohomology



This page collects introductory seminar notes to the concepts of generalized (Eilenberg-Steenrod) cohomology theory, basics of cobordism theory and complex oriented cohomology.


The category of those generalized cohomology theories that are equipped with a universal “complex orientation” happens to unify within it the abstract structure theory of stable homotopy theory with the concrete richness of the differential topology of cobordism theory and of the arithmetic geometry of formal group laws, such as elliptic curves. In the seminar we work through classical results in algebraic topology, organized such as to give in the end a first glimpse of the modern picture of chromatic homotopy theory.


For background on stable homotopy theory see Introduction to Stable homotopy theory.

For application to/of the Adams spectral sequence see Introduction to the Adams Spectral Sequence






Special and general types

Special notions


Extra structure



Manifolds and cobordisms



Outline. We start with two classical topics of algebraic topology that first run independently in parallel:

The development of either of these happens to give rise to the concept of spectra and via this concept it turns out that both topics are intimately related. The unification of both is our third topic


Literature. (Kochman 96).

Generalized cohomology

Idea. The concept that makes algebraic topology be about methods of homological algebra applied to topology is that of generalized homology and generalized cohomology: these are covariant functors or contravariant functors, respectively,

SpacesAb Spaces \longrightarrow Ab^{\mathbb{Z}}

from (sufficiently nice) topological spaces to \mathbb{Z}-graded abelian groups, such that a few key properties of the homotopy types of topological spaces is preserved as one passes them from Ho(Top) to the much more tractable abelian category Ab.

Literature. (Aguilar-Gitler-Prieto 02, chapters 7,8 and 12, Kochman 96, 3.4, 4.2, Schwede 12, II.6)

Generalized cohomology functors

Idea. A generalized (Eilenberg-Steenrod) cohomology theory is such a contravariant functor which satisfies the key properties exhibited by ordinary cohomology (as computed for instance by singular cohomology), notably homotopy invariance and excision, except that its value on the point is not required to be concentrated in degree 0. Dually for generalized homology. There are two versions of the axioms, one for reduced cohomology, and they are equivalent if properly set up.

An important example of a generalised cohomology theory other than ordinary cohomology is topological K-theory. The other two examples of key relevance below are cobordism cohomology and stable cohomotopy.

Literature. (Switzer 75, section 7, Aguilar-Gitler-Prieto 02, section 12 and section 9, Kochman 96, 3.4).


Reduced cohomology

The traditional formulation of reduced generalized cohomology in terms of point-set topology is this:


A reduced cohomology theory is

  1. a functor

    E˜ :(Top CW */) opAb \tilde E^\bullet \;\colon\; (Top^{\ast/}_{CW})^{op} \longrightarrow Ab^{\mathbb{Z}}

    from the opposite of pointed topological spaces (CW-complexes) to \mathbb{Z}-graded abelian groups (“cohomology groups”), in components

    E˜:(XfY)(E˜ (Y)f *E˜ (X)), \tilde E \;\colon\; (X \stackrel{f}{\longrightarrow} Y) \mapsto (\tilde E^\bullet(Y) \stackrel{f^\ast}{\longrightarrow} \tilde E^\bullet(X)) \,,
  2. equipped with a natural isomorphism of degree +1, to be called the suspension isomorphism, of the form

    σ E:E˜ ()E˜ +1(Σ) \sigma_E \;\colon\; \tilde E^\bullet(-) \overset{\simeq}{\longrightarrow} \tilde E^{\bullet +1}(\Sigma -)

such that:

  1. (homotopy invariance) If f 1,f 2:XYf_1,f_2 \colon X \longrightarrow Y are two morphisms of pointed topological spaces such that there is a (base point preserving) homotopy f 1f 2f_1 \simeq f_2 between them, then the induced homomorphisms of abelian groups are equal

    f 1 *=f 2 *. f_1^\ast = f_2^\ast \,.
  2. (exactness) For i:AXi \colon A \hookrightarrow X an inclusion of pointed topological spaces, with j:XCone(i)j \colon X \longrightarrow Cone(i) the induced mapping cone (def.), then this gives an exact sequence of graded abelian groups

    E˜ (Cone(i))j *E˜ (X)i *E˜ (A). \tilde E^\bullet(Cone(i)) \overset{j^\ast}{\longrightarrow} \tilde E^\bullet(X) \overset{i^\ast}{\longrightarrow} \tilde E^\bullet(A) \,.

(e.g. AGP 02, def. 12.1.4)

This is equivalent (prop. below) to the following more succinct homotopy-theoretic definition:


A reduced generalized cohomology theory is a functor

E˜ :Ho(Top */) opAb \tilde E^\bullet \;\colon\; Ho(Top^{\ast/})^{op} \longrightarrow Ab^{\mathbb{Z}}

from the opposite of the pointed classical homotopy category (def., def.), to \mathbb{Z}-graded abelian groups, and equipped with natural isomorphisms, to be called the suspension isomorphism of the form

σ:E˜ +1(Σ)E˜ () \sigma \;\colon\; \tilde E^{\bullet +1}(\Sigma -) \overset{\simeq}{\longrightarrow} \tilde E^\bullet(-)

such that:

As a consequence (prop. below), we find yet another equivalent definition:


A reduced generalized cohomology theory is a functor

E˜ :(Top */) opAb \tilde E^\bullet \;\colon\; (Top^{\ast/})^{op} \longrightarrow Ab^{\mathbb{Z}}

from the opposite of the category of pointed topological spaces to \mathbb{Z}-graded abelian groups, such that

and equipped with natural isomorphism, to be called the suspension isomorphism of the form

σ:E˜ +1(Σ)E˜ () \sigma \;\colon\; \tilde E^{\bullet +1}(\Sigma -) \overset{\simeq}{\longrightarrow} \tilde E^\bullet(-)

such that


The three definitions

  • def.

  • def.

  • def.

are indeed equivalent.


Regarding the equivalence of def. with def. :

By the existence of the classical model structure on topological spaces (thm.), the characterization of its homotopy category (cor.) and the existence of CW-approximations, the homotopy invariance axiom in def. is equivalent to the functor passing to the classical pointed homotopy category. In view of this and since on CW-complexes the standard topological mapping cone construction is a model for the homotopy cofiber (prop.), this gives the equivalence of the two versions of the exactness axiom.

Regarding the equivalence of def. with def. :

This is the universal property of the classical homotopy category (thm.) which identifies it with the localization (def.) of Top */Top^{\ast/} at the weak homotopy equivalences (thm.), together with the existence of CW approximations (rmk.): jointly this says that, up to natural isomorphism, there is a bijection between functors FF and F˜\tilde F in the following diagram (which is filled by a natural isomorphism itself):

Top op F Ab γ Top F˜ Ho(Top) op(Top CW)/ \array{ Top^{op} &\overset{F}{\longrightarrow}& Ab^{\mathbb{Z}} \\ {}^{\mathllap{\gamma_{Top}}}\downarrow & \nearrow_{\mathrlap{\tilde F}} \\ Ho(Top)^{op}\simeq (Top_{CW})/_\sim }

where FF sends weak homotopy equivalences to isomorphisms and where () (-)_\sim means identifying homotopic maps.

Prop. naturally suggests (e.g. Lurie 10, section 1.4) that the concept of generalized cohomology be formulated in the generality of any abstract homotopy theory (model category), not necessarily that of (pointed) topological spaces:


Let 𝒞\mathcal{C} be a model category (def.) with 𝒞 */\mathcal{C}^{\ast/} its pointed model category (prop.).

A reduced additive generalized cohomology theory on 𝒞\mathcal{C} is

  1. a functor

    E˜ :Ho(𝒞 */) opAb \tilde E^\bullet \;\colon \; Ho(\mathcal{C}^{\ast/})^{op} \longrightarrow Ab^{\mathbb{Z}}
  2. a natural isomorphism (“suspension isomorphisms”) of degree +1

    σ:E˜ E˜ +1Σ \sigma \; \colon \; \tilde E^\bullet \longrightarrow \tilde E^{\bullet+1} \circ \Sigma

such that

Finally we need the following terminology:


Let E˜ \tilde E^\bullet be a reduced cohomology theory according to either of def. , def. , def. or def. .

We say E˜ \tilde E^\bullet is additive if in addition

  • (wedge axiom) For {X i} iI\{X_i\}_{i \in I} any set of pointed CW-complexes, then the canonical morphism

    E˜ ( iIX i) iIE˜ (X i) \tilde E^\bullet(\vee_{i \in I} X_i) \longrightarrow \prod_{i \in I} \tilde E^\bullet(X_i)

    from the functor applied to their wedge sum (def.), to the product of its values on the wedge summands, is an isomorphism.

We say E˜ \tilde E^\bullet is ordinary if its value on the 0-sphere S 0S^0 is concentrated in degree 0:

  • (Dimension) E˜ 0(𝕊 0)0\tilde E^{\bullet\neq 0}(\mathbb{S}^0) \simeq 0.

If E˜ \tilde E^\bullet is not ordinary, one also says that it is generalized or extraordinary.

A homomorphism of reduced cohomology theories

η:E˜ F˜ \eta \;\colon\; \tilde E^\bullet \longrightarrow \tilde F^\bullet

is a natural transformation between the underlying functors which is compatible with the suspension isomorphisms in that all the following squares commute

E˜ (X) η X F˜ (X) σ E σ F E˜ +1(ΣX) η ΣX F˜ +1(ΣX). \array{ \tilde E^\bullet(X) &\overset{\eta_X}{\longrightarrow}& \tilde F^\bullet(X) \\ {}^{\mathllap{\sigma_E}}\downarrow && \downarrow^{\mathrlap{\sigma_F}} \\ \tilde E^{\bullet + 1}(\Sigma X) &\overset{\eta_{\Sigma X}}{\longrightarrow}& \tilde F^{\bullet + 1}(\Sigma X) } \,.

We now discuss some constructions and consequences implied by the concept of reduced cohomology theories:


Given a generalized cohomology theory (E ,δ)(E^\bullet,\delta) on some 𝒞\mathcal{C} as in def. , and given a homotopy cofiber sequence in 𝒞\mathcal{C} (prop.),

XfYgZcoker(g)ΣX, X \stackrel{f}{\longrightarrow} Y \stackrel{g}{\longrightarrow} Z \stackrel{coker(g)}{\longrightarrow} \Sigma X \,,

then the corresponding connecting homomorphism is the composite

:E (X)σE +1(ΣX)coker(g) *E +1(Z). \partial \;\colon\; E^\bullet(X) \stackrel{\sigma}{\longrightarrow} E^{\bullet+1}(\Sigma X) \stackrel{coker(g)^\ast}{\longrightarrow} E^{\bullet+1}(Z) \,.

The connecting homomorphisms of def. are parts of long exact sequences

E (Z)E (Y)E (X)E +1(Z). \cdots \stackrel{\partial}{\longrightarrow} E^{\bullet}(Z) \longrightarrow E^\bullet(Y) \longrightarrow E^\bullet(X) \stackrel{\partial}{\longrightarrow} E^{\bullet+1}(Z) \to \cdots \,.

By the defining exactness of E E^\bullet, def. , and the way this appears in def. , using that σ\sigma is by definition an isomorphism.

Unreduced cohomology

Given a reduced generalized cohomology theory as in def. , we may “un-reduce” it and evaluate it on unpointed topological spaces XX simply by evaluating it on X +X_+ (def.). It is conventional to further generalize to relative cohomology and evaluate on unpointed subspace inclusions i:AXi \colon A \hookrightarrow X, taken as placeholders for their mapping cones Cone(i +)Cone(i_+) (prop.).

In the following a pair (X,U)(X,U) refers to a subspace inclusion of topological spaces UXU \hookrightarrow X. Whenever only one space is mentioned, the subspace is assumed to be the empty set (X,)(X, \emptyset). Write Top CW Top_{CW}^{\hookrightarrow} for the category of such pairs (the full subcategory of the arrow category of Top CWTop_{CW} on the inclusions). We identify Top CWTop CW Top_{CW} \hookrightarrow Top_{CW}^{\hookrightarrow} by X(X,)X \mapsto (X,\emptyset).


A cohomology theory (unreduced, relative) is

  1. a functor

    E :(Top CW ) opAb E^\bullet : (Top_{CW}^{\hookrightarrow})^{op} \to Ab^{\mathbb{Z}}

    to the category of \mathbb{Z}-graded abelian groups,

  2. a natural transformation of degree +1, to be called the connecting homomorphism, of the form

    δ (X,A):E (A,)E +1(X,A). \delta_{(X,A)} \;\colon\; E^\bullet(A, \emptyset) \to E^{\bullet + 1}(X, A) \,.

such that:

  1. (homotopy invariance) For f:(X 1,A 1)(X 2,A 2)f \colon (X_1,A_1) \to (X_2,A_2) a homotopy equivalence of pairs, then

    E (f):E (X 2,A 2)E (X 1,A 1) E^\bullet(f) \;\colon\; E^\bullet(X_2,A_2) \stackrel{\simeq}{\longrightarrow} E^\bullet(X_1,A_1)

    is an isomorphism;

  2. (exactness) For AXA \hookrightarrow X the induced sequence

    E n(X,A)E n(X)E n(A)δE n+1(X,A) \cdots \to E^n(X, A) \longrightarrow E^n(X) \longrightarrow E^n(A) \stackrel{\delta}{\longrightarrow} E^{n+1}(X, A) \to \cdots

    is a long exact sequence of abelian groups.

  3. (excision) For UAXU \hookrightarrow A \hookrightarrow X such that U¯Int(A)\overline{U} \subset Int(A), then the natural inclusion of the pair i:(XU,AU)(X,A)i \colon (X-U, A-U) \hookrightarrow (X, A) induces an isomorphism

    E (i):E n(X,A)E n(XU,AU) E^\bullet(i) \;\colon\; E^n(X, A) \overset{\simeq}{\longrightarrow} E^n(X-U, A-U)

We say E E^\bullet is additive if it takes coproducts to products:

  • (additivity) If (X,A)= i(X i,A i)(X, A) = \coprod_i (X_i, A_i) is a coproduct, then the canonical comparison morphism

    E n(X,A) iE n(X i,A i) E^n(X, A) \overset{\simeq}{\longrightarrow} \prod_i E^n(X_i, A_i)

    is an isomorphism from the value on (X,A)(X,A) to the product of values on the summands.

We say E E^\bullet is ordinary if its value on the point is concentrated in degree 0

  • (Dimension): E 0(*,)=0E^{\bullet \neq 0}(\ast,\emptyset) = 0.

A homomorphism of unreduced cohomology theories

η:E F \eta \;\colon\; E^\bullet \longrightarrow F^\bullet

is a natural transformation of the underlying functors that is compatible with the connecting homomorphisms, hence such that all these squares commute:

E (A,) η (A,) F (A,) δ E δ F E +1(X,A) η (X,A) F +1(X,A). \array{ E^\bullet(A,\emptyset) &\overset{\eta_{(A,\emptyset)}}{\longrightarrow}& F^\bullet(A,\emptyset) \\ {}^{\mathllap{\delta_E}}\downarrow && \downarrow^{\mathrlap{\delta_F}} \\ E^{\bullet +1}(X,A) &\overset{\eta_{(X,A)}}{\longrightarrow}& F^{\bullet +1}(X,A) } \,.

e.g. (AGP 02, def. 12.1.1).


The excision axiom in def. is equivalent to the following statement:

For all A,BXA,B \hookrightarrow X with X=Int(A)Int(B)X = Int(A) \cup Int(B), then the inclusion

i:(A,AB)(X,B) i \colon (A, A \cap B) \longrightarrow (X,B)

induces an isomorphism,

i *:E (X,B)E (A,AB) i^\ast \;\colon\; E^\bullet(X, B) \overset{\simeq}{\longrightarrow} E^\bullet(A, A \cap B)

(e.g Switzer 75, 7.2)


In one direction, suppose that E E^\bullet satisfies the original excision axiom. Given A,BA,B with X=Int(A)Int(B)X = \Int(A) \cup Int(B), set UXAU \coloneqq X-A and observe that

U¯ =XA¯ =XInt(A) Int(B) \begin{aligned} \overline{U} & = \overline{X-A} \\ & = X- Int(A) \\ & \subset Int(B) \end{aligned}

and that

(XU,BU)=(A,AB). (X-U, B-U) = (A, A \cap B) \,.

Hence the excision axiom implies E (X,B)E (A,AB) E^\bullet(X, B) \overset{\simeq}{\longrightarrow} E^\bullet(A, A \cap B).

Conversely, suppose E E^\bullet satisfies the alternative condition. Given UAXU \hookrightarrow A \hookrightarrow X with U¯Int(A)\overline{U} \subset Int(A), observe that we have a cover

Int(XU)Int(A) =(XU¯)Int(A) (XInt(A))Int(A) =X \begin{aligned} Int(X-U) \cup Int(A) & = (X - \overline{U}) \cap \Int(A) \\ & \supset (X - Int(A)) \cap Int(A) \\ & = X \end{aligned}

and that

(XU,(XU)A)=(XU,AU). (X-U, (X-U) \cap A) = (X-U, A - U) \,.


E (XU,AU)E (XU,(XU)A)E (X,A). E^\bullet(X-U,A-U) \simeq E^\bullet(X-U, (X-U)\cap A) \simeq E^\bullet(X,A) \,.

The following lemma shows that the dependence in pairs of spaces in a generalized cohomology theory is really a stand-in for evaluation on homotopy cofibers of inclusions.


Let E E^\bullet be an cohomology theory, def. , and let AXA \hookrightarrow X. Then there is an isomorphism

E (X,A)E (XCone(A),*) E^\bullet(X,A) \stackrel{\simeq}{\longrightarrow} E^\bullet(X \cup Cone(A), \ast)

between the value of E E^\bullet on the pair (X,A)(X,A) and its value on the unreduced mapping cone of the inclusion (rmk.), relative to a basepoint.

If moreover AXA \hookrightarrow X is (the retract of) a relative cell complex inclusion, then also the morphism in cohomology induced from the quotient map p:(X,A)(X/A,*)p \;\colon\; (X,A)\longrightarrow (X/A, \ast) is an isomorphism:

E (p):E (X/A,*)E (X,A). E^\bullet(p) \;\colon\; E^\bullet(X/A,\ast) \longrightarrow E^\bullet(X,A) \,.

(e.g AGP 02, corollary 12.1.10)


Consider U(Cone(A)A×{0})Cone(A)U \coloneqq (Cone(A)-A \times \{0\}) \hookrightarrow Cone(A), the cone on AA minus the base AA. We have

(XCone(A)U,Cone(A)U)(X,A) ( X\cup Cone(A)-U, Cone(A)-U) \simeq (X,A)

and hence the first isomorphism in the statement is given by the excision axiom followed by homotopy invariance (along the contraction of the cone to the point).

Next consider the quotient of the mapping cone of the inclusion:

(XCone(A),Cone(A))(X/A,*). ( X\cup Cone(A), Cone(A) ) \longrightarrow (X/A,\ast) \,.

If AXA \hookrightarrow X is a cofibration, then this is a homotopy equivalence since Cone(A)Cone(A) is contractible and since by the dual factorization lemma (lem.) and by the invariance of homotopy fibers under weak equivalences (lem.), XCone(A)X/AX \cup Cone(A)\to X/A is a weak homotopy equivalence, hence, by the universal property of the classical homotopy category (thm.) a homotopy equivalence on CW-complexes.

Hence now we get a composite isomorphism

E (X/A,*)E (XCone(A),Cone(A))E (X,A). E^\bullet(X/A,\ast) \overset{\simeq}{\longrightarrow} E^\bullet( X\cup Cone(A), Cone(A) ) \overset{\simeq}{\longrightarrow} E^\bullet(X,A) \,.

As an important special case of : Let (X,x)(X,x) be a pointed CW-complex. For p:(Cone(X),X)(ΣX,{x})p\colon (Cone(X), X) \to (\Sigma X,\{x\}) the quotient map from the reduced cone on XX to the reduced suspension, then

E (p):E (Cone(X),X)E (ΣX,{x}) E^\bullet(p) \;\colon\; E^\bullet(Cone(X),X) \overset{\simeq}{\longrightarrow} E^\bullet(\Sigma X, \{x\})

is an isomorphism.


(exact sequence of a triple)

For E E^\bullet an unreduced generalized cohomology theory, def. , then every inclusion of two consecutive subspaces

ZYX Z \hookrightarrow Y \hookrightarrow X

induces a long exact sequence of cohomology groups of the form

E q1(Y,Z)δ¯E q(X,Y)E q(X,Z)E q(Y,Z) \cdots \to E^{q-1}(Y,Z) \stackrel{\bar \delta}{\longrightarrow} E^q(X,Y) \stackrel{}{\longrightarrow} E^q(X,Z) \stackrel{}{\longrightarrow} E^q(Y,Z) \to \cdots


δ¯:E q1(Y,Z)E q1(Y)δE q(X,Y). \bar \delta \;\colon \; E^{q-1}(Y,Z) \longrightarrow E^{q-1}(Y) \stackrel{\delta}{\longrightarrow} E^{q}(X,Y) \,.

Apply the braid lemma to the interlocking long exact sequences of the three pairs (X,Y)(X,Y), (X,Z)(X,Z), (Y,Z)(Y,Z):

(graphics from this Maths.SE comment, showing the dual situation for homology)

See here for details.


The exact sequence of a triple in prop. is what gives rise to the Cartan-Eilenberg spectral sequence for EE-cohomology of a CW-complex XX.


For (X,x)(X,x) a pointed topological space and Cone(X)=(X(I +))/XCone(X) = (X \wedge (I_+))/X its reduced cone, the long exact sequence of the triple ({x},X,Cone(X))(\{x\}, X, Cone(X)), prop. ,

0E q(Cone(X),{x})E q(X,{x})δ¯E q+1(Cone(X),X)E q+1(Cone(X),{x})0 0 \simeq E^q(Cone(X), \{x\}) \longrightarrow E^q(X,\{x\}) \overset{\bar \delta}{\longrightarrow} E^{q+1}(Cone(X),X) \longrightarrow E^{q+1}(Cone(X), \{x\}) \simeq 0

exhibits the connecting homomorphism δ¯\bar \delta here as an isomorphism

δ¯:E q(X,{x})E q+1(Cone(X),X). \bar \delta \;\colon\; E^q(X,\{x\}) \overset{\simeq}{\longrightarrow} E^{q+1}(Cone(X),X) \,.

This is the suspension isomorphism extracted from the unreduced cohomology theory, see def. below.


(Mayer-Vietoris sequence)

Given E E^\bullet an unreduced cohomology theory, def. . Given a topological space covered by the interior of two spaces as X=Int(A)Int(B)X = Int(A) \cup Int(B), then for each CABC \subset A \cap B there is a long exact sequence of cohomology groups of the form

E n1(AB,C)δ¯E n(X,C)E n(A,C)E n(B,C)E n(AB,C). \cdots \to E^{n-1}(A \cap B , C) \overset{\bar \delta}{\longrightarrow} E^n(X,C) \longrightarrow E^n(A,C) \oplus E^n(B,C) \longrightarrow E^n(A \cap B, C) \to \cdots \,.

e.g. (Switzer 75, theorem 7.19, Aguilar-Gitler-Prieto 02, theorem 12.1.22)

Relation between unreduced and reduced cohomology


(unreduced to reduced cohomology)

Let E E^\bullet be an unreduced cohomology theory, def. . Define a reduced cohomology theory, def. (E˜ ,σ)(\tilde E^\bullet, \sigma) as follows.

For x:*Xx \colon \ast \to X a pointed topological space, set

E˜ (X,x)E (X,{x}). \tilde E^\bullet(X,x) \coloneqq E^\bullet(X,\{x\}) \,.

This is clearly functorial. Take the suspension isomorphism to be the composite

σ:E˜ +1(ΣX)=E +1(ΣX,{x})E (p)E +1(Cone(X),X)δ¯ 1E (X,{x})=E˜ (X) \sigma \;\colon\; \tilde E^{\bullet+1}(\Sigma X) = E^{\bullet+1}(\Sigma X, \{x\}) \overset{E^\bullet(p)}{\longrightarrow} E^{\bullet+1}(Cone(X),X) \overset{\bar \delta^{-1}}{\longrightarrow} E^\bullet(X,\{x\}) = \tilde E^{\bullet}(X)

of the isomorphism E (p)E^\bullet(p) from example and the inverse of the isomorphism δ¯\bar \delta from example .


The construction in def. indeed gives a reduced cohomology theory.

(e.g Switzer 75, 7.34)


We need to check the exactness axiom given any AXA\hookrightarrow X. By lemma we have an isomorphism

E˜ (XCone(A))=E (XCone(A),{*})E (X,A). \tilde E^\bullet(X \cup Cone(A)) = E^\bullet(X \cup Cone(A), \{\ast\}) \overset{\simeq}{\longrightarrow} E^\bullet(X,A) \,.

Unwinding the constructions shows that this makes the following diagram commute:

E˜ (XCone(A)) E (X,A) E˜ (X) = E (X,{x}) E˜ (A) = E (A,{a}), \array{ \tilde E^\bullet(X\cup Cone(A)) &\overset{\simeq}{\longrightarrow}& E^\bullet(X,A) \\ \downarrow && \downarrow \\ \tilde E^\bullet(X) &=& E^\bullet(X,\{x\}) \\ \downarrow && \downarrow \\ \tilde E^\bullet(A) &=& E^\bullet(A,\{a\}) } \,,

where the vertical sequence on the right is exact by prop. . Hence the left vertical sequence is exact.


(reduced to unreduced cohomology)

Let (E˜ ,σ)(\tilde E^\bullet, \sigma) be a reduced cohomology theory, def. . Define an unreduced cohomolog theory E E^\bullet, def. , by

E (X,A)E˜ (X +Cone(A +)) E^\bullet(X,A) \coloneqq \tilde E^\bullet( X_+ \cup Cone(A_+))

and let the connecting homomorphism be as in def. .


The construction in def. indeed yields an unreduced cohomology theory.

e.g. (Switzer 75, 7.35)


Exactness holds by prop. . For excision, it is sufficient to consider the alternative formulation of lemma . For CW-inclusions, this follows immediately with lemma .


The constructions of def. and def. constitute a pair of functors between then categories of reduced cohomology theories, def. and unreduced cohomology theories, def. which exhbit an equivalence of categories.


(…careful with checking the respect for suspension iso and connecting homomorphism..)

To see that there are natural isomorphisms relating the two composites of these two functors to the identity:

One composite is

E (E˜ :(X,x)E (X,{x})) ((E) :(X,A)E (X +Cone(A +)),*), \begin{aligned} E^\bullet & \mapsto (\tilde E^\bullet \colon (X,x) \mapsto E^\bullet(X,\{x\})) \\ & \mapsto ((E')^\bullet \colon (X,A) \mapsto E^\bullet( X_+ \cup Cone(A_+) ), \ast) \end{aligned} \,,

where on the right we have, from the construction, the reduced mapping cone of the original inclusion AXA \hookrightarrow X with a base point adjoined. That however is isomorphic to the unreduced mapping cone of the original inclusion (prop.- P#UnreducedMappingConeAsReducedConeOfBasedPointAdjoined)). With this the natural isomorphism is given by lemma .

The other composite is

E˜ (E :(X,A)E˜ (X +Cone(A +))) ((E˜) :XE˜ (X +Cone(* +))) \begin{aligned} \tilde E^\bullet & \mapsto (E^\bullet \colon (X,A) \mapsto \tilde E^\bullet(X_+ \cup Cone(A_+))) \\ & \mapsto ((\tilde E')^\bullet \colon X \mapsto \tilde E^\bullet(X_+ \cup Cone(*_+))) \end{aligned}

where on the right we have the reduced mapping cone of the point inclusion with a point adoined. As before, this is isomorphic to the unreduced mapping cone of the point inclusion. That finally is clearly homotopy equivalent to XX, and so now the natural isomorphism follows with homotopy invariance.

Finally we record the following basic relation between reduced and unreduced cohomology:


Let E E^\bullet be an unreduced cohomology theory, and E˜ \tilde E^\bullet its reduced cohomology theory from def. . For (X,*)(X,\ast) a pointed topological space, then there is an identification

E (X)E˜ (X)E (*) E^\bullet(X) \simeq \tilde E^\bullet(X) \oplus E^\bullet(\ast)

of the unreduced cohomology of XX with the direct sum of the reduced cohomology of XX and the unreduced cohomology of the base point.


The pair *X\ast \hookrightarrow X induces the sequence

E 1(*)δE˜ (X)E (X)E (*)δE˜ +1(X) \cdots \to E^{\bullet-1}(\ast) \stackrel{\delta}{\longrightarrow} \tilde E^\bullet(X) \stackrel{}{\longrightarrow} E^\bullet(X) \stackrel{}{\longrightarrow} E^\bullet(\ast) \stackrel{\delta}{\longrightarrow} \tilde E^{\bullet+1}(X) \to \cdots

which by the exactness clause in def. is exact.

Now since the composite *X*\ast \to X \to \ast is the identity, the morphism E (X)E (*)E^\bullet(X) \to E^\bullet(\ast) has a section and so is in particular an epimorphism. Therefore, by exactness, the connecting homomorphism vanishes, δ=0\delta = 0 and we have a short exact sequence

0E˜ (X)E (X)E (*)0 0 \to \tilde E^\bullet(X) \stackrel{}{\longrightarrow} E^\bullet(X) \stackrel{}{\longrightarrow} E^\bullet(\ast) \to 0

with the right map an epimorphism. Hence this is a split exact sequence and the statement follows.

Generalized homology functors

All of the above has a dual version with generalized cohomology replaced by generalized homology. For ease of reference, we record these dual definitions:


A reduced homology theory is a functor

E˜ :(Top CW */)Ab \tilde E_\bullet \;\colon\; (Top^{\ast/}_{CW}) \longrightarrow Ab^{\mathbb{Z}}

from the category of pointed topological spaces (CW-complexes) to \mathbb{Z}-graded abelian groups (“homology groups”), in components

E˜ :(XfY)(E˜ (X)f *E˜ (Y)), \tilde E _\bullet \;\colon\; (X \stackrel{f}{\longrightarrow} Y) \mapsto (\tilde E_\bullet(X) \stackrel{f_\ast}{\longrightarrow} \tilde E_\bullet(Y)) \,,

and equipped with a natural isomorphism of degree +1, to be called the suspension isomorphism, of the form

σ:E˜ ()E˜ +1(Σ) \sigma \;\colon\; \tilde E_\bullet(-) \overset{\simeq}{\longrightarrow} \tilde E_{\bullet +1}(\Sigma -)

such that:

  1. (homotopy invariance) If f 1,f 2:XYf_1,f_2 \colon X \longrightarrow Y are two morphisms of pointed topological spaces such that there is a (base point preserving) homotopy f 1f 2f_1 \simeq f_2 between them, then the induced homomorphisms of abelian groups are equal

    f 1*=f 2*. f_1_\ast = f_2_\ast \,.
  2. (exactness) For i:AXi \colon A \hookrightarrow X an inclusion of pointed topological spaces, with j:XCone(i)j \colon X \longrightarrow Cone(i) the induced mapping cone, then this gives an exact sequence of graded abelian groups

    E˜ (A)i *E˜ (X)j *E˜ (Cone(i)). \tilde E_\bullet(A) \overset{i_\ast}{\longrightarrow} \tilde E_\bullet(X) \overset{j_\ast}{\longrightarrow} \tilde E_\bullet(Cone(i)) \,.

We say E˜ \tilde E_\bullet is additive if in addition

  • (wedge axiom) For {X i} iI\{X_i\}_{i \in I} any set of pointed CW-complexes, then the canonical morphism

    iIE˜ (X i)E˜ ( iIX i) \oplus_{i \in I} \tilde E_\bullet(X_i) \longrightarrow \tilde E^\bullet(\vee_{i \in I} X_i)

    from the direct sum of the value on the summands to the value on the wedge sum (prop.- P#WedgeSumAsCoproduct)), is an isomorphism.

We say E˜ \tilde E_\bullet is ordinary if its value on the 0-sphere S 0S^0 is concentrated in degree 0:

  • (Dimension) E˜ 0(𝕊 0)0\tilde E_{\bullet\neq 0}(\mathbb{S}^0) \simeq 0.

A homomorphism of reduced cohomology theories

η:E˜ F˜ \eta \;\colon\; \tilde E_\bullet \longrightarrow \tilde F_\bullet

is a natural transformation between the underlying functors which is compatible with the suspension isomorphisms in that all the following squares commute

E˜ (X) η X F˜ (X) σ E σ F E˜ +1(ΣX) η ΣX F˜ +1(ΣX). \array{ \tilde E_\bullet(X) &\overset{\eta_X}{\longrightarrow}& \tilde F_\bullet(X) \\ {}^{\mathllap{\sigma_E}}\downarrow && \downarrow^{\mathrlap{\sigma_F}} \\ \tilde E_{\bullet + 1}(\Sigma X) &\overset{\eta_{\Sigma X}}{\longrightarrow}& \tilde F_{\bullet + 1}(\Sigma X) } \,.

A homology theory (unreduced, relative) is a functor

E :(Top CW )Ab E_\bullet : (Top_{CW}^{\hookrightarrow}) \longrightarrow Ab^{\mathbb{Z}}

to the category of \mathbb{Z}-graded abelian groups, as well as a natural transformation of degree +1, to be called the connecting homomorphism, of the form

δ (X,A):E +1(X,A)E (A,). \delta_{(X,A)} \;\colon\; E_{\bullet + 1}(X, A) \longrightarrow E^\bullet(A, \emptyset) \,.

such that:

  1. (homotopy invariance) For f:(X 1,A 1)(X 2,A 2)f \colon (X_1,A_1) \to (X_2,A_2) a homotopy equivalence of pairs, then

    E (f):E (X 1,A 1)E (X 2,A 2) E_\bullet(f) \;\colon\; E_\bullet(X_1,A_1) \stackrel{\simeq}{\longrightarrow} E_\bullet(X_2,A_2)

    is an isomorphism;

  2. (exactness) For AXA \hookrightarrow X the induced sequence

    E n+1(X,A)δE n(A)E n(X)E n(X,A) \cdots \to E_{n+1}(X, A) \stackrel{\delta}{\longrightarrow} E_n(A) \longrightarrow E_n(X) \longrightarrow E_n(X, A) \to \cdots

    is a long exact sequence of abelian groups.

  3. (excision) For UAXU \hookrightarrow A \hookrightarrow X such that U¯Int(A)\overline{U} \subset Int(A), then the natural inclusion of the pair i:(XU,AU)(X,A)i \colon (X-U, A-U) \hookrightarrow (X, A) induces an isomorphism

    E (i):E n(XU,AU)E n(X,A) E_\bullet(i) \;\colon\; E_n(X-U, A-U) \overset{\simeq}{\longrightarrow} E_n(X, A)

We say E E^\bullet is additive if it takes coproducts to direct sums:

  • (additivity) If (X,A)= i(X i,A i)(X, A) = \coprod_i (X_i, A_i) is a coproduct, then the canonical comparison morphism

    iE n(X i,A i)E n(X,A) \oplus_i E^n(X_i, A_i) \overset{\simeq}{\longrightarrow} E^n(X, A)

    is an isomorphismfrom the direct sum of the value on the summands, to the value on the total pair.

We say E E_\bullet is ordinary if its value on the point is concentrated in degree 0

  • (Dimension): E 0(*,)=0E_{\bullet \neq 0}(\ast,\emptyset) = 0.

A homomorphism of unreduced homology theories

η:E F \eta \;\colon\; E_\bullet \longrightarrow F_\bullet

is a natural transformation of the underlying functors that is compatible with the connecting homomorphisms, hence such that all these squares commute:

E +1(X,A) η (X,A) F +1(X,A) δ E δ F E (A,) η (A,) F (A,). \array{ E_{\bullet +1}(X,A) &\overset{\eta_{(X,A)}}{\longrightarrow}& F_{\bullet +1}(X,A) \\ {}^{\mathllap{\delta_E}}\downarrow && \downarrow^{\mathrlap{\delta_F}} \\ E_\bullet(A,\emptyset) &\overset{\eta_{(A,\emptyset)}}{\longrightarrow}& F^\bullet(A,\emptyset) } \,.

Multiplicative cohomology theories

The generalized cohomology theories considered above assign cohomology groups. It is familiar from ordinary cohomology with coefficients not just in a group but in a ring, that also the cohomology groups inherit compatible ring structure. The generalization of this phenomenon to generalized cohomology theories is captured by the concept of multiplicative cohomology theories:


Let E 1,E 2,E 3E_1, E_2, E_3 be three unreduced generalized cohomology theories (def.). A pairing of cohomology theories

μ:E 1E 2E 3 \mu \;\colon\; E_1 \Box E_2 \longrightarrow E_3

is a natural transformation (of functors on (Top CW ×Top CW ) op(Top_{CW}^{\hookrightarrow}\times Top_{CW}^{\hookrightarrow})^{op} ) of the form

μ n 1,n 2:E 1 n 1(X,A)E 2 n 2(Y,B)E 3 n 1+n 2(X×Y,A×YX×B) \mu_{n_1,n_2} \;\colon\; E_1^{n_1}(X,A) \otimes E_2^{n_2}(Y,B) \longrightarrow E_3^{n_1 + n_2}(X\times Y \;,\; A\times Y \cup X \times B)

such that this is compatible with the connecting homomorphisms δ i\delta_i of E iE_i, in that the following are commuting squares

E 1 n 1(A)E 2 n 2(Y,B) δ 1id 2 E 1 n 1+1(X,A)E 2 n 2(Y,B) μ n 1,n 2 μ n 1+1,n 2 E 3 n 1+n 2(A×YX×B,X×B)E 3 n 1+n 2(A×Y,A×B) δ 3 E 3 n 1+n 2+1(X×Y,A×B) \array{ E_1^{n_1}(A) \otimes E_2^{n_2}(Y,B) &\overset{\delta_1 \otimes id_2}{\longrightarrow}& E_1^{n_1+1}(X,A) \otimes E_2^{n_2}(Y,B) \\ {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1+1, n_2}}} \\ \underoverset {E_3^{n_1 + n_2}(A \times Y \cup X \times B , X \times B)} {E_3^{n_1 + n_2}(A \times Y, A \times B)} {\simeq} &\overset{\delta_3}{\longrightarrow}& E_3^{n_1 + n_2+ 1}(X \times Y, A \times B) }


E 1 n 1(X,A)E 2 n 2(B) (1) n 1id 1δ 2 E 1 n 1+1(X,A)E 2 n 2(Y,B) μ n 1,n 2 μ n 1,n 2+1 E 3 n 1+n 2(A×YX×B,A×Y)E 3 n 1+n 2(X×B,A×B) δ 3 E 3 n 1+n 2+1(X×Y,A×B), \array{ E_1^{n_1}(X,A) \otimes E_2^{n_2}(B) &\overset{(-1)^{n_1} id_1 \otimes \delta_2}{\longrightarrow}& E_1^{n_1+1}(X,A) \otimes E_2^{n_2}(Y,B) \\ {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1, n_2 + 1}}} \\ \underoverset {E_3^{n_1 + n_2}(A \times Y \cup X \times B , A \times Y)} {E_3^{n_1 + n_2}(X \times B, A \times B)} {\simeq} &\overset{\delta_3}{\longrightarrow}& E_3^{n_1 + n_2+ 1}(X \times Y, A \times B) } \,,

where the isomorphisms in the bottom left are the excision isomorphisms.


An (unreduced) multiplicative cohomology theory is an unreduced generalized cohomology theory theory EE (def. ) equipped with

  1. (external multiplication) a pairing (def. ) of the form μ:EEE\mu \;\colon\; E \Box E \longrightarrow E;

  2. (unit) an element 1E 0(*)1 \in E^0(\ast)

such that

  1. (associativity) μ(idμ)=μ(μid)\mu \circ (id \otimes \mu) = \mu \circ (\mu \otimes id);

  2. (unitality) μ(1x)=μ(x1)=x\mu(1\otimes x) = \mu(x \otimes 1) = x for all xE n(X,A)x \in E^n(X,A).

The mulitplicative cohomology theory is called commutative (often considered by default) if in addition

  • (graded commutativity)

    E n 1(X,A)E n 2(Y,B) (uv)(1) n 1n 2(vu) E n 2(Y,B)E X,A n 1 μ n 1,n 2 μ n 2,n 1 E n 1+n 2(X×Y,A×YX×B) (switch (X,A),(Y,B)) * E n 1+n 2(Y×X,B×XY×A). \array{ E^{n_1}(X,A) \otimes E^{n_2}(Y,B) &\overset{(u \otimes v) \mapsto (-1)^{n_1 n_2} (v \otimes u) }{\longrightarrow}& E^{n_2}(Y,B) \otimes E^{n_1}_{X,A} \\ {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_2,n_1}}} \\ E^{n_1 + n_2}( X \times Y , A \times Y \cup X \times B) &\underset{(switch_{(X,A), (Y,B)})^\ast}{\longrightarrow}& E^{n_1 + n_2}( Y \times X , B \times X \cup Y \times A) } \,.

Given a multiplicative cohomology theory (E,μ,1)(E, \mu, 1), its cup product is the composite of the above external multiplication with pullback along the diagonal maps Δ (X,A):(X,A)(X×X,A×XX×A)\Delta_{(X,A)} \colon (X,A) \longrightarrow (X\times X, A \times X \cup X \times A);

()():E n 1(X,A)E n 2(X,A)μ n 1,n 2E n 1+n 2(X×X,A×XX×A)Δ (X,A) *E n 1+n 2(X,AB). (-) \cup (-) \;\colon\; E^{n_1}(X,A) \otimes E^{n_2}(X,A) \overset{\mu_{n_1,n_2}}{\longrightarrow} E^{n_1 + n_2}( X \times X, \; A \times X \cup X \times A) \overset{\Delta^\ast_{(X,A)}}{\longrightarrow} E^{n_1 + n_2}(X, \; A \cup B) \,.

e.g. (Tamaki-Kono 06, II.6)


Let (E,μ,1)(E,\mu,1) be a multiplicative cohomology theory, def. . Then

  1. For every space XX the cup product gives E (X)E^\bullet(X) the structure of a \mathbb{Z}-graded ring, which is graded-commutative if (E,μ,1)(E,\mu,1) is commutative.

  2. For every pair (X,A)(X,A) the external multiplication μ\mu gives E (X,A)E^\bullet(X,A) the structure of a left and right module over the graded ring E (*)E^\bullet(\ast).

  3. All pullback morphisms respect the left and right action of E (*)E^\bullet(\ast) and the connecting homomorphisms respect the right action and the left action up to multiplication by (1) n 1(-1)^{n_1}


Regarding the third point:

For pullback maps this is the naturality of the external product: let f:(X,A)(Y,B)f \colon (X,A) \longrightarrow (Y,B) be a morphism in Top CW Top_{CW}^{\hookrightarrow} then naturality says that the following square commutes:

E n 1(*)E n 2(Y,B) μ n 1,n 2 E n 1+n 2(Y,B) (id,f *) f * E n 1(*)E n 2(X,A) μ n 1,n 2 E n 1+n 2(Y,B). \array{ E^{n_1}(\ast) \otimes E^{n_2}(Y,B) &\overset{\mu_{n_1,n_2}}{\longrightarrow}& E^{n_1 + n_2}(Y, B) \\ {}^{\mathllap{(id,f^\ast)}}\downarrow && \downarrow^{\mathrlap{f^\ast}} \\ E^{n_1}(\ast) \otimes E^{n_2}(X,A) &\overset{\mu_{n_1,n_2}}{\longrightarrow}& E^{n_1 + n_2}(Y,B) } \,.

For connecting homomorphisms this is the (graded) commutativity of the squares in def. :

E n 1(*)E n 2(A) (1) n 1(id,δ) E n 1(*)E n 2+2(X) μ n 1,n 2 μ n 1,n 2 E n 1+n 2(A) δ E 3 n 1+n 2+1(X,B). \array{ E^{n_1}(\ast)\otimes E^{n_2}(A) &\overset{(-1)^{n_1} (id, \delta)}{\longrightarrow}& E^{n_1}(\ast) \otimes E^{n_2 + 2}(X) \\ {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1,n_2}}} \\ E^{n_1 + n_2}(A) &\overset{\delta}{\longrightarrow}& E_3^{n_1 + n_2+ 1}(X,B) } \,.

Brown representability theorem

Idea. Given any functor such as the generalized (co)homology functor above, an important question to ask is whether it is a representable functor. Due to the \mathbb{Z}-grading and the suspension isomorphisms, if a generalized (co)homology functor is representable at all, it must be represented by a \mathbb{Z}-indexed sequence of pointed topological spaces such that the reduced suspension of one is comparable to the next one in the list. This is a spectrum or more specifically: a sequential spectrum .

Whitehead observed that indeed every spectrum represents a generalized (co)homology theory. The Brown representability theorem states that, conversely, every generalized (co)homology theory is represented by a spectrum, subject to conditions of additivity.

As a first application, Eilenberg-MacLane spectra representing ordinary cohomology may be characterized via Brown representability.

Literature. (Switzer 75, section 9, Aguilar-Gitler-Prieto 02, section 12, Kochman 96, 3.4)

Traditional discussion

Write Top 1 */Top */Top_{{\geq 1}}^{\ast/} \hookrightarrow Top^{\ast/} for the full subcategory of connected pointed topological spaces. Write Set */Set^{\ast/} for the category of pointed sets.


A Brown functor is a functor

F:Ho(Top 1 */) opSet */ F\;\colon \; Ho(Top_{\geq 1}^{\ast/})^{op} \longrightarrow Set^{\ast/}

(from the opposite of the classical homotopy category (def., def.) of connected pointed topological spaces) such that

  1. (additivity) FF takes small coproducts (wedge sums) to products;

  2. (Mayer-Vietoris) If X=Int(A)Int(B)X = Int(A) \cup Int(B) then for all x AF(A)x_A \in F(A) and x BF(B)x_B \in F(B) such that (x A)| AB=(x B)| AB(x_A)|_{A \cap B} = (x_B)|_{A \cap B} then there exists x XF(X)x_X \in F(X) such that x A=(x X)| Ax_A = (x_X)|_A and x B=(x X)| Bx_B = (x_X)|_B.


For every additive reduced cohomology theory E˜ ():Ho(Top */) opSet */\tilde E^\bullet(-) \colon Ho(Top^{\ast/})^{op}\to Set^{\ast/} (def. ) and for each degree nn \in \mathbb{N}, the restriction of E˜ n()\tilde E^n(-) to connected spaces is a Brown functor (def. ).


Under the relation between reduced and unreduced cohomology above, this follows from the exactness of the Mayer-Vietoris sequence of prop. .


(Brown representability)

Every Brown functor FF (def. ) is representable, hence there exists XTop 1 */X \in Top_{\geq 1}^{\ast/} and a natural isomorphism

[,X] *F() [-,X]_{\ast} \overset{\simeq}{\longrightarrow} F(-)

(where [,] *[-,-]_\ast denotes the hom-functor of Ho(Top 1 */)Ho(Top_{\geq 1}^{\ast/}) (exmpl.)).

(e.g. AGP 02, theorem 12.2.22)


A key subtlety in theorem is the restriction to connected pointed topological spaces in def. . This comes about since the proof of the theorem requires that continuous functions f:XYf \colon X \longrightarrow Y that induce isomorphisms on pointed homotopy classes

[S n,X] *[S n,Y] * [S^n,X]_\ast \longrightarrow [S^n,Y]_\ast

for all nn are weak homotopy equivalences (For instance in AGP 02 this is used in the proof of theorem 12.2.19 there). But [S n,X] *=π n(X,x)[S^n,X]_\ast = \pi_n(X,x) gives the nnth homotopy group of XX only for the canonical basepoint, while for a weak homotopy equivalence in general one needs to consider the homotopy groups at all possible basepoints, at least one for each connected component. But so if one does assume that all spaces involved are connected, hence only have one connected component, then indeed weak homotopy equivalences are equivalently those maps XYX\to Y making all the [S n,X] *[S n,Y] *[S^n,X]_\ast \longrightarrow [S^n,Y]_\ast into isomorphisms.

See also example below.

The representability result applied degreewise to an additive reduced cohomology theory will yield (prop. below) the following concept.


An Omega-spectrum XX (def.) is

  1. a sequence {X n} n\{X_n\}_{n \in \mathbb{N}} of pointed topological spaces X nTop */X_n \in Top^{\ast/}

  2. weak homotopy equivalences

    σ˜ n:X nW clσ˜ nΩX n+1 \tilde \sigma_n \;\colon\; X_n \underoverset{\in W_{cl}}{\tilde \sigma_n}{\longrightarrow} \Omega X_{n+1}

    for each nn \in \mathbb{N}, form each space to the loop space of the following space.


Every additive reduced cohomology theory E˜ ():(Top CW *) opAb \tilde E^\bullet(-) \colon (Top_{CW}^\ast)^{op} \longrightarrow Ab^{\mathbb{Z}} according to def. , is represented by an Omega-spectrum EE (def. ) in that in each degree nn \in \mathbb{N}

  1. E˜ n()\tilde E^n(-) is represented by some E nHo(Top */)E_n \in Ho(Top^{\ast/});

  2. the suspension isomorphism σ n\sigma_n of E˜ \tilde E^\bullet is represented by the structure map σ˜ n\tilde \sigma_n of the Omega-spectrum in that for all XTop */X \in Top^{\ast/} the following diagram commutes:

    E˜ n(X) σ n(X) E˜ n+1(ΣX) [X,E n] * [X,σ˜ n] * [X,ΩE n+1] * [ΣX,E n+1] *, \array{ \tilde E^{n}(X) &\overset{\sigma_n(X)}{\longrightarrow}& &\longrightarrow& \tilde E^{n+1}(\Sigma X) \\ {}^{\mathllap{\simeq}}\downarrow && && \downarrow^{\mathrlap{\simeq}} \\ [X,E_n]_\ast &\overset{[X,\tilde \sigma_n]_\ast}{\longrightarrow}& [X, \Omega E_{n+1}]_\ast &\simeq& [\Sigma X, E_{n+1}]_\ast } \,,

    where [,] *Hom Ho(Top 1 */)[-,-]_\ast \coloneqq Hom_{Ho(Top_{\geq 1}^{\ast/})} denotes the hom-sets in the classical pointed homotopy category (def.) and where in the bottom right we have the (ΣΩ)(\Sigma\dashv \Omega)-adjunction isomorphism (prop.).


If it were not for the connectedness clause in def. (remark ), then theorem with prop. would immediately give the existence of the {E n} n\{E_n\}_{n \in \mathbb{N}} and the remaining statement would follow immediately with the Yoneda lemma, which says in particular that morphisms between representable functors are in natural bijection with the morphisms of objects that represent them.

The argument with the connectivity condition in Brown representability taken into account is essentially the same, just with a little bit more care:

For XX a pointed topological space, write X (0)X^{(0)} for the connected component of its basepoint. Observe that the loop space of a pointed topological space only depends on this connected component:

ΩXΩ(X (0)). \Omega X \simeq \Omega (X^{(0)}) \,.

Now for nn \in \mathbb{N}, to show that E˜ n()\tilde E^n(-) is representable by some E nHo(Top */)E_n \in Ho(Top^{\ast/}), use first that the restriction of E˜ n+1\tilde E^{n+1} to connected spaces is represented by some E n+1 (0)E_{n+1}^{(0)}. Observe that the reduced suspension of any XTop */X \in Top^{\ast/} lands in Top 1 */Top_{\geq 1}^{\ast/}. Therefore the (ΣΩ)(\Sigma\dashv \Omega)-adjunction isomorphism (prop.) implies that E˜ n+1(Σ())\tilde E^{n+1}(\Sigma(-)) is represented on all of Top */Top^{\ast/} by ΩE n+1 (0)\Omega E_{n+1}^{(0)}:

E˜ n+1(ΣX)[ΣX,E n+1 (0)] *[X,ΩE n+1 (0)] *[X,ΩE n+1] *, \tilde E^{n+1}(\Sigma X) \simeq [\Sigma X, E_{n+1}^{(0)}]_\ast \simeq [X, \Omega E_{n+1}^{(0)}]_\ast \simeq [X, \Omega E_{n+1}]_\ast \,,

where E n+1E_{n+1} is any pointed topological space with the given connected component E n+1 (0)E_{n+1}^{(0)}.

Now the suspension isomorphism of E˜\tilde E says that E nHo(Top */)E_n \in Ho(Top^{\ast/}) representing E˜ n\tilde E^n exists and is given by ΩE n+1 (0)\Omega E_{n+1}^{(0)}:

E˜ n(X)E˜ n+1(Σ,X)[X,ΩE n+1] \tilde E^n(X) \simeq \tilde E^{n+1}(\Sigma, X)\simeq [X,\Omega E_{n+1}]

for any E n+1E_{n+1} with connected component E n+1 (0)E_{n+1}^{(0)}.

This completes the proof. Notice that running the same argument next for (n+1)(n+1) gives a representing space E n+1E_{n+1} such that its connected component of the base point is E n+1 (0)E_{n+1}^{(0)} found before. And so on.



Every Omega-spectrum EE, def. , represents an additive reduced cohomology theory def. E˜ \tilde E^\bullet by

E˜ n(X)[X,E n] * \tilde E^n(X) \coloneqq [X,E_n]_\ast

with suspension isomorphism given by

σ n:E˜ n(X)=[X,E n] *[X,σ˜ n][X,ΩE n+1] *[ΣX,E n+1]=E˜ n+1(ΣX). \sigma_n \;\colon\; \tilde E^n(X) = [X,E_n]_\ast \overset{[X,\tilde \sigma_n]}{\longrightarrow} [X, \Omega E_{n+1}]_\ast \overset{\simeq}{\to} [\Sigma X, E_{n+1}] = \tilde E^{n+1}(\Sigma X) \,.

The additivity is immediate from the construction. The exactnes follows from the long exact sequences of homotopy cofiber sequences given by this prop..


If we consider the stable homotopy category Ho(Spectra)Ho(Spectra) of spectra (def.) and consider any topological space XX in terms of its suspension spectrum Σ XHo(Spectra)\Sigma^\infty X \in Ho(Spectra) (exmpl.), then the statement of prop. is more succinctly summarized by saying that the graded reduced cohomology groups of a topological space XX represented by an Omega-spectrum EE are the hom-groups

E˜ (X)[Σ X,Σ E] \tilde E^\bullet(X) \;\simeq\; [\Sigma^\infty X, \Sigma^\bullet E]

in the stable homotopy category, into all the suspensions (thm.) of EE.

This means that more generally, for XHo(Spectra)X \in Ho(Spectra) any spectrum, it makes sense to consider

E˜ (X)[X,Σ E] \tilde E^\bullet(X) \;\coloneqq\; [X,\Sigma^\bullet E]

to be the graded reduced generalized EE-cohomology groups of the spectrum XX.

See also in part 1 this example.

Application to ordinary cohomology


Let AA be an abelian group. Consider singular cohomology H n(,A)H^n(-,A) with coefficients in AA. The corresponding reduced cohomology evaluated on n-spheres satisfies

H˜ n(S q,A){A ifq=n 0 otherwise \tilde H^n(S^q,A) \simeq \left\{ \array{ A & if \; q = n \\ 0 & otherwise } \right.

Hence singular cohomology is a generalized cohomology theory which is “ordinary cohomology” in the sense of def. .

Applying the Brown representability theorem as in prop. hence produces an Omega-spectrum (def. ) whose nnth component space is characterized as having homotopy groups concentrated in degree nn on AA. These are called Eilenberg-MacLane spaces K(A,n)K(A,n)

π q(K(A,n)){A ifq=n 0 otherwise. \pi_q(K(A,n)) \simeq \left\{ \array{ A & if \; q = n \\ 0 & otherwise } \right. \,.

Here for n>0n \gt 0 then K(A,n)K(A,n) is connected, therefore with an essentially unique basepoint, while K(A,0)K(A,0) is (homotopy equivalent to) the underlying set of the group AA.

Such spectra are called Eilenberg-MacLane spectra HAH A:

(HA) nK(A,n). (H A)_n \simeq K(A,n) \,.

As a consequence of example one obtains the uniqueness result of Eilenberg-Steenrod:


Let E˜ 1\tilde E_1 and E˜ 2\tilde E_2 be ordinary (def. ) generalized (Eilenberg-Steenrod) cohomology theories. If there is an isomorphism

E˜ 1(S 0)E˜ 2(S 0) \tilde E_1(S^0) \simeq \tilde E_2(S^0)

of cohomology groups of the 0-sphere, then there is an isomorphism of cohomology theories

E˜ 1E˜ 2. \tilde E_1 \overset{\simeq}{\longrightarrow} \tilde E_2 \,.

(e.g. Aguilar-Gitler-Prieto 02, theorem 12.3.6)

Homotopy-theoretic discussion

Using abstract homotopy theory in the guise of model category theory (see the lecture notes on classical homotopy theory), the traditional proof and further discussion of the Brown representability theorem above becomes more transparent (Lurie 10, section 1.4.1, for exposition see also Mathew 11).

This abstract homotopy-theoretic proof uses the general concept of homotopy colimits in model categories as well as the concept of derived hom-spaces (“∞-categories”). Even though in the accompanying Lecture notes on classical homotopy theory these concepts are only briefly indicated, the following is included for the interested reader.


Let 𝒞\mathcal{C} be a model category. A functor

F:Ho(𝒞) opSet F \;\colon\; Ho(\mathcal{C})^{op} \longrightarrow Set

(from the opposite of the homotopy category of 𝒞\mathcal{C} to Set)

is called a Brown functor if

  1. it sends small coproducts to products;

  2. it sends homotopy pushouts in 𝒞Ho(𝒞)\mathcal{C}\to Ho(\mathcal{C}) to weak pullbacks in Set (see remark ).


A weak pullback is a diagram that satisfies the existence clause of a pullback, but not necessarily the uniqueness condition. Hence the second clause in def. says that for a homotopy pushout square

Z X Y XZY \array{ Z &\longrightarrow& X \\ \downarrow &\swArrow& \downarrow \\ Y &\longrightarrow& X \underset{Z}{\sqcup}Y }

in 𝒞\mathcal{C}, then the induced universal morphism

F(XZY)epiF(X)×F(Z)F(Y) F\left(X \underset{Z}{\sqcup}Y\right) \stackrel{epi}{\longrightarrow} F(X) \underset{F(Z)}{\times} F(Y)

into the actual pullback is an epimorphism.


Say that a model category 𝒞\mathcal{C} is compactly generated by cogroup objects closed under suspensions if

  1. 𝒞\mathcal{C} is generated by a set

    {S i𝒞} iI \{S_i \in \mathcal{C}\}_{i \in I}

    of compact objects (i.e. every object of 𝒞\mathcal{C} is a homotopy colimit of the objects S iS_i.)

  2. each S iS_i admits the structure of a cogroup object in the homotopy category Ho(𝒞)Ho(\mathcal{C});

  3. the set {S i}\{S_i\} is closed under forming reduced suspensions.


(suspensions are H-cogroup objects)

Let 𝒞\mathcal{C} be a model category and 𝒞 */\mathcal{C}^{\ast/} its pointed model category (prop.) with zero object (rmk.). Write Σ:X0X0\Sigma \colon X \mapsto 0 \underset{X}{\coprod} 0 for the reduced suspension functor.

Then the fold map

ΣXΣX0X0X00XXX00X0ΣX \Sigma X \coprod \Sigma X \simeq 0 \underset{X}{\sqcup} 0 \underset{X}{\sqcup} 0 \longrightarrow 0 \underset{X}{\sqcup} X \underset{X}{\sqcup} 0 \simeq 0 \underset{X}{\sqcup} 0 \simeq \Sigma X

exhibits cogroup structure on the image of any suspension object ΣX\Sigma X in the homotopy category.

This is equivalently the group-structure of the first (fundamental) homotopy group of the values of functor co-represented by ΣX\Sigma X:

Ho(𝒞)(ΣX,):YHo(𝒞)(ΣX,Y)Ho(𝒞)(X,ΩY)π 1Ho(𝒞)(X,Y). Ho(\mathcal{C})(\Sigma X, -) \;\colon\; Y \mapsto Ho(\mathcal{C})(\Sigma X, Y) \simeq Ho(\mathcal{C})(X, \Omega Y) \simeq \pi_1 Ho(\mathcal{C})(X, Y) \,.

In bare pointed homotopy types 𝒞=Top Quillen */\mathcal{C} = Top^{\ast/}_{Quillen}, the (homotopy types of) n-spheres S nS^n are cogroup objects for n1n \geq 1, but not for n=0n = 0, by example . And of course they are compact objects.

So while {S n} n\{S^n\}_{n \in \mathbb{N}} generates all of the homotopy theory of Top */Top^{\ast/}, the latter is not an example of def. due to the failure of S 0S^0 to have cogroup structure.

Removing that generator, the homotopy theory generated by {S n} nn1\{S^n\}_{{n \in \mathbb{N}} \atop {n \geq 1}} is Top 1 */Top^{\ast/}_{\geq 1}, that of connected pointed homotopy types. This is one way to see how the connectedness condition in the classical version of Brown representability theorem arises. See also remark above.

See also (Lurie 10, example

In homotopy theories compactly generated by cogroup objects closed under forming suspensions, the following strenghtening of the Whitehead theorem holds.


In a homotopy theory compactly generated by cogroup objects {S i} iI\{S_i\}_{i \in I} closed under forming suspensions, according to def. , a morphism f:XYf\colon X \longrightarrow Y is an equivalence precisely if for each iIi \in I the induced function of maps in the homotopy category

Ho(𝒞)(S i,f):Ho(𝒞)(S i,X)Ho(𝒞)(S i,Y) Ho(\mathcal{C})(S_i,f) \;\colon\; Ho(\mathcal{C})(S_i,X) \longrightarrow Ho(\mathcal{C})(S_i,Y)

is an isomorphism (a bijection).

(Lurie 10, p. 114, Lemma star)


By the ∞-Yoneda lemma, the morphism ff is a weak equivalence precisely if for all objects A𝒞A \in \mathcal{C} the induced morphism of derived hom-spaces

𝒞(A,f):𝒞(A,X)𝒞(A,Y) \mathcal{C}(A,f) \;\colon\; \mathcal{C}(A,X) \longrightarrow \mathcal{C}(A,Y)

is an equivalence in Top QuillenTop_{Quillen}. By assumption of compact generation and since the hom-functor 𝒞(,)\mathcal{C}(-,-) sends homotopy colimits in the first argument to homotopy limits, this is the case precisely already if it is the case for A{S i} iIA \in \{S_i\}_{i \in I}.

Now the maps

𝒞(S i,f):𝒞(S i,X)𝒞(S i,Y) \mathcal{C}(S_i,f) \;\colon\; \mathcal{C}(S_i,X) \longrightarrow \mathcal{C}(S_i,Y)

are weak equivalences in Top QuillenTop_{Quillen} if they are weak homotopy equivalences, hence if they induce isomorphisms on all homotopy groups π n\pi_n for all basepoints.

It is this last condition of testing on all basepoints that the assumed cogroup structure on the S iS_i allows to do away with: this cogroup structure implies that 𝒞(S i,)\mathcal{C}(S_i,-) has the structure of an HH-group, and this implies (by group multiplication), that all connected components have the same homotopy groups, hence that all homotopy groups are independent of the choice of basepoint, up to isomorphism.

Therefore the above morphisms are equivalences precisely if they are so under applying π n\pi_n based on the connected component of the zero morphism

π n𝒞(S i,f):π n𝒞(S i,X)π n𝒞(S i,Y). \pi_n\mathcal{C}(S_i,f) \;\colon\; \pi_n \mathcal{C}(S_i,X) \longrightarrow \pi_n\mathcal{C}(S_i,Y) \,.

Now in this pointed situation we may use that

π n𝒞(,) π 0𝒞(,Ω n()) π 0𝒞(Σ n(),) Ho(𝒞)(Σ n(),) \begin{aligned} \pi_n \mathcal{C}(-,-) & \simeq \pi_0 \mathcal{C}(-,\Omega^n(-)) \\ & \simeq \pi_0\mathcal{C}(\Sigma^n(-),-) \\ & \simeq Ho(\mathcal{C})(\Sigma^n(-),-) \end{aligned}

to find that ff is an equivalence in 𝒞\mathcal{C} precisely if the induced morphisms

Ho(𝒞)(Σ nS i,f):Ho(𝒞)(Σ nS i,X)Ho(𝒞)(Σ nS i,Y) Ho(\mathcal{C})(\Sigma^n S_i, f) \;\colon\; Ho(\mathcal{C})(\Sigma^n S_i,X) \longrightarrow Ho(\mathcal{C})(\Sigma^n S_i,Y)

are isomorphisms for all iIi \in I and nn \in \mathbb{N}.

Finally by the assumption that each suspension Σ nS i\Sigma^n S_i of a generator is itself among the set of generators, the claim follows.


(Brown representability)

Let 𝒞\mathcal{C} be a model category compactly generated by cogroup objects closed under forming suspensions, according to def. . Then a functor

F:Ho(𝒞) opSet F \;\colon\; Ho(\mathcal{C})^{op} \longrightarrow Set

(from the opposite of the homotopy category of 𝒞\mathcal{C} to Set) is representable precisely if it is a Brown functor, def. .

(Lurie 10, theorem


Due to the version of the Whitehead theorem of prop. we are essentially reduced to showing that Brown functors FF are representable on the S iS_i. To that end consider the following lemma. (In the following we notationally identify, via the Yoneda lemma, objects of 𝒞\mathcal{C}, hence of Ho(𝒞)Ho(\mathcal{C}), with the functors they represent.)

Lemma (\star): Given X𝒞X \in \mathcal{C} and ηF(X)\eta \in F(X), hence η:XF\eta \colon X \to F, then there exists a morphism f:XXf \colon X \to X' and an extension η:XF\eta' \colon X' \to F of η\eta which induces for each S iS_i a bijection η():PSh(Ho(𝒞))(S i,X)Ho(𝒞)(S i,F)F(S i)\eta'\circ (-) \colon PSh(Ho(\mathcal{C}))(S_i,X') \stackrel{\simeq}{\longrightarrow} Ho(\mathcal{C})(S_i,F) \simeq F(S_i).

To see this, first notice that we may directly find an extension η 0\eta_0 along a map XX oX\to X_o such as to make a surjection: simply take X 0X_0 to be the coproduct of all possible elements in the codomain and take

η 0:X(iI,γ:S iFS i)F \eta_0 \;\colon\; X \sqcup \left( \underset{{i \in I,} \atop {\gamma \colon S_i \stackrel{}{\to} F}}{\coprod} S_i \right) \longrightarrow F

to be the canonical map. (Using that FF, by assumption, turns coproducts into products, we may indeed treat the coproduct in 𝒞\mathcal{C} on the left as the coproduct of the corresponding functors.)

To turn the surjection thus constructed into a bijection, we now successively form quotients of X 0X_0. To that end proceed by induction and suppose that η n:X nF\eta_n \colon X_n \to F has been constructed. Then for iIi \in I let

K iker(Ho(𝒞)(S i,X n)η n()F(S i)) K_i \coloneqq ker \left( Ho(\mathcal{C})(S_i, X_n) \stackrel{\eta_n \circ (-)}{\longrightarrow} F(S_i) \right)

be the kernel of η n\eta_n evaluated on S iS_i. These K iK_i are the pieces that need to go away in order to make a bijection. Hence define X n+1X_{n+1} to be their joint homotopy cofiber

X n+1coker((iI,γK iS i)(γ) iIγK iX n). X_{n+1} \coloneqq coker\left( \left( \underset{{i \in I,} \atop {\gamma \in K_i}}{\sqcup} S_i \right) \overset{(\gamma)_{{i \in I} \atop {\gamma\in K_i}}}{\longrightarrow} X_n \right) \,.

Then by the assumption that FF takes this homotopy cokernel to a weak fiber (as in remark ), there exists an extension η n+1\eta_{n+1} of η n\eta_n along X nX n+1X_n \to X_{n+1}:

Then by the assumption that FF takes this homotopy cokernel to a weak fiber (as in remark ), there exists an extension η n+1\eta_{n+1} of η n\eta_n along X nX n+1X_n \to X_{n+1}:

(iIγK iS i) (γ) iIγK i X n η n F (po h) η n+1 * X n+1 F(X n+1) * η n+1 epi * η n ker((γ *) iIγK i) * η n (pb) F(X n) (γ *) iIγK i iIγK iF(S i). \array{ \left( \underset{{i \in I}\atop {\gamma \in K_i}}{\sqcup} S_i \right) &\overset{(\gamma)_{{i \in I}\atop \gamma \in K_i}}{\longrightarrow}& X_n &\overset{\eta_n}{\longrightarrow}& F \\ \downarrow &(po^{h})& \downarrow & \nearrow_{\mathrlap{\exists \eta_{n+1}}} \\ \ast &\longrightarrow& X_{n+1} } \;\;\;\;\;\;\;\;\;\;\; \Leftrightarrow \;\;\;\;\;\;\;\;\;\;\; \array{ && F(X_{n+1}) &\longrightarrow& \ast \\ &{}^{\mathllap{\exists \eta_{n+1}}}\nearrow& \downarrow^{\mathrlap{epi}} && \downarrow \\ \ast &\overset{\eta_n}{\longrightarrow}& ker\left((\gamma^\ast\right)_{{i \in I} \atop {\gamma \in K_i}}) &\longrightarrow& \ast \\ &{}_{\mathllap{\eta_n}}\searrow& \downarrow &(pb)& \downarrow \\ && F(X_n) &\underset{(\gamma^\ast)_{{i \in I} \atop {\gamma \in K_i}} }{\longrightarrow}& \underset{{i \in I}\atop {\gamma\in K_i}}{\prod}F(S_i) } \,.

It is now clear that we want to take

Xlim nX n X' \coloneqq \underset{\rightarrow}{\lim}_n X_n

and extend all the η n\eta_n to that colimit. Since we have no condition for evaluating FF on colimits other than pushouts, observe that this sequential colimit is equivalent to the following pushout:

nX n nX 2n nX 2n+1 X, \array{ \underset{n}{\sqcup} X_n &\longrightarrow& \underset{n}{\sqcup} X_{2n} \\ \downarrow && \downarrow \\ \underset{n}{\sqcup} X_{2n+1} &\longrightarrow& X' } \,,

where the components of the top and left map alternate between the identity on X nX_n and the above successor maps X nX n+1X_n \to X_{n+1}. Now the excision property of FF applies to this pushout, and we conclude the desired extension η:XF\eta' \colon X' \to F:

nX n nX 2n+1 X nX 2n (η 2n+1) n η (η 2n) n F F(X) η epi *(η n) n lim nF(X n) nF(X 2n+1) n(X 2n) nF(X n), \array{ && \underset{n}{\sqcup} X_n \\ & \swarrow && \searrow \\ \underset{n}{\sqcup} X_{2n+1} &\longrightarrow& X' &\longleftarrow& \underset{n}{\sqcup} X_{2n} \\ & {}_{\mathllap{(\eta_{2n+1})_{n}}}\searrow& \downarrow^{\mathrlap{\exists \eta}} & \swarrow_{\mathrlap{(\eta_{2n})_n}} \\ && F } \;\;\;\;\;\;\;\;\; \Leftrightarrow \;\;\;\;\;\;\;\;\; \array{ && F(X') \\ &{}^{\mathllap{\exists \eta}}\nearrow& \downarrow^{\mathrlap{epi}} \\ &\ast \overset{(\eta_n)_n}{\longrightarrow}& \underset{\longleftarrow}{\lim}_n F(X_n) \\ & \swarrow && \searrow \\ \underset{n}{\prod}F(X_{2n+1}) && && \underset{n}{\prod}(X_{2n}) \\ & \searrow && \swarrow \\ && \underset{n}{\prod}F(X_n) } \,,

It remains to confirm that this indeed gives the desired bijection. Surjectivity is clear. For injectivity use that all the S iS_i are, by assumption, compact, hence they may be taken inside the sequential colimit:

X n(γ) γ^ S i γ X=lim nX n. \array{ && X_{n(\gamma)} \\ &{}^{\mathllap{ \exists \hat \gamma}}\nearrow& \downarrow \\ S_i &\overset{\gamma}{\longrightarrow}& X' = \underset{\longrightarrow}{\lim}_n X_n } \,.

With this, injectivity follows because by construction we quotiented out the kernel at each stage. Because suppose that γ\gamma is taken to zero in F(S i)F(S_i), then by the definition of X n+1X_{n+1} above there is a factorization of γ\gamma through the point:

0: S i γ^ X n(γ) η n F * X n(γ)+1 X \array{ 0 \colon & S_i &\overset{\hat \gamma}{\longrightarrow}& X_{n(\gamma)} &\overset{\eta_n}{\longrightarrow}& F \\ & \downarrow && \downarrow & \\ & \ast &\longrightarrow& X_{n(\gamma)+1} \\ & && \downarrow \\ & && X' }

This concludes the proof of Lemma (\star).

Now apply the construction given by this lemma to the case X 00X_0 \coloneqq 0 and the unique η 0:0!F\eta_0 \colon 0 \stackrel{\exists !}{\to} F. Lemma ()(\star) then produces an object XX' which represents FF on all the S iS_i, and we want to show that this XX' actually represents FF generally, hence that for every Y𝒞Y \in \mathcal{C} the function

θη():Ho(𝒞)(Y,X)F(Y) \theta \coloneqq \eta'\circ (-) \;\colon\; Ho(\mathcal{C})(Y,X') \stackrel{}{\longrightarrow} F(Y)

is a bijection.

First, to see that θ\theta is surjective, we need to find a preimage of any ρ:YF\rho \colon Y \to F. Applying Lemma ()(\star) to (η,ρ):XYF(\eta',\rho)\colon X'\sqcup Y \longrightarrow F we get an extension κ\kappa of this through some XYZX' \sqcup Y \longrightarrow Z and the morphism on the right of the following commuting diagram:

Ho(𝒞)(,X) Ho(𝒞)(,Z) η() κ() F(). \array{ Ho(\mathcal{C})(-,X') && \longrightarrow && Ho(\mathcal{C})(-, Z) \\ & {}_{\mathllap{\eta'\circ(-)}}\searrow && \swarrow_{\mathrlap{\kappa \circ (-)}} \\ && F(-) } \,.

Moreover, Lemma ()(\star) gives that evaluated on all S iS_i, the two diagonal morphisms here become isomorphisms. But then prop. implies that XZX' \longrightarrow Z is in fact an equivalence. Hence the component map YZZY \to Z \simeq Z is a lift of κ\kappa through θ\theta.

Second, to see that θ\theta is injective, suppose f,g:YXf,g \colon Y \to X' have the same image under θ\theta. Then consider their homotopy pushout

YY (f,g) X Y Z \array{ Y \sqcup Y &\stackrel{(f,g)}{\longrightarrow}& X' \\ \downarrow && \downarrow \\ Y &\longrightarrow& Z }

along the codiagonal of YY. Using that FF sends this to a weak pullback by assumption, we obtain an extension η¯\bar \eta of η\eta' along XZX' \to Z. Applying Lemma ()(\star) to this gives a further extension η¯:ZZ\bar \eta' \colon Z' \to Z which now makes the following diagram

Ho(𝒞)(,X) Ho(𝒞)(,Z) η() η¯() F() \array{ Ho(\mathcal{C})(-,X') && \longrightarrow && Ho(\mathcal{C})(-, Z) \\ & {}_{\mathllap{\eta'\circ(-)}}\searrow && \swarrow_{\mathrlap{\bar \eta' \circ (-)}} \\ && F(-) }

such that the diagonal maps become isomorphisms when evaluated on the S iS_i. As before, it follows via prop. that the morphism h:XZh \colon X' \longrightarrow Z' is an equivalence.

Since by this construction hfh\circ f and hgh\circ g are homotopic

YY (f,g) X h Y Z Z \array{ Y \sqcup Y &\stackrel{(f,g)}{\longrightarrow}& X' \\ \downarrow && \downarrow & \searrow^{\mathrlap{\stackrel{h}{\simeq}}} \\ Y &\longrightarrow& Z &\longrightarrow& Z' }

it follows with hh being an equivalence that already ff and gg were homotopic, hence that they represented the same element.


Given a reduced additive cohomology functor H :Ho(𝒞) opAb H^\bullet \colon Ho(\mathcal{C})^{op}\to Ab^{\mathbb{Z}}, def. , its underlying Set-valued functors H n:Ho(𝒞) opAbSetH^n \colon Ho(\mathcal{C})^{op}\to Ab\to Set are Brown functors, def. .


The first condition on a Brown functor holds by definition of H H^\bullet. For the second condition, given a homotopy pushout square

X 1 f 1 Y 1 X 2 f 2 Y 2 \array{ X_1 &\stackrel{f_1}{\longrightarrow}& Y_1 \\ \downarrow^{} && \downarrow \\ X_2 &\stackrel{f_2}{\longrightarrow}& Y_2 }

in 𝒞\mathcal{C}, consider the induced morphism of the long exact sequences given by prop.

H (coker(f 2)) H (Y 2) f 2 * H (X 2) H +1(Σcoker(f 2)) H (coker(f 1)) H (Y 1) f 1 * H (X 1) H +1(Σcoker(f 1)) \array{ H^\bullet(coker(f_2)) &\longrightarrow& H^\bullet(Y_2) &\stackrel{f^\ast_2}{\longrightarrow}& H^\bullet(X_2) &\stackrel{}{\longrightarrow}& H^{\bullet+1}(\Sigma coker(f_2)) \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow && \downarrow && \downarrow^{\mathrlap{\simeq}} \\ H^\bullet(coker(f_1)) &\longrightarrow& H^\bullet(Y_1) &\stackrel{f^\ast_1}{\longrightarrow}& H^\bullet(X_1) &\stackrel{}{\longrightarrow}& H^{\bullet+1}(\Sigma coker(f_1)) }

Here the outer vertical morphisms are isomorphisms, as shown, due to the pasting law (see also at fiberwise recognition of stable homotopy pushouts). This means that the four lemma applies to this diagram. Inspection shows that this implies the claim.


Let 𝒞\mathcal{C} be a model category which satisfies the conditions of theorem , and let (H ,δ)(H^\bullet, \delta) be a reduced additive generalized cohomology functor on 𝒞\mathcal{C}, def. . Then there exists a spectrum object EStab(𝒞)E \in Stab(\mathcal{C}) such that

  1. HH\bullet is degreewise represented by EE:

    H Ho(𝒞)(,E ), H^\bullet \simeq Ho(\mathcal{C})(-,E_\bullet) \,,
  2. the suspension isomorphism δ\delta is given by the structure morphisms σ˜ n:E nΩE n+1\tilde \sigma_n \colon E_n \to \Omega E_{n+1} of the spectrum, in that

    δ:H n()Ho(𝒞)(,E n)Ho(𝒞)(,σ˜ n)Ho(𝒞)(,ΩE n+1)Ho(𝒞)(Σ(),E n+1)H n+1(Σ()). \delta \colon H^n(-) \simeq Ho(\mathcal{C})(-,E_n) \stackrel{Ho(\mathcal{C})(-,\tilde\sigma_n) }{\longrightarrow} Ho(\mathcal{C})(-,\Omega E_{n+1}) \simeq Ho(\mathcal{C})(\Sigma (-), E_{n+1}) \simeq H^{n+1}(\Sigma(-)) \,.

Via prop. , theorem gives the first clause. With this, the second clause follows by the Yoneda lemma.

Milnor exact sequence

Idea. One tool for computing generalized cohomology groups via “inverse limits” are Milnor exact sequences. For instance the generalized cohomology of the classifying space BU(1)B U(1) plays a key role in the complex oriented cohomology-theory discussed below, and via the equivalence BU(1)P B U(1) \simeq \mathbb{C}P^\infty to the homotopy type of the infinite complex projective space (def. ), which is the direct limit of finite dimensional projective spaces P n\mathbb{C}P^n, this is an inverse limit of the generalized cohomology groups of the P n\mathbb{C}P^ns. But what really matters here is the derived functor of the limit-operation – the homotopy limit – and the Milnor exact sequence expresses how the naive limits receive corrections from higher “lim^1-terms”. In practice one mostly proceeds by verifying conditions under which these corrections happen to disappear, these are the Mittag-Leffler conditions.

We need this for instance for the computation of Conner-Floyd Chern classes below.

Literature. (Switzer 75, section 7 from def. 7.57 on, Kochman 96, section 4.2, Goerss-Jardine 99, section VI.2, )

Lim 1Lim^1


Given a tower A A_\bullet of abelian groups

A 3f 2A 2f 1A 1f 0A 0 \cdots \to A_3 \stackrel{f_2}{\to} A_2 \stackrel{f_1}{\to} A_1 \stackrel{f_0}{\to} A_0


:nA nnA n \partial \;\colon\; \underset{n}{\prod} A_n \longrightarrow \underset{n}{\prod} A_n

for the homomorphism given by

:(a n) n(a nf n(a n+1)) n. \partial \;\colon\; (a_n)_{n \in \mathbb{N}} \mapsto (a_n - f_n(a_{n+1}))_{n \in \mathbb{N}}.

The [[limit]] of a sequence as in def. – hence the group lim nA n\underset{\longleftarrow}{\lim}_n A_n universally equipped with morphisms lim nA np nA n\underset{\longleftarrow}{\lim}_n A_n \overset{p_n}{\to} A_n such that all

lim nA n p n+1 p n A n+1 f n A n \array{ && \underset{\longleftarrow}{\lim}_n A_n \\ & {}^{\mathllap{p_{n+1}}}\swarrow && \searrow^{\mathrlap{p_n}} \\ A_{n+1} && \overset{f_n}{\longrightarrow} && A_n }

[[commuting diagram|commute]] – is equivalently the [[kernel]] of the morphism \partial in def. .


Given a [[tower]] A A_\bullet of [[abelian groups]]

A 3f 2A 2f 1A 1f 0A 0 \cdots \to A_3 \stackrel{f_2}{\to} A_2 \stackrel{f_1}{\to} A_1 \stackrel{f_0}{\to} A_0

then lim 1A \underset{\longleftarrow}{\lim}^1 A_\bullet is the [[cokernel]] of the map \partial in def. , hence the group that makes a [[long exact sequence]] of the form

0lim nA nnA nnA nlim n 1A n0, 0 \to \underset{\longleftarrow}{\lim}_n A_n \longrightarrow \underset{n}{\prod} A_n \stackrel{\partial}{\longrightarrow} \underset{n}{\prod} A_n \longrightarrow \underset{\longleftarrow}{\lim}^1_n A_n \to 0 \,,

The [[functor]] lim 1:Ab (,)Ab\underset{\longleftarrow}{\lim}^1 \colon Ab^{(\mathbb{N}, \geq)} \longrightarrow Ab (def. ) satisfies

  1. for every [[short exact sequence]] 0A B C 0Ab (,)0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0 \;\;\; \in Ab^{(\mathbb{N}, \geq)} then the induced sequence

    0lim nA nlim nB nlim nC nlim n 1A nlim n 1B nlim n 1C n0 0 \to \underset{\longleftarrow}{\lim}_n A_n \to \underset{\longleftarrow}{\lim}_n B_n \to \underset{\longleftarrow}{\lim}_n C_n \to \underset{\longleftarrow}{\lim}_n^1 A_n \to \underset{\longleftarrow}{\lim}_n^1 B_n \to \underset{\longleftarrow}{\lim}_n^1 C_n \to 0

    is a [[long exact sequence]] of abelian groups;

  2. if A A_\bullet is a tower such that all maps are [[surjections]], then lim n 1A n0\underset{\longleftarrow}{\lim}^1_n A_n \simeq 0.

(e.g. Switzer 75, prop. 7.63, Goerss-Jardine 96, section VI. lemma 2.11)


For the first property: Given A A_\bullet a tower of abelian groups, write

L (A )[0nA ndeg0nA ndeg10] L^\bullet(A_\bullet) \coloneqq \left[ 0 \to \underset{deg \, 0}{\underbrace{\underset{n}{\prod} A_n}} \overset{\partial}{\longrightarrow} \underset{deg\, 1}{\underbrace{\underset{n}{\prod} A_n}} \to 0 \right]

for the homomorphism from def. regarded as the single non-trivial differential in a [[cochain complex]] of abelian groups. Then by remark and def. we have H 0(L(A ))limA H^0(L(A_\bullet)) \simeq \underset{\longleftarrow}{\lim} A_\bullet and H 1(L(A ))lim 1A H^1(L(A_\bullet)) \simeq \underset{\longleftarrow}{\lim}^1 A_\bullet.

With this, then for a short exact sequence of towers 0A B C 00 \to A_\bullet \to B_\bullet \to C_\bullet \to 0 the long exact sequence in question is the [[long exact sequence in homology]] of the corresponding short exact sequence of complexes

0L (A )L (B )L (C )0. 0 \to L^\bullet(A_\bullet) \longrightarrow L^\bullet(B_\bullet) \longrightarrow L^\bullet(C_\bullet) \to 0 \,.

For the second statement: If all the f kf_k are surjective, then inspection shows that the homomorphism \partial in def. is surjective. Hence its [[cokernel]] vanishes.


The category Ab (,)Ab^{(\mathbb{N}, \geq)} of [[towers]] of [[abelian groups]] has [[enough injectives]].


The functor () n:Ab (,)Ab(-)_n \colon Ab^{(\mathbb{N}, \geq)} \to Ab that picks the nn-th component of the tower has a [[right adjoint]] r nr_n, which sends an abelian group AA to the tower

r n[idAidA=(r n) n+1idA=(r n) nid0=(r n) n1000]. r_n \coloneqq \left[ \cdots \overset{id}{\to} A \overset{id}{\to} \underset{= (r_n)_{n+1}}{\underbrace{A}} \overset{id}{\to} \underset{= (r_n)_n}{\underbrace{A}} \overset{id}{\to} \underset{= (r_n)_{n-1}}{\underbrace{0}} \to 0 \to \cdots \to 0 \to 0 \right] \,.

Since () n(-)_n itself is evidently an [[exact functor]], its right adjoint preserves injective objects (prop.).

So with A Ab (,)A_\bullet \in Ab^{(\mathbb{N}, \geq)}, let A nA˜ nA_n \hookrightarrow \tilde A_n be an injective resolution of the abelian group A nA_n, for each nn \in \mathbb{N}. Then

A (η n) nnr nA nnr nA˜ n A_\bullet \overset{(\eta_n)_{n \in \mathbb{N}}}{\longrightarrow} \underset{n \in \mathbb{R}}{\prod} r_n A_n \hookrightarrow \underset{n \in \mathbb{N}}{\prod} r_n \tilde A_n

is an injective resolution for A A_\bullet.


The [[functor]] lim 1:Ab (,)Ab\underset{\longleftarrow}{\lim}^1 \colon Ab^{(\mathbb{N}, \geq)} \longrightarrow Ab (def. ) is the [[derived functor in homological algebra|first right derived functor]] of the [[limit]] functor lim:Ab (,)Ab\underset{\longleftarrow}{\lim} \colon Ab^{(\mathbb{N},\geq)} \longrightarrow Ab.


By lemma there are [[enough injectives]] in Ab (,)Ab^{(\mathbb{N}, \geq)}. So for A Ab (,)A_\bullet \in Ab^{(\mathbb{N}, \geq)} the given tower of abelian groups, let

0A j 0J 0j 1J 1j 2J 2 0 \to A_\bullet \overset{j^0}{\longrightarrow} J^0_\bullet \overset{j^1}{\longrightarrow} J^1_\bullet \overset{j^2}{\longrightarrow} J^2_\bullet \overset{}{\longrightarrow} \cdots

be an [[injective resolution]]. We need to show that

lim 1A ker(lim(j 2))/im(lim(j 1)). \underset{\longleftarrow}{\lim}^1 A_\bullet \simeq ker(\underset{\longleftarrow}{\lim}(j^2))/im(\underset{\longleftarrow}{\lim}(j^1)) \,.

Since limits preserve [[kernels]], this is equivalently

lim 1A (lim(ker(j 2) ))/im(lim(j 1)) \underset{\longleftarrow}{\lim}^1 A_\bullet \simeq (\underset{\longleftarrow}{\lim}(ker(j^2)_\bullet))/im(\underset{\longleftarrow}{\lim}(j^1))

Now observe that each injective J qJ^q_\bullet is a tower of epimorphism. This follows by the defining [[right lifting property]] applied against the monomorphisms of towers of the following form

0 0 0 id id id id id incl id id id 0 0 id id id id \array{ \cdots &\to & 0 &\to& 0 &\longrightarrow& 0 &\longrightarrow& \mathbb{Z} &\overset{id}{\longrightarrow}& \cdots &\overset{id}{\longrightarrow}& \mathbb{Z} &\overset{id}{\longrightarrow}& \mathbb{Z} \\ \cdots && \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{incl}} && \downarrow^{\mathrlap{id}} && && \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{id}} \\ \cdots &\to& 0 &\to& 0 &\to & \mathbb{Z} &\underset{id}{\longrightarrow}& \mathbb{Z} &\underset{id}{\longrightarrow}& \cdots &\underset{id}{\longrightarrow}& \mathbb{Z} &\underset{id}{\longrightarrow}& \mathbb{Z} }

Therefore by the second item of prop. the long exact sequence from the first item of prop. applied to the [[short exact sequence]]

0A j 0J 0j 1ker(j 2) 0 0 \to A_\bullet \overset{j^0}{\longrightarrow} J^0_\bullet \overset{j^1}{\longrightarrow} ker(j^2)_\bullet \to 0


0limA limj 0limJ 0limj 1lim(ker(j 2) )lim 1A 0. 0 \to \underset{\longleftarrow}{\lim} A_\bullet \overset{\underset{\longleftarrow}{\lim} j^0}{\longrightarrow} \underset{\longleftarrow}{\lim} J^0_\bullet \overset{\underset{\longleftarrow}{\lim}j^1}{\longrightarrow} \underset{\longleftarrow}{\lim}(ker(j^2)_\bullet) \longrightarrow \underset{\longleftarrow}{\lim}^1 A_\bullet \longrightarrow 0 \,.

Exactness of this sequence gives the desired identification lim 1A (lim(ker(j 2) ))/im(lim(j 1)). \underset{\longleftarrow}{\lim}^1 A_\bullet \simeq (\underset{\longleftarrow}{\lim}(ker(j^2)_\bullet))/im(\underset{\longleftarrow}{\lim}(j^1)) \,.


The [[functor]] lim 1:Ab (,)Ab\underset{\longleftarrow}{\lim}^1 \colon Ab^{(\mathbb{N}, \geq)} \longrightarrow Ab (def. ) is in fact the unique functor, up to [[natural isomorphism]], satisfying the conditions in prop. .


The proof of prop. only used the conditions from prop. , hence any functor satisfying these conditions is the first right derived functor of lim\underset{\longleftarrow}{\lim}, up to natural isomorphism.

The following is a kind of double dual version of the lim 1\lim^1 construction which is sometimes useful:


Given a [[cotower]]

A =(A 0f 0A 1f 1A 2) A_\bullet = (A_0 \overset{f_0}{\to} A _1 \overset{f_1}{\to} A_2 \to \cdots)

of [[abelian groups]], then for every abelian group BAbB \in Ab there is a [[short exact sequence]] of the form

0lim n 1Hom(A n,B)Ext 1(lim nA n,B)lim nExt 1(A n,B)0, 0 \to \underset{\longleftarrow}{\lim}^1_n Hom(A_n, B) \longrightarrow Ext^1( \underset{\longrightarrow}{\lim}_n A_n, B ) \longrightarrow \underset{\longleftarrow}{\lim}_n Ext^1( A_n, B) \to 0 \,,

where Hom(,)Hom(-,-) denotes the [[hom-object|hom-group]], Ext 1(,)Ext^1(-,-) denotes the first [[Ext]]-group (and so Hom(,)=Ext 0(,)Hom(-,-) = Ext^0(-,-)).


Consider the homomorphism

˜:nA nnA n \tilde \partial \;\colon\; \underset{n}{\oplus} A_n \longrightarrow \underset{n}{\oplus} A_n

which sends a nA na_n \in A_n to a nf n(a n)a_n - f_n(a_n). Its [[cokernel]] is the [[colimit]] over the cotower, but its [[kernel]] is trivial (in contrast to the otherwise [[formal dual|formally dual]] situation in remark ). Hence (as opposed to the long exact sequence in def. ) there is a [[short exact sequence]] of the form

0nA n˜nA nlim nA n0. 0 \to \underset{n}{\oplus} A_n \overset{\tilde \partial}{\longrightarrow} \underset{n}{\oplus} A_n \overset{}{\longrightarrow} \underset{\longrightarrow}{lim}_n A_n \to 0 \,.

Every short exact sequence gives rise to a [[long exact sequence]] of [[derived functors]] (prop.) which in the present case starts out as

0Hom(lim nA n,B)nHom(A n,B)nHom(A n,B)Ext 1(lim nA n,B)nExt 1(A n,B)nExt 1(A n,B) 0 \to Hom(\underset{\longrightarrow}{\lim}_n A_n,B) \longrightarrow \underset{n}{\prod} Hom( A_n, B ) \overset{\partial}{\longrightarrow} \underset{n}{\prod} Hom( A_n, B ) \longrightarrow Ext^1(\underset{\longrightarrow}{\lim}_n A_n,B) \longrightarrow \underset{n}{\prod} Ext^1( A_n, B ) \overset{\partial}{\longrightarrow} \underset{n}{\prod} Ext^1( A_n, B ) \longrightarrow \cdots

where we used that [[direct sum]] is the [[coproduct]] in abelian groups, so that homs out of it yield a [[product]], and where the morphism \partial is the one from def. corresponding to the [[tower]]

Hom(A ,B)=(Hom(A 2,B)Hom(A 1,B)Hom(A 0,B)). Hom(A_\bullet,B) = ( \cdots \to Hom(A_2,B) \to Hom(A_1,B) \to Hom(A_0,B) ) \,.

Hence truncating this long sequence by forming kernel and cokernel of \partial, respectively, it becomes the short exact sequence in question.

Mittag-Leffler condition


A tower A A_\bullet of [[abelian groups]]

A 3A 2A 1A 0 \cdots \to A_3 \to A_2 \to A_1 \to A_0

is said to satify the [[Mittag-Leffler condition]] if for all kk there exists iki \geq k such that for all jikj \geq i \geq k the [[image]] of the [[homomorphism]] A iA kA_i \to A_k equals that of A jA kA_j \to A_k

im(A iA k)im(A jA k). im(A_i \to A_k) \simeq im(A_j \to A_k) \,.

(e.g. Switzer 75, def. 7.74)


The Mittag-Leffler condition, def. , is satisfied in particular when all morphisms A i+1A iA_{i+1}\to A_i are [[epimorphisms]] (hence [[surjections]] of the underlying [[sets]]).


If a tower A A_\bullet satisfies the [[Mittag-Leffler condition]], def. , then its lim 1\underset{\leftarrow}{\lim}^1 vanishes:

lim 1A =0. \underset{\longleftarrow}{\lim}^1 A_\bullet = 0 \,.

e.g. (Switzer 75, theorem 7.75, Kochmann 96, prop. 4.2.3, Weibel 94, prop. 3.5.7)

Proof idea

One needs to show that with the Mittag-Leffler condition, then the [[cokernel]] of \partial in def. vanishes, hence that \partial is an [[epimorphism]] in this case, hence that every (a n) nnA n(a_n)_{n \in \mathbb{N}} \in \underset{n}{\prod} A_n has a preimage under \partial. So use the Mittag-Leffler condition to find pre-images of a na_n by [[induction]] over nn.

Mapping telescopes

Given a sequence

X =(X 0f 0X 1f 1X 2f 2) X_\bullet = \left( X_0 \overset{f_0}{\longrightarrow} X_1 \overset{f_1}{\longrightarrow} X_2 \overset{f_2}{\longrightarrow} \cdots \right)

of ([[pointed topological space|pointed]]) [[topological spaces]], then its mapping telescope is the result of forming the (reduced) [[mapping cylinder]] Cyl(f n)Cyl(f_n) for each nn and then attaching all these cylinders to each other in the canonical way



X =(X 0f 0X 1f 1X 2f 2) X_\bullet = \left( X_0 \overset{f_0}{\longrightarrow} X_1 \overset{f_1}{\longrightarrow} X_2 \overset{f_2}{\longrightarrow} \cdots \right)

a sequence in [[Top]], its mapping telescope is the [[quotient topological space]] of the [[disjoint union]] of [[product topological spaces]]

Tel(X )(n(X n×[n,n+1]))/ Tel(X_\bullet) \coloneqq \left( \underset{n \in \mathbb{N}}{\sqcup} \left( X_n \times [n,n+1] \right) \right)/_\sim

where the [[equivalence relation]] quotiented out is

(x n,n)(f(x n),n+1) (x_n, n) \sim (f(x_n), n+1)

for all nn\in \mathbb{N} and x nX nx_n \in X_n.

Analogously for X X_\bullet a sequence of [[pointed topological spaces]] then use [[reduced cylinders]] (exmpl.) to set

Tel(X )(n(X n[n,n+1] +))/ . Tel(X_\bullet) \coloneqq \left( \underset{n \in \mathbb{N}}{\sqcup} \left( X_n \wedge [n,n+1]_+ \right) \right)/_\sim \,.

For X X_\bullet the sequence of stages of a ([[pointed topological space|pointed]]) [[CW-complex]] X=lim nX nX = \underset{\longleftarrow}{\lim}_n X_n, then the canonical map

Tel(X )X Tel(X_\bullet) \longrightarrow X

from the [[mapping telescope]], def. , is a [[weak homotopy equivalence]].


Write in the following Tel(X)Tel(X) for Tel(X )Tel(X_\bullet) and write Tel(X n)Tel(X_n) for the mapping telescop of the substages of the finite stage X nX_n of XX. It is intuitively clear that each of the projections at finite stage

Tel(X n)X n Tel(X_n) \longrightarrow X_n

is a [[homotopy equivalence]], hence (prop.) a weak homotopy equivalence. A concrete construction of a homotopy inverse is given for instance in (Switzer 75, proof of prop. 7.53).

Moreover, since spheres are [[compact object|compact]], so that elements of [[homotopy groups]] π q(Tel(X))\pi_q(Tel(X)) are represented at some finite stage π q(Tel(X n))\pi_q(Tel(X_n)) it follows that

lim nπ q(Tel(X n))π q(Tel(X)) \underset{\longrightarrow}{\lim}_n \pi_q(Tel(X_n)) \overset{\simeq}{\longrightarrow} \pi_q(Tel(X))

are [[isomorphisms]] for all qq\in \mathbb{N} and all choices of basepoints (not shown).

Together these two facts imply that in the following commuting square, three morphisms are isomorphisms, as shown.

lim nπ q(Tel(X n)) π q(Tel(X)) lim nπ q(X n) π q(X). \array{ \underset{\longleftarrow}{\lim}_n \pi_q(Tel(X_n)) &\overset{\simeq}{\longrightarrow}& \pi_q(Tel(X)) \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow \\ \underset{\longleftarrow}{\lim}_n \pi_q(X_n) &\underset{\simeq}{\longrightarrow}& \pi_q(X) } \,.

Therefore also the remaining morphism is an isomorphism ([[two-out-of-three]]). Since this holds for all qq and all basepoints, it is a weak homotopy equivalence.

Milnor exact sequences


(Milnor exact sequence for homotopy groups)


X 3p 2X 2p 1X 1p 0X 0 \cdots \to X_3 \overset{p_2}{\longrightarrow} X_2 \overset{p_1}{\longrightarrow} X_1 \overset{p_0}{\longrightarrow} X_0

be a [[tower of fibrations]] ([[Serre fibrations]] (def.)). Then for each qq \in \mathbb{N} there is a [[short exact sequence]]

0lim i 1π q+1(X i)π q(lim iX i)lim iπ q(X i)0, 0 \to \underset{\longleftarrow}{\lim}^1_i \pi_{q+1}(X_i) \longrightarrow \pi_q(\underset{\longleftarrow}{\lim}_i X_i) \longrightarrow \underset{\longleftarrow}{\lim}_i \pi_q(X_i) \to 0 \,,

for π \pi_\bullet the [[homotopy group]]-functor (exact as [[pointed sets]] for i=0i = 0, as [[groups]] for i1i \geq 1) which says that

  1. the failure of the [[limit]] over the homotopy groups of the stages of the tower to equal the homotopy groups of the [[limit]] of the tower is at most in the [[kernel]] of the canonical comparison map;

  2. that kernel is the lim 1\underset{\longleftarrow}{\lim}^1 (def. ) of the homotopy groups of the stages.

An elementary but tedious proof is indicated in (Bousfield-Kan 72, chapter IX, theorem 3.1. The following is a neat [[model category]]-theoretic proof following (Goerss-Jardine 96, section VI. prop. 2.15), which however requires the concept of [[homotopy limit]] over towers.


With respect to the [[classical model structure on simplicial sets]] or the [[classical model structure on topological spaces]], a tower of fibrations as stated is a fibrant object in the injective [[model structure on functors]] [(,),sSet] inj[(\mathbb{N},\geq), sSet]_{inj} ([(,),Top] inj[(\mathbb{N},\geq), Top]_{inj}) (prop). Hence the plain [[limit]] over this diagram represents the [[homotopy limit]]. By the discussion there, up to weak equivalence that homotopy limit is also the pullback in

holimX nPath(X n) (pb) nX n (id,p n) n nX n×X n, \array{ holim X_\bullet &\longrightarrow& \underset{n}{\prod} Path(X_n) \\ \downarrow &(pb)& \downarrow \\ \underset{n}{\prod} X_n &\underset{(id,p_n)_n}{\longrightarrow}& \underset{n}{\prod} X_ n \times X_n } \,,

where on the right we have the product over all the canonical fibrations out of the [[path space objects]]. Hence also the left vertical morphism is a fibration, and so by taking its [[fiber]] over a basepoint, the [[pasting law]] gives a [[homotopy fiber sequence]]

nΩX nholimX nX n. \underset{n}{\prod} \Omega X_n \longrightarrow holim X_\bullet \longrightarrow \underset{n}{\prod} X_n \,.

The [[long exact sequence of homotopy groups]] of this fiber sequence goes

nπ q+1(X n)nπ q+1(X n)π q(limX )nπ q(X n)nπ q(X n). \cdots \to \underset{n}{\prod} \pi_{q+1}(X_n) \longrightarrow \underset{n}{\prod} \pi_{q+1}(X_n) \longrightarrow \pi_q (\underset{\longleftarrow}{\lim} X_\bullet) \longrightarrow \underset{n}{\prod} \pi_q(X_n) \longrightarrow \underset{n}{\prod} \pi_q(X_n) \to \cdots \,.

Chopping that off by forming kernel and cokernel yields the claim for positive qq. For q=0q = 0 it follows by inspection.


(Milnor exact sequence for generalized cohomology)

Let XX be a [[pointed topological space|pointed]] [[CW-complex]], X=lim nX nX = \underset{\longrightarrow}{\lim}_n X_n and let E˜ \tilde E^\bullet an additive [[reduced cohomology theory]], def. .

Then the canonical morphisms make a [[short exact sequence]]

0lim n 1E˜ 1(X n)E˜ (X)lim nE˜ (X n)0, 0 \to \underset{\longleftarrow}{\lim}^1_n \tilde E^{\bullet-1}(X_n) \longrightarrow \tilde E^{\bullet}(X) \longrightarrow \underset{\longleftarrow}{\lim}_n \tilde E^{\bullet}(X_n) \to 0 \,,

saying that

  1. the failure of the canonical comparison map E˜ (X)limE˜ (X n)\tilde E^\bullet(X) \to \underset{\longleftarrow}{\lim} \tilde E^\bullet(X_n) to the [[limit]] of the [[cohomology groups]] on the finite stages to be an [[isomorphism]] is at most in a non-vanishing [[kernel]];

  2. this kernel is precisely the lim 1\lim^1 (def. ) of the cohomology groups at the finite stages in one degree lower.

e.g. (Switzer 75, prop. 7.66, Kochmann 96, prop. 4.2.2)



X =(X 0i 0X 1i 1X 2i 1) X_\bullet = \left( X_0 \overset{i_0}{\hookrightarrow} X_1 \overset{i_1}{\hookrightarrow} X_2 \overset{i_1}{\hookrightarrow} \cdots \right)

the sequence of stages of the ([[pointed topological space|pointed]]) [[CW-complex]] X=lim nX nX = \underset{\longleftarrow}{\lim}_n X_n, write

A X nX 2n×[2n,2n+1]; B X nX (2n+1)×[2n+1,2n+2]. \begin{aligned} A_X &\coloneqq \underset{n \in \mathbb{N}}{\sqcup} X_{2n} \times [2n,{2n}+1]; \\ B_X &\coloneqq \underset{n \in \mathbb{N}}{\sqcup} X_{(2n+1)} \times [2n+1,{2n}+2]. \end{aligned}

for the [[disjoint unions]] of the [[cylinders]] over all the stages in even and all those in odd degree, respectively.

These come with canonical inclusion maps into the [[mapping telescope]] Tel(X )Tel(X_\bullet) (def.), which we denote by

A X B X ι A x ι B x Tel(X ). \array{ A_X && && B_X \\ & {}_{\mathllap{\iota_{A_x}}}\searrow && \swarrow_{\mathrlap{\iota_{B_x}}} \\ && Tel(X_\bullet) } \,.

Observe that

  1. A XB XTel(X )A_X \cup B_X \simeq Tel(X_\bullet);

  2. A XB XnX nA_X \cap B_X \simeq \underset{n \in \mathbb{N}}{\sqcup} X_n;

and that there are [[homotopy equivalences]]

  1. A XnX 2n+1A_X \simeq \underset{n \in \mathbb{N}}{\sqcup} X_{2n+1}

  2. B XnX 2nB_X \simeq \underset{n \in \mathbb{N}}{\sqcup} X_{2n}

  3. Tel(X )XTel(X_\bullet) \simeq X.

The first two are obvious, the third is this proposition.

This implies that the [[Mayer-Vietoris sequence]] (prop.) for E˜ \tilde E^\bullet on the cover ABXA \sqcup B \to X is isomorphic to the bottom horizontal sequence in the following diagram:

E˜ 1(A X)E˜ 1(B X) E˜ 1(A XB X) E˜ (X) (ι A x) *(ι B x) * E˜ (A X)E˜ (B X) E˜ (A XB X) = (id,id) nE˜ 1(X n) nE˜ 1(X n) E˜ (X) (i n *) n nE˜ (X n) nE˜ (X n), \array{ \tilde E^{\bullet-1}(A_X)\oplus \tilde E^{\bullet-1}(B_X) &\longrightarrow& \tilde E^{\bullet-1}(A_X \cap B_X) &\longrightarrow& \tilde E^\bullet(X) &\overset{(\iota_{A_x})^\ast - (\iota_{B_x})^\ast}{\longrightarrow}& \tilde E^\bullet(A_X)\oplus \tilde E^\bullet(B_X) &\overset{}{\longrightarrow}& \tilde E^\bullet(A_X \cap B_X) \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{=}} && {}^{\mathllap{(id, -id)}}\downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ \underset{n}{\prod}\tilde E^{\bullet-1}(X_n) &\underset{\partial}{\longrightarrow}& \underset{n}{\prod}\tilde E^{\bullet-1}(X_n) &\longrightarrow& \tilde E^\bullet(X) &\overset{(i_n^\ast)_{n}}{\longrightarrow}& \underset{n}{\prod}\tilde E^\bullet(X_n) &\underset{\partial}{\longrightarrow}& \underset{n}{\prod}\tilde E^\bullet(X_n) } \,,

hence that the bottom sequence is also a [[long exact sequence]].

To identify the morphism \partial, notice that it comes from pulling back EE-cohomology classes along the inclusions ABAA \cap B \to A and ABBA\cap B \to B. Comonentwise these are the inclusions of each X nX_n into the left and the right end of its cylinder inside the [[mapping telescope]], respectively. By the construction of the [[mapping telescope]], one of these ends is embedded via i n:X nX n+1i_n \colon X_n \hookrightarrow X_{n+1} into the cylinder over X n+1X_{n+1}. In conclusion, \partial acts by

:(a n) n(a ni n *(a n+1)). \partial \;\colon\; (a_n)_{n \in \mathbb{N}} \mapsto ( a_n - i_n^\ast(a_{n+1}) ) \,.

(The relative sign is the one in (ι A x) *(ι B x) *(\iota_{A_x})^\ast - (\iota_{B_x})^\ast originating in the definition of the [[Mayer-Vietoris sequence]] and properly propagated to the bottom sequence while ensuring that E˜ (X) nE˜ (X n)\tilde E^\bullet(X)\to \prod_n \tilde E^\bullet(X_n) is really (i n *) n(i_n^\ast)_n and not (1) n(i n *) n(-1)^n(i_n^\ast)_n, as needed for the statement to be proven.)

This is the morphism from def. for the sequence

E˜ (X n+1)i n *E˜ (X n)i n *E˜ (X n1). \cdots \to \tilde E^\bullet(X_{n+1}) \overset{i_n^\ast}{\longrightarrow} \tilde E^\bullet(X_n) \overset{i_n^\ast}{\longrightarrow} \tilde E^{\bullet}(X_{n-1}) \to \cdots \,.

Hence truncating the above long exact sequence by forming kernel and cokernel of \partial, the result follows via remark and definition .

In contrast:


Let XX be a [[pointed topological space|pointed]] [[CW-complex]], X=lim nX nX = \underset{\longleftarrow}{\lim}_n X_n.

For E˜ \tilde E_\bullet an additive reduced [[generalized homology theory]], then

lim nE˜ (X n)E˜ (X) \underset{\longrightarrow}{\lim}_n \tilde E_\bullet(X_n) \overset{\simeq}{\longrightarrow} \tilde E_\bullet(X)

is an [[isomorphism]].

(Switzer 75, prop. 7.53)

There is also a version for cohomology of spectra:

For X,EHo(Spectra)X, E \in Ho(Spectra) two [[spectra]], then the EE-generalized cohomology of XX is the graded group of homs in the [[stable homotopy category]] (def., exmpl.)

E (X) [X,E] [Σ X,E]. \begin{aligned} E^\bullet(X) & \coloneqq [X,E]_{-\bullet} \\ & \coloneqq [\Sigma^\bullet X, E] \end{aligned} \,.

The [[stable homotopy category]] is, in particular, the [[homotopy category of a model category|homotopy category]] of the stable [[model structure on orthogonal spectra]], in that its [[localization]] at the [[stable weak homotopy equivalences]] is of the form

γ:OrthSpec(Top cg) stableHo(Spectra). \gamma \;\colon\; OrthSpec(Top_{cg})_{stable} \longrightarrow Ho(Spectra) \,.

In the following when considering an [[orthogonal spectrum]] XOrthSpec(Top cg)X \in OrthSpec(Top_{cg}), we use, for brevity, the same symbol for its image under γ\gamma.


For X,EOrthSpec(Top cg)X, E \in OrthSpec(Top_{cg}) two [[orthogonal spectra]] (or two [[symmetric spectra]] such that XX is a [[semistable symmetric spectrum]]) then there is a [[short exact sequence]] of the form

0lim n 1E +n1(X n)E (X)lim nE +n(X n)0 0 \to \underset{\longleftarrow}{\lim}^1_n E^{\bullet + n -1}(X_{n}) \longrightarrow E^\bullet(X) \longrightarrow \underset{\longleftarrow}{\lim}_n E^{\bullet + n}(X_n) \to 0

where lim 1\underset{\longleftarrow}{\lim}^1 denotes the [[lim^1]], and where this and the limit on the right are taken over the following structure morphisms

E +n+1(X n+1)E +1n+1(Σ n X)E +n+1(X nS 1)E +n(X n). E^{\bullet + n + 1}(X_{n+1}) \overset{E^{\bullet+1n+1}(\Sigma^X_n)}{\longrightarrow} E^{\bullet+n+1}(X_n \wedge S^1) \overset{\simeq}{\longrightarrow} E^{\bullet + n}(X_n) \,.

(Schwede 12, chapter II prop. 6.5 (ii)) (using that symmetric spectra underlying orthogonal spectra are semistable (Schwede 12, p. 40))


For X,EHo(Spectra)X,E \in Ho(Spectra) two [[spectra]] such that the tower nE n1(X n)n \mapsto E^{n -1}(X_{n}) satisfies the [[Mittag-Leffler condition]] (def. ), then two morphisms of spectra XEX \longrightarrow E are homotopic already if all their morphisms of component spaces X nE nX_n \to E_n are.


By prop. the assumption implies that the lim 1lim^1-term in prop. vanishes, hence by exactness it follows that in this case there is an [[isomorphism]]

[X,E]E 0(X)lim n[X n,E n]. [X,E] \simeq E^0(X) \overset{\simeq}{\longrightarrow} \underset{\longleftarrow}{\lim}_n [X_n, E_n] \,.

Serre-Atiyah-Hirzebruch spectral sequence

Idea. Another important tool for computing [[generalized cohomology]] is to reduce it to the computation of [[ordinary cohomology]] with [[coefficients]]. Given a [[generalized cohomology theory]] EE, there is a [[spectral sequence]] known as the [[Atiyah-Hirzebruch spectral sequence]] (AHSS) which serves to compute EE-cohomology of FF-[[fiber bundles]] over a [[simplicial complex]] XX in terms of [[ordinary cohomology]] with [[coefficients]] in the generalized cohomology E (F)E^\bullet(F) of the fiber. For E=E = [[HA]] this is known as the [[Serre spectral sequence]].

The [[Atiyah-Hirzebruch spectral sequence]] in turn is a consequence of the “[[Cartan-Eilenberg spectral sequence]]” which arises from the [[exact couple]] of [[relative cohomology]] groups of the skeleta of the CW-complex, and whose first page is the relative cohomology groups for codimension-1 skeleta.

We need the AHSS for instance for the computation of [[Conner-Floyd Chern classes]] below.

Literature. (Kochman 96, section 2.2 and 4.2)

See also the accompanying [[Introduction to Stable homotopy theory – I|lecture notes on spectral sequences]].

Converging spectral sequences


A cohomology [[spectral sequence]] {E r p,q,d r}\{E_r^{p,q}, d_r\} is

  1. a sequence {E r ,}\{E_r^{\bullet,\bullet}\} (for rr \in \mathbb{N}, r1r \geq 1) of [[bigraded object|bigraded]] [[abelian groups]] (the “pages”);

  2. a sequence of [[linear maps]] (the “[[differentials]]”)

    {d r:E r ,E r +r,r+1} \{d_r \;\colon\; E_r^{\bullet,\bullet} \longrightarrow E_r^{\bullet+r, \bullet-r+1}\}

such that

  • H r+1 ,H_{r+1}^{\bullet,\bullet} is the [[cochain cohomology]] of d rd_r, i.e. E r+1 ,=H(E r ,,d r)E_{r+1}^{\bullet, \bullet} = H(E_r^{\bullet,\bullet},d_r), for all rr \in \mathbb{N}, r1r \geq 1.

Given a \mathbb{Z}-[[graded abelian group]]_ C C^\bullet equipped with a decreasing [[filtration]]

C F sC F s+1C 0 C^\bullet \supset \cdots \supset F^s C^\bullet \supset F^{s+1} C^\bullet \supset \cdots \supset 0

such that

C =sF sC and0=sF sC C^\bullet = \underset{s}{\cup} F^s C^\bullet \;\;\;\; and \;\;\;\; 0 = \underset{s}{\cap} F^s C^\bullet

then the spectral sequence is said to converge to C C^\bullet, denoted,

E 2 ,C E_2^{\bullet,\bullet} \Rightarrow C^\bullet


  1. in each bidegree (s,t)(s,t) the sequence {E r s,t} r\{E_r^{s,t}\}_r eventually becomes constant on a group

    E s,tE 1 s,tE_\infty^{s,t} \coloneqq E_{\gg 1}^{s,t};

  2. E ,E_\infty^{\bullet,\bullet} is the [[associated graded]] of the filtered C C^\bullet in that

    E s,tF sC s+t/F s+1C s+tE_\infty^{s,t} \simeq F^s C^{s+t} / F^{s+1}C^{s+t}.

The converging spectral sequence is called a [[multiplicative spectral sequence]] if

  1. {E 2 ,}\{E_2^{\bullet,\bullet}\} is equipped with the structure of a [[bigraded object|bigraded]] [[associative algebra|algebra]];

  2. F C F^\bullet C^\bullet is equipped with the structure of a filtered [[graded algebra]] (F pC kF qC lF p+qC k+lF^p C^k \cdot F^q C^l \subset F^{p+q} C^{k+l});

such that

  1. each d rd_{r} is a [[derivation]] with respect to the (induced) algebra structure on E r ,{E_r^{\bullet,\bullet}}, graded of degree 1 with respect to total degree;

  2. the multiplication on E ,E_\infty^{\bullet,\bullet} is compatible with that on C C^\bullet.


The point of [[spectral sequences]] is that by subdividing the data in any [[graded abelian group]] C C^\bullet into filtration stages, with each stage itself subdivided into bidegrees, such that each consecutive stage depends on the previous one in way tightly controled by the bidegrees, then this tends to give much control on the computation of C C^\bullet. For instance it often happens that one may argue that the differentials in some spectral sequence all vanish from some page on (one says that the spectral sequence collapses at that page) by pure degree reasons, without any further computation.


The archetypical example of (co-)homology spectral sequences as in def. are induced from a [[filtered chain complex|filtering]] on a (co-)chain complex, converging to the (co-)[[chain homology]] of the chain complex by consecutively computing relative (co-)chain homologies, relative to decreasing (increasing) filtering degrees. For more on such [[spectral sequences of filtered complexes]] see at [[Introduction to Stable homotopy theory – I|Interlude – Spectral sequences]] the section For filtered complexes.

A useful way to generate spectral sequences is via [[exact couples]]:


An [[exact couple]] is three [[homomorphisms]] of [[abelian groups]] of the form

D g D f h E \array{ D && \stackrel{g}{\longrightarrow} && D \\ & {}_{\mathllap{f}}\nwarrow && \swarrow_{\mathrlap{h}} \\ && E }

such that the [[image]] of one is the [[kernel]] of the next.

im(h)=ker(f),im(f)=ker(g),im(g)=ker(f). im(h) = ker(f)\,,\;\;\; im(f) = ker(g)\,, \;\;\; im(g) = ker(f) \,.

Given an exact couple, then its derived exact couple is

im(g) g im(g) f hg 1 H(E,hf), \array{ im(g) && \stackrel{g}{\longrightarrow} && im(g) \\ & {}_{\mathllap{f}}\nwarrow && \swarrow_{\mathrlap{h \circ g^{-1}}} \\ && H(E, h \circ f) } \,,

where g 1g^{-1} denotes the operation of sending one equivalence class to the equivalenc class of any preimage under gg of any of its representatives.


(cohomological spectral sequence of an exact couple)

Given an exact couple, def. ,

D 1 g 1 D 1 f 1 h 1 E 1 \array{ D_1 && \stackrel{g_1}{\longrightarrow} && D_1 \\ & {}_{\mathllap{f_1}}\nwarrow && \swarrow_{\mathrlap{h_1}} \\ && E_1 }

its derived exact couple

D 2 g 2 D 2 f 2 h 2 E 2 \array{ D_2 && \stackrel{g_2}{\longrightarrow} && D_2 \\ & {}_{\mathllap{f_2}}\nwarrow && \swarrow_{\mathrlap{h_2}} \\ && E_2 }

is itself an exact couple. Accordingly there is induced a sequence of exact couples

D r g r D r f r h r E r. \array{ D_r && \stackrel{g_r}{\longrightarrow} && D_r \\ & {}_{\mathllap{f_r}}\nwarrow && \swarrow_{\mathrlap{h_r}} \\ && E_r } \,.

If the abelian groups DD and EE are equipped with [[bigraded object|bigrading]] such that

deg(f)=(0,0),deg(g)=(1,1),deg(h)=(1,0) deg(f) = (0,0)\,,\;\;\;\; deg(g) = (-1,1)\,,\;\;\; deg(h) = (1,0)

then {E r ,,d r}\{E_r^{\bullet,\bullet}, d_r\} with

d r h rf r =hg r+1f \begin{aligned} d_r & \coloneqq h_r \circ f_r \\ & = h \circ g^{-r+1} \circ f \end{aligned}

is a cohomological spectral sequence, def. .

(As before in prop. , the notation g ng^{-n} with nn \in \mathbb{N} denotes the function given by choosing, on representatives, a [[preimage]] under g n=gggntimesg^n = \underset{n\;times}{\underbrace{g \circ \cdots \circ g \circ g}}, with the implicit claim that all possible choices represent the same equivalence class.)

If for every bidegree (s,t)(s,t) there exists R s,t1R_{s,t} \gg 1 such that for all rR s,tr \geq R_{s,t}

  1. g:D s+R,tRD s+R1,tR1g \colon D^{s+R,t-R} \stackrel {\simeq}{\longrightarrow} D^{s+R -1, t-R-1};

  2. g:D sR+1,t+R20D sR,t+R1g\colon D^{s-R+1, t+R-2} \stackrel{0}{\longrightarrow} D^{s-R,t+R-1}

then this spectral sequence converges to the [[inverse limit]] group

G lim(gD s,sgD s1,s+1g) G^\bullet \coloneqq \underset{}{\lim} \left( \cdots \stackrel{g}{\to} D^{s,\bullet-s} \stackrel{g}{\longrightarrow} D^{s-1, \bullet - s + 1} \stackrel{g}{\to} \cdots \right)

filtered by

F pG ker(G D p1,p+1). F^p G^\bullet \coloneqq ker(G^\bullet \to D^{p-1, \bullet - p+1}) \,.

(e.g. Kochmann 96, lemma 2.6.2)


We check the claimed form of the E E_\infty-page:

Since ker(h)=im(g)ker(h) = im(g) in the exact couple, the kernel

ker(d r1)ker(hg r+2f) ker(d_{r-1}) \coloneqq ker(h \circ g^{-r+2} \circ f)

consists of those elements xx such that g r+2(f(x))=g(y)g^{-r+2} (f(x)) = g(y), for some yy, hence

ker(d r1) s,tf 1(g r1(D s+r1,tr+1)). ker(d_{r-1})^{s,t} \simeq f^{-1}(g^{r-1}(D^{s+r-1,t-r+1})) \,.

By assumption there is for each (s,t)(s,t) an R s,tR_{s,t} such that for all rR s,tr \geq R_{s,t} then ker(d r1) s,tker(d_{r-1})^{s,t} is independent of rr.

Moreover, im(d r1)im(d_{r-1}) consists of the image under hh of those xD s1,tx \in D^{s-1,t} such that g r2(x)g^{r-2}(x) is in the image of ff, hence (since im(f)=ker(g)im(f) = ker(g) by exactness of the exact couple) such that g r2(x)g^{r-2}(x) is in the kernel of gg, hence such that xx is in the kernel of g r1g^{r-1}. If r>Rr \gt R then by assumption g r1| D s1,t=0g^{r-1}|_{D^{s-1,t}} = 0 and so then im(d r1)=im(h)im(d_{r-1}) = im(h).

(Beware this subtlety: while g R s,t| D s1,tg^{R_{s,t}}|_{D^{s-1,t}} vanishes by the convergence assumption, the expression g R s,t| D s+r1,tr+1g^{R_{s,t}}|_{D^{s+r-1,t-r+1}} need not vanish yet. Only the higher power g R s,t+R s+1,t+2+2| D s+r1,tr+1g^{R_{s,t}+ R_{s+1,t+2}+2}|_{D^{s+r-1,t-r+1}} is again guaranteed to vanish. )

It follows that

E p,np =ker(d R)/im(d R) f 1(im(g R1))/im(h) fim(g R1)im(f) im(g R1)ker(g) \begin{aligned} E_\infty^{p,n-p} & = ker(d_R)/im(d_R) \\ & \simeq f^{-1}(im(g^{R-1}))/im(h) \\ & \underoverset{\simeq}{f}{\longrightarrow} im(g^{R-1}) \cap im(f) \\ & \simeq im(g^{R-1}) \cap ker(g) \end{aligned}

where in last two steps we used once more the exactness of the exact couple.

(Notice that the above equation means in particular that the E E_\infty-page is a sub-group of the image of the E 1E_1-page under ff.)

The last group above is that of elements xG nx \in G^n which map to zero in D p1,np+1D^{p-1,n-p+1} and where two such are identified if they agree in D p,npD^{p,n-p}, hence indeed

E p,npF pG n/F p+1G n. E_\infty^{p,n-p} \simeq F^p G^n / F^{p+1} G^n \,.

Given a [[spectral sequence]] (def. ), then even if it converges strongly, computing its infinity-page still just gives the [[associated graded]] of the [[filtered object]] that it converges to, not the filtered object itself. The latter is in each filter stage an [[extension]] of the previous stage by the corresponding stage of the infinity-page, but there are in general several possible extensions (the trivial extension or some twisted extensions). The problem of determining these extensions and hence the problem of actually determining the filtered object from a spectral sequence converging to it is often referred to as the extension problem.

More in detail, consider, for definiteness, a cohomology spectral sequence converging to some [[filtered object|filtered]] F H F^\bullet H^\bullet

E p,qH . E^{p,q} \;\Rightarrow\; H^\bullet \,.

Then by definition of convergence there are isomorphisms

E p,F pH p+/F p+1H p+. E_\infty^{p,\bullet} \simeq F^p H^{p + \bullet} / F^{p+1} H^{p + \bullet} \,.

Equivalently this means that there are [[short exact sequences]] of the form

0F p+1H p+F pH p+E p,0. 0 \to F^{p+1}H^{p +\bullet} \hookrightarrow F^p H^{p +\bullet} \longrightarrow E_\infty^{p,\bullet} \to 0 \,.

for all pp. The extension problem then is to inductively deduce F pH F^p H^\bullet from knowledge of F p+1H F^{p+1}H^\bullet and E p,E_\infty^{p,\bullet}.

In good cases these short exact sequences happen to be [[split exact sequences]], which means that the extension problem is solved by the [[direct sum]]

F pH p+F p+1H p+E p,. F^p H^{p+\bullet} \simeq F^{p+1} H^{p+\bullet} \oplus E_\infty^{p,\bullet} \,.

But in general this need not be the case.

One sufficient condition that these exact sequences split is that they consist of homomorphisms of RR-[[modules]], for some [[ring]] RR, and that E p,E_\infty^{p,\bullet} are [[projective modules]] (for instance [[free modules]]) over RR. Because then the [[Ext]]-group Ext R 1(E p,,)Ext^1_R(E_\infty^{p,\bullet},-) vanishes, and hence all extensions are trivial, hence split.

So for instance for every spectral sequence in [[vector spaces]] the extension problem is trivial (since every vector space is a free module).


The following proposition requires, in general, to evaluate cohomology functors not just on [[CW-complexes]], but on all topological spaces. Hence we invoke prop. to regard a [[reduced cohomology theory]] as a contravariant functor on all pointed topological spaces, which sends [[weak homotopy equivalences]] to isomorphisms (def. ).


(Serre-Cartan-Eilenberg-Whitehead-Atiyah-Hirzebruch spectral sequence)

Let A A^\bullet be a an additive unreduced [[generalized (Eilenberg-Steenrod) cohomology|generalized cohomology functor]] (def.). Let BB be a [[CW-complex]] and let XπBX \stackrel{\pi}{\to} B be a [[Serre fibration]] (def.), such that all its [[fibers]] are [[weakly contractible topological space|weakly contractible]] or such that BB is [[simply connected topological space|simply connected]]. In either case all [[fibers]] are identified with a typical fiber FF up to [[weak homotopy equivalence]] by connectedness (this example), and well defined up to unique iso in the homotopy category by simply connectedness:

F X Fib cl B. \array{ F &\longrightarrow& X \\ && \downarrow^{\mathrlap{\in Fib_{cl}}} \\ && B } \,.

If at least one of the following two conditions is met

  • BB is [[finite number|finite]]-dimensional as a [[CW-complex]];

  • A (F)A^\bullet(F) is bounded below in degree and the sequences A p(X n+1)A p(X n)\cdots \to A^p(X_{n+1}) \to A^p(X_n) \to \cdots satisfy the [[Mittag-Leffler condition]] (def. ) for all pp;

then there is a cohomology [[spectral sequence]], def. , whose E 2E_2-page is the [[ordinary cohomology]] H (B,A (F))H^\bullet(B,A^\bullet(F)) of BB with [[coefficients]] in the AA-[[cohomology groups]] A (F)A^\bullet(F) of the fiber, and which converges to the AA-cohomology groups of the total space

E 2 p,q=H p(B,A q(F))A (X) E_2^{p,q} = H^p(B, A^q(F)) \; \Rightarrow \; A^\bullet(X)

with respect to the filtering given by

F pA (X)ker(A (X)A (X p1)), F^p A^\bullet(X) \coloneqq ker\left( A^\bullet(X) \to A^\bullet(X_{p-1}) \right) \,,

where X pπ 1(B p)X_{p} \coloneqq \pi^{-1}(B_{p}) is the fiber over the ppth stage of the [[CW-complex]] B=lim nB nB = \underset{\longleftarrow}{\lim}_n B_n.


The exactness axiom for AA gives an [[exact couple]], def. , of the form

s,tA s+t(X s) s,tA s+t(X s) s,tA s+t(X s,X s1)(A s+t(X s) A s+t(X s1) δ A s+t(X s,X s1) A s+t+1(X s,X s1)), \array{ \underset{s,t}{\prod} A^{s+t}(X_{s}) && \stackrel{}{\longrightarrow} && \underset{s,t}{\prod} A^{s+t}(X_{s}) \\ & \nwarrow && \swarrow \\ && \underset{s,t}{\prod} A^{s+t}(X_{s}, X_{s-1}) } \;\;\;\;\;\;\; \left( \array{ A^{s+t}(X_s) & \longrightarrow & A^{s+t}(X_{s-1}) \\ \uparrow && \downarrow_{\mathrlap{\delta}} \\ A^{s+t}(X_s, X_{s-1}) && A^{s+t+1}(X_{s}, X_{s-1}) } \right) \,,

where we take X 1=XX_{\gg 1} = X and X <0=X_{\lt 0} = \emptyset.

In order to determine the E 2E_2-page, we analyze the E 1E_1-page: By definition

E 1 s,t=A s+t(X s,X s1) E_1^{s,t} = A^{s+t}(X_s, X_{s-1})

Let C(s)C(s) be the set of ss-dimensional cells of BB, and notice that for σC(s)\sigma \in C(s) then

(π 1(σ),π 1(σ))(D n,S n1)×F σ, (\pi^{-1}(\sigma), \pi^{-1}(\partial \sigma)) \simeq (D^n, S^{n-1}) \times F_\sigma \,,

where F σF_\sigma is [[weak homotopy equivalence|weakly homotopy equivalent]] to FF (exmpl.).

This implies that

E 1 s,t A s+t(X s,X s1) A˜ s+t(X s/X s1) A˜ s+t(σC(n)S sF +) σC(s)A˜ s+t(S sF +) σC(s)A˜ t(F +) σC(s)A t(F) C cell s(B,A t(F)), \begin{aligned} E_1^{s,t} & \coloneqq A^{s+t}(X_s, X_{s-1}) \\ & \simeq \tilde A^{s+t}(X_s/X_{s-1}) \\ & \simeq \tilde A^{s+t}(\underset{\sigma \in C(n)}{\vee} S^s \wedge F_+) \\ & \simeq \underset{\sigma \in C(s)}{\prod} \tilde A^{s+t}(S^s \wedge F_+) \\ & \simeq \underset{\sigma \in C(s)}{\prod} \tilde A^t(F_+) \\ & \simeq \underset{\sigma \in C(s)}{\prod} A^t(F) \\ & \simeq C^s_{cell}(B,A^t(F)) \end{aligned} \,,

where we used the relation to [[reduced cohomology]] A˜\tilde A, prop. together with lemma , then the wedge axiom and the suspension isomorphism of the latter.

The last group C cell s(B,A t(F))C^s_{cell}(B,A^t(F)) appearing in this sequence of isomorphisms is that of [[cellular cohomology|cellular cochains]] (def.) of degree ss on BB with [[coefficients]] in the group A t(F)A^t(F).

Since [[cellular cohomology]] of a [[CW-complex]] agrees with its [[singular cohomology]] (thm.), hence with its [[ordinary cohomology]], to conclude that the E 2E_2-page is as claimed, it is now sufficient to show that the differential d 1d_1 coincides with the differential in the [[cellular cochain complex]] (def.).

We discuss this now for π=id\pi = id, hence X=BX = B and F=*F = \ast. The general case works the same, just with various factors of FF appearing in the following:

Consider the following diagram, which [[commuting diagram|commutes]] due to the [[natural transformation|naturality]] of the connecting homomorphism δ\delta of A A^\bullet:

*: C cell s1(X,A t(*)) = iI s1A t(*) iI sA t(*) = C cell s(X,A t(*)) iI s1A˜ s+t1(S s1) iI sA˜ s+t(S s) d 1: A s+t1(X s1,X s2) A s+t1(X s1) δ A s+t(X s,X s1) A s+t1(S s1,) A s+t1(S s1) δ A s+t(D s,S s1). \array{ \partial^\ast \colon & C^{s-1}_{cell}(X,A^t(\ast)) & =& \underset{i \in I_{s-1}}{\prod} A^t(\ast) && \longrightarrow && \underset{i \in I_s}{\prod} A^t(\ast) & = & C_{cell}^{s}(X,A^t(\ast)) \\ && & {}^{\mathllap{\simeq}}\downarrow && && \downarrow^{\mathrlap{\simeq}} \\ && & \underset{i \in I_{s-1}}{\prod} \tilde A^{s+t-1}(S^{s-1}) && && \underset{i \in I_s}{\prod} \tilde A^{s+t}(S^{s}) \\ && & {}^{\mathllap{\simeq}}\downarrow && && \downarrow^{\mathrlap{\simeq}} \\ && d_1 \colon & A^{s+t-1}(X_{s-1}, X_{s-2}) &\overset{}{\longrightarrow}& A^{s+t-1}(X_{s-1}) &\overset{\delta}{\longrightarrow}& A^{s+t}(X_s, X_{s-1}) \\ && & \downarrow && \downarrow && \downarrow \\ && & A^{s+t-1}(S^{s-1}, \emptyset) &\overset{}{\longrightarrow}& A^{s+t-1}(S^{s-1}) &\overset{\delta}{\longrightarrow}& A^{s+t}(D^s , S^{s-1}) } \,.

Here the bottom vertical morphisms are those induced from any chosen cell inclusion (D s,S s1)(X s,X s1)(D^s , S^{s-1}) \hookrightarrow (X_s, X_{s-1}).

The differential d 1d_1 in the spectral sequence is the middle horizontal composite. From this the vertical isomorphisms give the top horizontal map. But the bottom horizontal map identifies this top horizontal morphism componentwise with the restriction to the boundary of cells. Hence the top horizontal morphism is indeed the coboundary operator *\partial^\ast for the [[cellular cohomology]] of XX with coefficients in A (*)A^\bullet(\ast) (def.). This cellular cohomology coincides with [[singular cohomology]] of the [[CW-complex]] XX (thm.), hence computes the [[ordinary cohomology]] of XX.

Now to see the convergence. If BB is finite dimensional then the convergence condition as stated in prop. is met. Alternatively, if A (F)A^\bullet(F) is bounded below in degree, then by the above analysis the E 1E_1-page has a horizontal line below which it vanishes. Accordingly the same is then true for all higher pages, by each of them being the cohomology of the previous page. Since the differentials go right and down, eventually they pass beneath this vanishing line and become 0. This is again the condition needed in the proof of prop. to obtain convergence.

By that proposition the convergence is to the [[inverse limit]]

lim(A (X s+1)A (X s)). \underset{\longleftarrow}{\lim} \left( \cdots \stackrel{}{\to} A^\bullet(X_{s+1}) \longrightarrow A^\bullet(X_{s}) \to \cdots \right) \,.

If XX is finite dimensional or more generally if the sequences that this limit is over satisfy the [[Mittag-Leffler condition]] (def. ), then this limit is A (X)A^\bullet(X), by prop. .

Multiplicative structure


For E E^\bullet a [[multiplicative cohomology theory]] (def. ), then the Atiyah-Hirzebruch spectral sequences (prop. ) for E (X)E^\bullet(X) are [[multiplicative spectral sequences]].

A decent proof is spelled out in (Kochman 96, prop. 4.2.9). Use the graded commutativity of smash products of spheres to get the sign in the graded derivation law for the differentials. See also the proof via [[Cartan-Eilenberg systems]] at multiplicative spectral sequence – Examples – AHSS for multiplicative cohomology.


Given a multiplicative cohomology theory (A,μ,1)(A,\mu,1) (def. ), then for every [[Serre fibration]] XBX \to B (def.) all the differentials in the corresponding [[Atiyah-Hirzebruch spectral sequence]] of prop.

H (B,A (F))A (X) H^\bullet(B,A^\bullet(F)) \;\Rightarrow\; A^\bullet(X)

are linear over A (*)A^\bullet(\ast).


By the proof of prop. , the differentials are those induced by the [[exact couple]]

s,tA s+t(X s) s,tA s+t(X s) s,tA s+t(X s,X s1)(A s+t(X s) A s+t(X s1) δ A s+t(X s,X s1) A s+t+1(X s,X s1)). \array{ \underset{s,t}{\prod} A^{s+t}(X_{s}) && \stackrel{}{\longrightarrow} && \underset{s,t}{\prod} A^{s+t}(X_{s}) \\ & \nwarrow && \swarrow \\ && \underset{s,t}{\prod} A^{s+t}(X_{s}, X_{s-1}) } \;\;\;\;\;\;\; \left( \array{ A^{s+t}(X_s) & \longrightarrow & A^{s+t}(X_{s-1}) \\ \uparrow && \downarrow_{\mathrlap{\delta}} \\ A^{s+t}(X_s, X_{s-1}) && A^{s+t+1}(X_{s}, X_{s-1}) } \right) \,.

consisting of the pullback homomorphisms and the connecting homomorphisms of AA.

By prop. its differentials on page rr are the composites of one pullback homomorphism, the preimage of (r1)(r-1) pullback homomorphisms, and one connecting homomorphism of AA. Hence the statement follows with prop. .


For EE a [[homotopy commutative ring spectrum]] (def.) and XX a finite [[CW-complex]], then the [[Kronecker pairing]]

, X:E 1(X)E 2(X)π 2 1(E) \langle-,-\rangle_X \;\colon\; E^{\bullet_1}(X) \otimes E_{\bullet_2}(X) \longrightarrow \pi_{\bullet_2-\bullet_1}(E)

extends to a compatible pairing of [[Atiyah-Hirzebruch spectral sequences]].

(Kochman 96, prop. 4.2.10)

Cobordism theory

Idea. As one passes from [[abelian groups]] to [[spectra]], a miracle happens: even though the latter are just the proper embodiment of [[linear algebra]] in the context of [[homotopy theory]] (“[[higher algebra]]”) their inspection reveals that spectra natively know about deep phenomena of [[differential topology]], [[index theory]] and in fact [[string theory]] (for instance via a close relation between [[genera and partition functions - table|genera and partition functions]]).

A strong manifestation of this phenomenon comes about in [[complex oriented cohomology theory]]/[[chromatic homotopy theory]] that we eventually come to below. It turns out to be higher algebra over the complex Thom spectrum [[MU]].

Here we first concentrate on its real avatar, the [[Thom spectrum]] [[MO]]. The seminal result of [[Thom’s theorem]] says that the [[stable homotopy groups]] of [[MO]] form the [[cobordism ring]] of [[cobordism]]-[[equivalence classes]] of [[manifolds]]. In the course of discussing this [[cobordism theory]] one encounters various phenomena whose complex version also governs the complex oriented cohomology theory that we are interested in below.

Literature. (Kochman 96, chapter I and sections II.2, II6). A quick efficient account is in (Malkiewich 11). See also (Aguilar-Gitler-Prieto 02, section 11).

Classifying spaces and GG-Structure

Idea. Every [[manifold]] XX of [[dimension]] nn carries a canonical [[vector bundle]] of [[rank]] nn: its [[tangent bundle]]. There is a [[universal vector bundle]] of rank nn, of which all others arise by [[pullback]], up to [[isomorphism]]. The base space of this universal bundle is hence called the [[classifying space]] and denoted BGL(n)BO(n)B GL(n) \simeq B O(n) (for O(n)O(n) the [[orthogonal group]]). This may be realized as the [[homotopy type]] of a [[direct limit]] of [[Grassmannian manifolds]]. In particular the tangent bundle of a manifold XX is classified by a map XBO(n)X \longrightarrow B O(n), unique up to homotopy. For GG a [[subgroup]] of O(n)O(n), then a lift of this map through the canonical map BGBO(n)B G \longrightarrow B O(n) of classifying spaces is a [[G-structure]] on XX

BG X BO(n) \array{ && B G \\ &\nearrow& \downarrow \\ X &\longrightarrow& B O(n) }

for instance an [[orientation]] for the inclusion SO(n)O(n)SO(n) \hookrightarrow O(n) of the [[special orthogonal group]], or an [[almost complex structure]] for the inclusion U(n)O(2n)U(n) \hookrightarrow O(2n) of the [[unitary group]].

All this generalizes, for instance from tangent bundles to [[normal bundles]] with respect to any [[embedding]]. It also behaves well with respect to passing to the [[boundary]] of manifolds, hence to [[bordism]]-classes of manifolds. This is what appears in [[Thom’s theorem]] below.

Literature. (Kochman 96, 1.3-1.4), for stable normal structures also (Stong 68, beginning of chapter II)

Coset spaces


For XX a [[smooth manifold]] and GG a [[compact Lie group]] equipped with a [[free action|free]] smooth [[action]] on XX, then the [[quotient]] [[projection]]

XX/G X \longrightarrow X/G

is a GG-[[principal bundle]] (hence in particular a [[Serre fibration]]).

This is originally due to (Gleason 50). See e.g. (Cohen, theorem 1.3)


For GG a [[Lie group]] and HGH \subset G a [[compact Lie group|compact]] [[subgroup]], then the [[coset]] [[quotient]] [[projection]]

GG/H G \longrightarrow G/H

is an HH-[[principal bundle]] (hence in particular a [[Serre fibration]]).


For GG a [[compact Lie group]] and KHGK \subset H \subset G [[closed subspace|closed]] [[subgroups]], then the [[projection]] map on [[coset spaces]]

p:G/KG/H p \;\colon\; G/K \longrightarrow G/H

is a locally trivial H/KH/K-[[fiber bundle]] (hence in particular a [[Serre fibration]]).


Observe that the projection map in question is equivalently

G× H(H/K)G/H, G \times_H (H/K) \longrightarrow G/H \,,

(where on the left we form the [[Cartesian product]] and then divide out the [[diagonal action]] by HH). This exhibits it as the H/KH/K-[[fiber bundle]] [[associated bundle|associated]] to the HH-[[principal bundle]] of corollary .

Orthogonal and Unitary groups


The orthogonal group O(n)O(n) is [[compact topological space]], hence in particular a [[compact Lie group]].


The unitary group U(n)U(n) is [[compact topological space]], hence in particular a [[compact Lie group]].


The [[n-spheres]] are [[coset]] spaces of [[orthogonal groups]]:

S nO(n+1)/O(n). S^n \simeq O(n+1)/O(n) \,.

The odd-dimensional spheres are also coset spaces of [[unitary groups]]:

S 2n+1U(n+1)/U(n) S^{2n+1} \simeq U(n+1)/U(n)

Regarding the first statement:

Fix a unit vector in n+1\mathbb{R}^{n+1}. Then its [[orbit]] under the defining O(n+1)O(n+1)-[[action]] on n+1\mathbb{R}^{n+1} is clearly the canonical embedding S n n+1S^n \hookrightarrow \mathbb{R}^{n+1}. But precisely the subgroup of O(n+1)O(n+1) that consists of rotations around the axis formed by that unit vector [[stabilizer group|stabilizes]] it, and that subgroup is isomorphic to O(n)O(n), hence S nO(n+1)/O(n)S^n \simeq O(n+1)/O(n).

The second statement follows by the same kind of reasoning:

Clearly U(n+1)U(n+1) [[transitive action|acts transitively]] on the unit sphere S 2n+1S^{2n+1} in n+1\mathbb{C}^{n+1}. It remains to see that its [[stabilizer subgroup]] of any point on this sphere is U(n)U(n). If we take the point with [[coordinates]] (1,0,0,,0)(1,0, 0, \cdots,0) and regard elements of U(n+1)U(n+1) as [[matrices]], then the stabilizer subgroup consists of matrices of the block diagonal form

(1 0 0 A) \left( \array{ 1 & \vec 0 \\ \vec 0 & A } \right)

where AU(n)A \in U(n).


For n,kn,k \in \mathbb{N}, nkn \leq k, then the canonical inclusion of [[orthogonal groups]]

O(n)O(k) O(n) \hookrightarrow O(k)

is an [[n-equivalence|(n-1)-equivalence]], hence induces an [[isomorphism]] on [[homotopy groups]] in degrees <n1\lt n-1 and a [[surjection]] in degree n1n-1.


Consider the [[coset]] [[quotient]] [[projection]]

O(n)O(n+1)O(n+1)/O(n). O(n) \longrightarrow O(n+1) \longrightarrow O(n+1)/O(n) \,.

By prop. and by corollary , the projection O(n+1)O(n+1)/O(n)O(n+1)\to O(n+1)/O(n) is a [[Serre fibration]]. Furthermore, example identifies the [[coset]] with the [[n-sphere]]

S nO(n+1)/O(n). S^{n}\simeq O(n+1)/O(n) \,.

Therefore the [[long exact sequence of homotopy groups]] (exmpl.)of the [[fiber sequence]] O(n)O(n+1)S nO(n)\to O(n+1)\to S^n has the form

π +1(S n)π (O(n))π (O(n+1))π (S n) \cdots \to \pi_{\bullet+1}(S^n) \longrightarrow \pi_\bullet(O(n)) \longrightarrow \pi_\bullet(O(n+1)) \longrightarrow \pi_\bullet(S^n) \to \cdots

Since π <n(S n)=0\pi_{\lt n}(S^n) = 0, this implies that

π <n1(O(n))π <n1(O(n+1)) \pi_{\lt n-1}(O(n)) \overset{\simeq}{\longrightarrow} \pi_{\lt n-1}(O(n+1))

is an isomorphism and that

π n1(O(n))π n1(O(n+1)) \pi_{n-1}(O(n)) \overset{\simeq}{\longrightarrow} \pi_{n-1}(O(n+1))

is surjective. Hence now the statement follows by [[induction]] over knk-n.



For n,kn,k \in \mathbb{N}, nkn \leq k, then the canonical inclusion of [[unitary groups]]

U(n)U(k) U(n) \hookrightarrow U(k)

is a [[n-equivalence|2n-equivalence]], hence induces an [[isomorphism]] on [[homotopy groups]] in degrees <2n\lt 2n and a [[surjection]] in degree 2n2n.


Consider the [[coset]] [[quotient]] [[projection]]

U(n)U(n+1)U(n+1)/U(n). U(n) \longrightarrow U(n+1) \longrightarrow U(n+1)/U(n) \,.

By prop. and corollary , the projection U(n+1)U(n+1)/U(n)U(n+1)\to U(n+1)/U(n) is a [[Serre fibration]]. Furthermore, example identifies the [[coset]] with the [[n-sphere|(2n+1)-sphere]]

S 2n+1U(n+1)/U(n). S^{2n+1}\simeq U(n+1)/U(n) \,.

Therefore the [[long exact sequence of homotopy groups]] (exmpl.)of the [[fiber sequence]] U(n)U(n+1)S 2n+1U(n)\to U(n+1) \to S^{2n+1} is of the form

π +1(S 2n+1)π (U(n))π (U(n+1))π (S 2n+1) \cdots \to \pi_{\bullet+1}(S^{2n+1}) \longrightarrow \pi_\bullet(U(n)) \longrightarrow \pi_\bullet(U(n+1)) \longrightarrow \pi_\bullet(S^{2n+1}) \to \cdots

Since π 2n(S 2n+1)=0\pi_{\leq 2n}(S^{2n+1}) = 0, this implies that

π <2n(U(n))π <2n(U(n+1)) \pi_{\lt 2n}(U(n)) \overset{\simeq}{\longrightarrow} \pi_{\lt 2n}(U(n+1))

is an isomorphism and that

π 2n(U(n))π 2n(U(n+1)) \pi_{2n}(U(n)) \overset{\simeq}{\longrightarrow} \pi_{2n}(U(n+1))

is surjective. Hence now the statement follows by induction over knk-n.

Stiefel manifolds and Grassmannians

Throughout we work in the [[category]] Top cgTop_{cg} of [[compactly generated topological spaces]] (def.). For these the [[Cartesian product]] X×()X \times (-) is a [[left adjoint]] (prop.) and hence preserves [[colimits]].


For n,kn, k \in \mathbb{N} and nkn \leq k, then the nnth real [[Stiefel manifold]] of k\mathbb{R}^k is the [[coset]] [[topological space]].

V n( k)O(k)/O(kn), V_n(\mathbb{R}^k) \coloneqq O(k)/O(k-n) \,,

where the [[action]] of O(kn)O(k-n) is via its canonical embedding O(kn)O(k)O(k-n)\hookrightarrow O(k).

Similarly the nnth complex Stiefel manifold of k\mathbb{C}^k is

V n( k)U(k)/U(kn), V_n(\mathbb{C}^k) \coloneqq U(k)/U(k-n) \,,

here the [[action]] of U(kn)U(k-n) is via its canonical embedding U(kn)U(k)U(k-n)\hookrightarrow U(k).


For n,kn, k \in \mathbb{N} and nkn \leq k, then the nnth real [[Grassmannian]] of k\mathbb{R}^k is the [[coset]] [[topological space]].

Gr n( k)O(k)/(O(n)×O(kn)), Gr_n(\mathbb{R}^k) \coloneqq O(k)/(O(n) \times O(k-n)) \,,

where the [[action]] of the [[product group]] is via its canonical embedding O(n)×O(kn)O(n)O(n)\times O(k-n) \hookrightarrow O(n) into the [[orthogonal group]].

Similarly the nnth complex [[Grassmannian]] of k\mathbb{C}^k is the [[coset]] [[topological space]].

Gr n( k)U(k)/(U(n)×U(kn)), Gr_n(\mathbb{C}^k) \coloneqq U(k)/(U(n) \times U(k-n)) \,,

where the [[action]] of the [[product group]] is via its canonical embedding U(n)×U(kn)U(n)U(n)\times U(k-n) \hookrightarrow U(n) into the [[unitary group]].

  • G 1( n+1)P nG_1(\mathbb{R}^{n+1}) \simeq \mathbb{R}P^n is [[real projective space]] of [[dimension]] nn.

  • G 1( n+1)P nG_1(\mathbb{C}^{n+1}) \simeq \mathbb{C}P^n is [[complex projective space]] of [[dimension]] nn (def. ).


For all nkn \leq k \in \mathbb{N}, the canonical [[projection]] from the [[Stiefel manifold]] (def. ) to the [[Grassmannian]] is a O(n)O(n)-[[principal bundle]]

O(n) V n( k) Gr n( k) \array{ O(n) &\hookrightarrow& V_n(\mathbb{R}^k) \\ && \downarrow \\ && Gr_n(\mathbb{R}^k) }

and the projection from the complex Stiefel manifold to the Grassmannian us a U(n)U(n)-[[principal bundle]]:

U(n) V n( k) Gr n( k). \array{ U(n) &\hookrightarrow& V_n(\mathbb{C}^k) \\ && \downarrow \\ && Gr_n(\mathbb{C}^k) } \,.

By prop and prop .


The real [[Grassmannians]] Gr n( k)Gr_n(\mathbb{R}^k) and the complex Grassmannians Gr n( k)Gr_n(\mathbb{C}^k) of def. admit the structure of [[CW-complexes]]. Moreover the canonical inclusions

Gr n( k)Gr n( k+1) Gr_n(\mathbb{R}^k) \hookrightarrow Gr_n(\mathbb{R}^{k+1})

are subcomplex incusion (hence [[relative cell complex]] inclusions).

Accordingly there is an induced CW-complex structure on the [[classifying space]] (def. ).

BO(n)lim kGr n( k). B O(n) \simeq \underset{\longrightarrow}{\lim}_k Gr_n(\mathbb{R}^k) \,.

A proof is spelled out in (Hatcher, section 1.2 (pages 31-34)).


The [[Stiefel manifolds]] V n( k)V_n(\mathbb{R}^k) and V n( k)V_n(\mathbb{C}^k) from def. admits the structure of a [[CW-complex]].

e.g. (James 59, p. 3, James 76, p. 5 with p. 21, Blaszczyk 07)

(And I suppose with that cell structure the inclusions V n( k)V n( k+1)V_n(\mathbb{R}^k) \hookrightarrow V_n(\mathbb{R}^{k+1}) are subcomplex inclusions.)


The real [[Stiefel manifold]] V n( k)V_n(\mathbb{R}^k) (def. ) is [[n-connected topological space|(k-n-1)-connected]].


Consider the [[coset]] [[quotient]] [[projection]]

O(kn)O(k)O(k)/O(kn)=V n( k). O(k-n) \longrightarrow O(k) \longrightarrow O(k)/O(k-n) = V_n(\mathbb{R}^k) \,.

By prop. and by corollary , the projection O(k)O(k)/O(kn)O(k)\to O(k)/O(k-n) is a [[Serre fibration]]. Therefore there is induced the [[long exact sequence of homotopy groups]] of this [[fiber sequence]], and by prop. it has the following form in degrees bounded by nn:

π kn1(O(kn))epiπ kn1(O(k))0π kn1(V n( k))0π 1<kn1(O(k))π 1<kn1(O(kn)). \cdots \to \pi_{\bullet \leq k-n-1}(O(k-n)) \overset{epi}{\longrightarrow} \pi_{\bullet \leq k-n-1}(O(k)) \overset{0}{\longrightarrow} \pi_{\bullet \leq k-n-1}(V_n(\mathbb{R}^k)) \overset{0}{\longrightarrow} \pi_{\bullet-1 \lt k-n-1}(O(k)) \overset{\simeq}{\longrightarrow} \pi_{\bullet-1 \lt k-n-1}(O(k-n)) \to \cdots \,.

This implies the claim. (Exactness of the sequence says that every element in π n1(V n( k))\pi_{\bullet \leq n-1}(V_n(\mathbb{R}^k)) is in the kernel of zero, hence in the image of 0, hence is 0 itself.)



The complex [[Stiefel manifold]] V n( k)V_n(\mathbb{C}^k) (def. ) is [[n-connected topological space|2(k-n)-connected]].


Consider the [[coset]] [[quotient]] [[projection]]

U(kn)U(k)U(k)/U(kn)=V n( k). U(k-n) \longrightarrow U(k) \longrightarrow U(k)/U(k-n) = V_n(\mathbb{C}^k) \,.

By prop. and by corollary the projection U(k)U(k)/U(kn)U(k)\to U(k)/U(k-n) is a [[Serre fibration]]. Therefore there is induced the [[long exact sequence of homotopy groups]] of this [[fiber sequence]], and by prop. it has the following form in degrees bounded by nn:

π 2(kn)(U(kn))epiπ 2(kn)(U(k))0π 2(kn)(V n( k))0π 1<2(kn)(U(k))π 1<2(kn)(U(kn)). \cdots \to \pi_{\bullet \leq 2(k-n)}(U(k-n)) \overset{epi}{\longrightarrow} \pi_{\bullet \leq 2(k-n)}(U(k)) \overset{0}{\longrightarrow} \pi_{\bullet \leq 2(k-n)}(V_n(\mathbb{C}^k)) \overset{0}{\longrightarrow} \pi_{\bullet-1 \lt 2(k-n)}(U(k)) \overset{\simeq}{\longrightarrow} \pi_{\bullet-1 \lt 2(k-n)}(U(k-n)) \to \cdots \,.

This implies the claim.

Classifying spaces


By def. there are canonical inclusions

Gr n( k)Gr n( k+1) Gr_n(\mathbb{R}^k) \hookrightarrow Gr_n(\mathbb{R}^{k+1})


Gr n( k)Gr n( k+1) Gr_n(\mathbb{C}^k) \hookrightarrow Gr_n(\mathbb{C}^{k+1})

for all kk \in \mathbb{N}. The [[colimit]] (in [[Top]], see there, or rather in Top cgTop_{cg}, see this cor.) over these inclusions is denoted

BO(n)lim kGr n( k) B O(n) \coloneqq \underset{\longrightarrow}{\lim}_k Gr_n(\mathbb{R}^k)


BU(n)lim kGr n( k), B U(n) \coloneqq \underset{\longrightarrow}{\lim}_k Gr_n(\mathbb{C}^k) \,,


Moreover, by def. there are canonical inclusions

V n( k)V n( k+1) V_n(\mathbb{R}^k) \hookrightarrow V_n(\mathbb{R}^{k+1})


V n( k)V n( k+1) V_n(\mathbb{C}^k) \hookrightarrow V_n(\mathbb{C}^{k+1})

that are compatible with the O(n)O(n)-[[action]] and with the U(n)U(n)-action, respectively. The [[colimit]] (in [[Top]], see there, or rather in Top cgTop_{cg}, see this cor.) over these inclusions, regarded as equipped with the induced O(n)O(n)-[[action]], is denoted

EO(n)lim kV n( k) E O(n) \coloneqq \underset{\longrightarrow}{\lim}_k V_n(\mathbb{R}^k)


EU(n)lim kV n( k), E U(n) \coloneqq \underset{\longrightarrow}{\lim}_k V_n(\mathbb{C}^k) \,,


The inclusions are in fact compatible with the bundle structure from prop. , so that there are induced projections

(EO(n) BO(n))lim k(V n( k) Gr n( k)) \left( \array{ E O(n) \\ \downarrow \\ B O(n) } \right) \;\; \simeq \;\; \underset{\longrightarrow}{\lim}_k \left( \array{ V_n(\mathbb{R}^k) \\ \downarrow \\ Gr_n(\mathbb{R}^k) } \right)


(EU(n) BU(n))lim k(V n( k) Gr n( k)), \left( \array{ E U(n) \\ \downarrow \\ B U(n) } \right) \;\; \simeq \;\; \underset{\longrightarrow}{\lim}_k \left( \array{ V_n(\mathbb{C}^k) \\ \downarrow \\ Gr_n(\mathbb{C}^k) } \right) \,,

respectively. These are the standard models for the [[universal principal bundles]] for OO and UU, respectively. The corresponding [[associated bundles|associated]] [[vector bundles]]

EO(n)×O(n) n E O(n) \underset{O(n)}{\times} \mathbb{R}^n


EU(n)×U(n) n E U(n) \underset{U(n)}{\times} \mathbb{C}^n

are the corresponding [[universal vector bundles]].

Since the [[Cartesian product]] O(n)×()O(n)\times (-) in [[compactly generated topological spaces]] preserves colimits, it follows that the colimiting bundle is still an O(n)O(n)-[[principal bundle]]

(EO(n))/O(n) (lim kV n( k))/O(n) lim k(V n( k)/O(n)) lim kGr n( k) BO(n), \begin{aligned} (E O(n))/O(n) & \simeq (\underset{\longrightarrow}{\lim}_k V_{n}(\mathbb{R}^k))/O(n) \\ & \simeq \underset{\longrightarrow}{\lim}_k (V_n(\mathbb{R}^k)/O(n)) \\ & \simeq \underset{\longrightarrow}{\lim}_k Gr_n(\mathbb{R}^k) \\ & \simeq B O(n) \end{aligned} \,,

and anlogously for EU(n)E U(n).

As such this is the standard presentation for the O(n)O(n)-[[universal principal bundle]] and U(n)U(n)-[[universal principal bundle]], respectively. Its base space BO(n)B O(n) is the corresponding [[classifying space]].


There are canonical inclusions

Gr n( k)Gr n+1( k+1) Gr_n(\mathbb{R}^k) \hookrightarrow Gr_{n+1}(\mathbb{R}^{k+1})


Gr n( k)Gr n+1( k+1) Gr_n(\mathbb{C}^k) \hookrightarrow Gr_{n+1}(\mathbb{C}^{k+1})

given by adjoining one coordinate to the ambient space and to any subspace. Under the colimit of def. these induce maps of classifying spaces

BO(n)BO(n+1) B O(n) \longrightarrow B O(n+1)


BU(n)BU(n+1). B U(n) \longrightarrow B U(n+1) \,.

There are canonical maps

Gr n 1( k 1)×Gr n 2( k 2)Gr n 1+n 2( k 1+k 2) Gr_{n_1}(\mathbb{R}^{k_1}) \times Gr_{n_2}(\mathbb{R}^{k_2}) \longrightarrow Gr_{n_1 + n_2}(\mathbb{R}^{k_1 + k_2})


Gr n 1( k 1)×Gr n 2( k 2)Gr n 1+n 2( k 1+k 2) Gr_{n_1}(\mathbb{C}^{k_1}) \times Gr_{n_2}(\mathbb{C}^{k_2}) \longrightarrow Gr_{n_1 + n_2}(\mathbb{C}^{k_1 + k_2})

given by sending ambient spaces and subspaces to their [[direct sum]].

Under the colimit of def. these induce maps of classifying spaces

BO(n 1)×BO(n 2)BO(n 1+n 2) B O(n_1) \times B O(n_2) \longrightarrow B O(n_1 + n_2)


BU(n 1)×BU(n 2)BU(n 1+n 2) B U(n_1) \times B U(n_2) \longrightarrow B U(n_1 + n_2)

The colimiting space EO(n)=lim kV n( k)E O(n) = \underset{\longrightarrow}{\lim}_k V_n(\mathbb{R}^k) from def. is [[weakly contractible topological space|weakly contractible]].

The colimiting space EU(n)=lim kV n( k)E U(n) = \underset{\longrightarrow}{\lim}_k V_n(\mathbb{C}^k) from def. is [[weakly contractible topological space|weakly contractible]].


By propositions , and , the Stiefel manifolds are more and more highly connected as kk increases. Since the inclusions are relative cell complex inclusions by prop. , the claim follows.


The [[homotopy groups]] of the classifying spaces BO(n)B O(n) and BU(n)B U(n) (def. ) are those of the [[orthogonal group]] O(n)O(n) and of the [[unitary group]] U(n)U(n), respectively, shifted up in degree: there are [[isomorphisms]]

π +1(BO(n))π O(n) \pi_{\bullet+1}(B O(n)) \simeq \pi_\bullet O(n)


π +1(BU(n))π U(n) \pi_{\bullet+1}(B U(n)) \simeq \pi_\bullet U(n)

(for homotopy groups based at the canonical basepoint).


Consider the sequence

O(n)EO(n)BO(n) O(n) \longrightarrow E O(n) \longrightarrow B O(n)

from def. , with O(n)O(n) the [[fiber]]. Since (by prop. ) the second map is a [[Serre fibration]], this is a [[fiber sequence]] and so it induces a [[long exact sequence of homotopy groups]] of the form

π (O(n))π (EO(n))π (BO(n))π 1(O(n))π 1(EO(n)). \cdots \to \pi_\bullet(O(n)) \longrightarrow \pi_\bullet(E O(n)) \longrightarrow \pi_\bullet(B O(n)) \longrightarrow \pi_{\bullet-1}(O (n)) \longrightarrow \pi_{\bullet-1}(E O(n)) \to \cdots \,.

Since by cor. π (EO(n))=0\pi_\bullet(E O(n))= 0, exactness of the sequence implies that

π (BO(n))π 1(O(n)) \pi_\bullet(B O(n)) \overset{\simeq}{\longrightarrow} \pi_{\bullet-1}(O (n))

is an isomorphism.

The same kind of argument applies to the complex case.


For nn \in \mathbb{N} there are [[homotopy fiber sequence]] (def.)

S nBO(n)BO(n+1) S^n \longrightarrow B O(n) \longrightarrow B O(n+1)


S 2n+1BU(n)BU(n+1) S^{2n+1} \longrightarrow B U(n) \longrightarrow B U(n+1)

exhibiting the [[n-sphere]] ((2n+1)(2n+1)-sphere) as the [[homotopy fiber]] of the canonical maps from def. .

This means (thm.), that there is a replacement of the canonical inclusion BO(n)BO(n+1)B O(n) \hookrightarrow B O(n+1) (induced via def. ) by a [[Serre fibration]]

BO(n) BO(n+1) weakhomotopyequivalence Serrefib. B˜O(n) \array{ B O(n) &\hookrightarrow& B O(n+1) \\ {}^{\mathllap{{weak \, homotopy} \atop equivalence}}\downarrow & \nearrow_{\mathrlap{Serre \, fib.}} \\ \tilde B O(n) }

such that S nS^n is the ordinary [[fiber]] of BO(n)B˜O(n+1)B O(n)\to \tilde B O(n+1), and analogously for the complex case.


Take B˜O(n)(EO(n+1))/O(n)\tilde B O(n) \coloneqq (E O(n+1))/O(n).

To see that the canonical map BO(n)(EO(n+1))/O(n)B O(n)\longrightarrow (E O(n+1))/O(n) is a [[weak homotopy equivalence]] consider the [[commuting diagram]]

O(n) id O(n) EO(n) EO(n+1) BO(n) (EO(n+1))/O(n). \array{ O(n) &\overset{id}{\longrightarrow}& O(n) \\ \downarrow && \downarrow \\ E O(n) &\longrightarrow& E O(n+1) \\ \downarrow && \downarrow \\ B O(n) &\longrightarrow& (E O(n+1))/O(n) } \,.

By prop. both bottom vertical maps are [[Serre fibrations]] and so both vertical sequences are [[fiber sequences]]. By prop. part of the induced morphisms of [[long exact sequences of homotopy groups]] looks like this

π (BO(n)) π ((EO(n+1))/O(n)) π 1(O(n)) = π 1(O(n)), \array{ \pi_\bullet(B O(n)) &\overset{}{\longrightarrow}& \pi_\bullet( (E O(n+1))/O(n) ) \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}} \\ \pi_{\bullet-1}(O(n)) &\overset{=}{\longrightarrow}& \pi_{\bullet-1}(O(n)) } \,,

where the vertical and the bottom morphism are isomorphisms. Hence also the to morphisms is an isomorphism.

That BO(n)B˜O(n+1)B O(n)\to \tilde B O(n+1) is indeed a [[Serre fibration]] follows again with prop. , which gives the [[fiber sequence]]

O(n+1)/O(n)(EO(n+1))/O(n)(EO(n+1))/O(n+1). O(n+1)/O(n) \longrightarrow (E O(n+1))/O(n) \longrightarrow (E O(n+1))/O(n+1) \,.

The claim then follows with the identification

O(n+1)/O(n)S n O(n+1)/O(n) \simeq S^n

of example .

The argument for the complex case is directly analogous, concluding instead with the identification

U(n+1)/U(n)S 2n+1 U(n+1)/U(n)\simeq S^{2n+1}

from example .

GG-Structure on the Stable normal bundle


Given a [[smooth manifold]] XX of [[dimension]] nn and equipped with an [[embedding]]

i:X k i \;\colon\; X \hookrightarrow \mathbb{R}^k

for some kk \in \mathbb{N}, then the classifying map of its normal bundle is the function

g i:XGr kn( k)BO(kn) g_i \;\colon\; X \to Gr_{k-n}(\mathbb{R}^k) \hookrightarrow B O(k-n)

which sends xXx \in X to the normal of the [[tangent space]]

N xX=(T xX) k N_x X = (T_x X)^{\perp} \hookrightarrow \mathbb{R}^k

regarded as a point in G kn( k)G_{k-n}(\mathbb{R}^k).

The [[normal bundle]] of ii itself is the subbundle of the [[tangent bundle]]

T k k× k T \mathbb{R}^k \simeq \mathbb{R}^k \times \mathbb{R}^k

consisting of those vectors which are [[orthogonal]] to the [[tangent vectors]] of XX:

N i{xX,vT i(x) k|vi *T xXT i(x) k}. N_i \coloneqq \left\{ x\in X, v \in T_{i(x)}\mathbb{R}^k \;\vert\; v \,\perp\, i_\ast T_x X \subset T_{i(x)}\mathbb{R}^k \right\} \,.

A (B,f)(B,f)-structure is

  1. for each nn\in \mathbb{N} a [[pointed topological space|pointed]] [[CW-complex]] B nTop CW */B_n \in Top_{CW}^{\ast/}

  2. equipped with a pointed [[Serre fibration]]

    B n f n BO(n) \array{ B_n \\ \downarrow^{\mathrlap{f_n}} \\ B O(n) }

    to the [[classifying space]] BO(n)B O(n) (def.);

  3. for all n 1n 2n_1 \leq n_2 a pointed continuous function

    g n 1,n 2:B n 1B n 2g_{n_1, n_2} \;\colon\; B_{n_1} \longrightarrow B_{n_2}

    which is the identity for n 1=n 2n_1 = n_2;

such that for all n 1n 2n_1 \leq n_2 \in \mathbb{N} these [[commuting square|squares commute]]

B n 1 g n 1,n 2 B n 2 f n 1 f n 2 BO(n 1) BO(n 2), \array{ B_{n_1} &\overset{g_{n_1,n_2}}{\longrightarrow}& B_{n_2} \\ {}^{\mathllap{f_{n_1}}}\downarrow && \downarrow^{\mathrlap{f_{n_2}}} \\ B O(n_1) &\longrightarrow& B O(n_2) } \,,

where the bottom map is the canonical one from def. .

The (B,f)(B,f)-structure is multiplicative if it is moreover equipped with a system of maps μ n 1,n 2:B n 1×B n 2B n 1+n 2\mu_{n_1,n_2} \colon B_{n_1}\times B_{n_2} \to B_{n_1 + n_2} which cover the canonical multiplication maps (def.)

B n 1×B n 2 μ n 1,n 2 B n 1+n 2 f n 1×f n 2 f n 1+n 2 BO(n 1)×BO(n 2) BO(n 1+n 2) \array{ B_{n_1} \times B_{n_2} &\overset{\mu_{n_1, n_2}}{\longrightarrow}& B_{n_1 + n_2} \\ {}^{\mathllap{f_{n_1} \times f_{n_2}}}\downarrow && \downarrow^{\mathrlap{f_{n_1 + n_2}}} \\ B O(n_1) \times B O(n_2) &\longrightarrow& B O(n_1 + n_2) }

and which satisfy the evident [[associativity]] and [[unitality]], for B 0=*B_0 = \ast the unit, and, finally, which commute with the maps gg in that all n 1,n 2,n 3n_1,n_2, n_3 \in \mathbb{N} these squares commute:

B n 1×B n 2 id×g n 2,n 2+n 3 B n 1×B n 2+n 3 μ n 1,n 2 μ n 1,n 2+n 3 B n 1+n 2 g n 1+n 2,n 1+n 2+n 3 B n 1+n 2+n 3 \array{ B_{n_1} \times B_{n_2} &\overset{id \times g_{n_2,n_2+n_3}}{\longrightarrow}& B_{n_1} \times B_{n_2 + n_3} \\ {}^{\mathllap{\mu_{n_1, n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1,n_2 + n_3}}} \\ B_{n_1 + n_2} &\underset{g_{n_1+n_2, n_1 + n_2 + n_3}}{\longrightarrow}& B_{n_1 + n_2 + n_3} }


B n 1×B n 2 g n 1,n 1+n 3×id B n 1+n 3×B n 2 μ n 1,n 2 μ n 1+n 3,n 2 B n 1+n 2 g n 1+n 2,n 1+n 2+n 3 B n 1+n 2+n 3. \array{ B_{n_1} \times B_{n_2} &\overset{g_{n_1,n_1+n_3} \times id}{\longrightarrow}& B_{n_1+n_3} \times B_{n_2 } \\ {}^{\mathllap{\mu_{n_1, n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1 + n_3 , n_2}}} \\ B_{n_1 + n_2} &\underset{g_{n_1+n_2, n_1 + n_2 + n_3}}{\longrightarrow}& B_{n_1 + n_2 + n_3} } \,.

Similarly, an S 2S^2-(B,f)(B,f)-structure is a compatible system

f 2n:B 2nBO(2n) f_{2n} \colon B_{2n} \longrightarrow B O(2n)

indexed only on the even natural numbers.

Generally, an S kS^k-(B,f)(B,f)-structure for kk \in \mathbb{N}, k1k \geq 1 is a compatible system

f kn:B knBO(kn) f_{k n} \colon B_{k n} \longrightarrow B O(k n)

for all nn \in \mathbb{N}, hence for all knkk n \in k \mathbb{N}.


Examples of (B,f)(B,f)-structures (def. ) include the following:

  1. B n=BO(n)B_n = B O(n) and f n=idf_n = id is orthogonal structure (or “no structure”);

  2. B n=EO(n)B_n = E O(n) and f nf_n the [[universal principal bundle]]-projection is [[framing]]-structure;

  3. B n=BSO(n)=EO(n)/SO(n)B_n = B SO(n) = E O(n)/SO(n) the classifying space of the [[special orthogonal group]] and f nf_n the canonical projection is [[orientation]] structure;

  4. B n=BSpin(n)=EO(n)/Spin(n)B_n = B Spin(n) = E O(n)/Spin(n) the classifying space of the [[spin group]] and f nf_n the canonical projection is [[spin structure]].

Examples of S 2S^2-(B,f)(B,f)-structures (def. ) include

  1. B 2n=BU(n)=EO(2n)/U(n)B_{2n} = B U(n) = E O(2n)/U(n) the classifying space of the [[unitary group]], and f 2nf_{2n} the canonical projection is [[almost complex structure]] (or rather: [[almost Hermitian structure]]).

  2. B 2n=BSp(2n)=EO(2n)/Sp(2n)B_{2n} = B Sp(2n) = E O(2n)/Sp(2n) the classifying space of the [[symplectic group]], and f 2nf_{2n} the canonical projection is [[almost symplectic structure]].

Examples of S 4S^4-(B,f)(B,f)-structures (def. ) include

  1. B 4n=BU (n)=EO(4n)/U (n)B_{4n} = B U_{\mathbb{H}}(n) = E O(4n)/U_{\mathbb{H}}(n) the classifying space of the [[quaternionic unitary group]], and f 4nf_{4n} the canonical projection is [[almost quaternionic structure]].

Given a [[smooth manifold]] XX of [[dimension]] nn, and given a (B,f)(B,f)-structure as in def. , then a (B,f)(B,f)-structure on the stable normal bundle of the manifold is an [[equivalence class]] of the following structure:

  1. an [[embedding]] i X:X ki_X \; \colon \; X \hookrightarrow \mathbb{R}^k for some kk \in \mathbb{N};

  2. a [[homotopy class]] of a [[lift]] g^\hat g of the classifying map gg of the normal bundle (def. )

    B kn g^ f kn X g BO(kn). \array{ && B_{k-n} \\ &{}^{\mathllap{\hat g}}\nearrow& \downarrow^{\mathrlap{f_{k-n}}} \\ X &\overset{g}{\longrightarrow}& B O(k-n) } \,.

The equivalence relation on such structures is to be that generated by the relation ((i X) 1,g^ 1)((i X) ,g^ 2)((i_{X})_1, \hat g_1) \sim ((i_{X})_,\hat g_2) if

  1. k 2k 1k_2 \geq k_1

  2. the second inclusion factors through the first as

    (i X) 2:X(i X) 1 k 1 k 2 (i_X)_2 \;\colon\; X \overset{(i_X)_1}{\hookrightarrow} \mathbb{R}^{k_1} \hookrightarrow \mathbb{R}^{k_2}
  3. the lift of the classifying map factors accordingly (as homotopy classes)

    g^ 2:Xg^ 1B k 1ng k 1n,k 2nB k 2n. \hat g_2 \;\colon\; X \overset{\hat g_1}{\longrightarrow} B_{k_1-n} \overset{g_{k_1-n, k_2-n}}{\longrightarrow} B_{k_2-n} \,.

Thom spectra

Idea. Given a [[vector bundle]] VV of rank nn over a [[compact topological space]], then its [[one-point compactification]] is equivalently the result of forming the bundle D(V)VD(V) \hookrightarrow V of unit [[n-balls]], and identifying with one single point all the boundary unit [[n-spheres]] S(V)VS(V)\hookrightarrow V. Generally, this construction Th(C)D(V)/S(V)Th(C) \coloneqq D(V)/S(V) is called the [[Thom space]] of VV.

Thom spaces occur notably as codomains for would-be [[left inverses]] of [[embeddings]] of [[manifolds]] XYX \hookrightarrow Y. The [[Pontrjagin-Thom collapse map]] YTh(NX)Y \to Th(N X) of such an embedding is a continuous function going the other way around, but landing not quite in XX but in the [[Thom space]] of the [[normal bundle]] of XX in YY. Composing this further with the classifying map of the [[normal bundle]] lands in the Thom space of the [[universal vector bundle]] over the [[classifying space]] BO(k)B O(k), denoted MO(k)M O(k). In particular in the case that Y=S nY = S^n is an [[n-sphere]] (and every manifold embeds into a large enough nn-sphere, see also at [[Whitney embedding theorem]]), the [[Pontryagin-Thom collapse map]] hence associates with every manifold an element of a [[homotopy group]] of a universal Thom space MO(k)M O(k).

This curious construction turns out to have excellent formal properties: as the dimension ranges, the universal Thom spaces arrange into a [[spectrum]], called the [[Thom spectrum]], and the homotopy groups defined by the Pontryagin-Thom collapse pass along to the [[stable homotopy groups]] of this spectrum.

Moreover, via [[Whitney sum]] of [[vector bundle]] the [[Thom spectrum]] naturally is a [[homotopy commutative ring spectrum]] (def.), and under the Pontryagin-Thom collapse the [[Cartesian product]] of manifolds is compatible with this ring structure.

Literature. (Kochman 96, 1.5, Schwede 12, chapter I, example 1.16)

Thom spaces


Let XX be a [[topological space]] and let VXV \to X be a [[vector bundle]] over XX of [[rank]] nn, which is [[associated bundle|associated]] to an [[orthogonal group|O(n)]]-[[principal bundle]]. Equivalently this means that VXV \to X is the [[pullback]] of the [[universal vector bundle]] E nBO(n)E_n \to B O(n) (def. ) over the [[classifying space]]. Since O(n)O(n) preserves the [[metric]] on n\mathbb{R}^n, by definition, such VV inherits the structure of a [[metric space]]-[[fiber bundle]]. With respect to this structure:

  1. the unit disk bundle D(V)XD(V) \to X is the subbundle of elements of [[norm]] 1\leq 1;

  2. the unit sphere bundle S(V)XS(V)\to X is the subbundle of elements of norm =1= 1;

    S(V)i VD(V)VS(V) \overset{i_V}{\hookrightarrow} D(V) \hookrightarrow V;

  3. the [[Thom space]] Th(V)Th(V) is the [[cofiber]] (formed in [[Top]] (prop.)) of i Vi_V

    Th(V)cofib(i V) Th(V) \coloneqq cofib(i_V)

    canonically regarded as a [[pointed topological space]].

S(V) i V D(V) (po) * Th(V). \array{ S(V) &\overset{i_V}{\longrightarrow}& D(V) \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& Th(V) } \,.

If VXV \to X is a general real vector bundle, then there exists an isomorphism to an O(n)O(n)-[[associated bundle]] and the Thom space of VV is, up to based [[homeomorphism]], that of this orthogonal bundle.


If the [[rank]] of VV is positive, then S(V)S(V) is non-empty and then the Thom space (def. ) is the [[quotient topological space]]

Th(V)D(V)/S(V). Th(V) \simeq D(V)/S(V) \,.

However, in the degenerate case that the [[rank]] of VV vanishes, hence the case that V=X× 0XV = X\times \mathbb{R}^0 \simeq X, then D(V)VXD(V) \simeq V \simeq X, but S(V)=S(V) = \emptyset. Hence now the [[pushout]] defining the cofiber is

i V X (po) * Th(V)X *, \array{ \emptyset &\overset{i_V}{\longrightarrow}& X \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& Th(V) \simeq X_* } \,,

which exhibits Th(V)Th(V) as the [[coproduct]] of XX with the point, hence as XX with a basepoint freely adjoined.

Th(X× 0)=Th(X)X +. Th(X \times \mathbb{R}^0) = Th(X) \simeq X_+ \,.

Let VXV \to X be a [[vector bundle]] over a [[CW-complex]] XX. Then the Thom space Th(V)Th(V) (def. ) is equivalently the [[homotopy cofiber]] (def.) of the inclusion S(V)D(V)S(V) \longrightarrow D(V) of the sphere bundle into the disk bundle.


The Thom space is defined as the ordinary [[cofiber]] of S(V)D(V)S(V)\to D(V). Under the given assumption, this inclusion is a [[relative cell complex]] inclusion, hence a cofibration in the [[classical model structure on topological spaces]] (thm.). Therefore in this case the ordinary cofiber represents the homotopy cofiber (def.).

The equivalence to the following alternative model for this homotopy cofiber is relevant when discussing [[Thom isomorphisms]] and [[orientation in generalized cohomology]]:


Let VXV \to X be a [[vector bundle]] over a [[CW-complex]] XX. Write VXV-X for the complement of its 0-[[section]]. Then the Thom space Th(V)Th(V) (def. ) is [[homotopy equivalence|homotopy equivalent]] to the [[mapping cone]] of the inclusion (VX)V(V-X) \hookrightarrow V (hence to the pair (V,VX)(V,V-X) in the language of [[generalized (Eilenberg-Steenrod) cohomology]]).


The [[mapping cone]] of any map out of a [[CW-complex]] represents the [[homotopy cofiber]] of that map (exmpl.). Moreover, transformation by (weak) homotopy equivalences between morphisms induces a (weak) homotopy equivalence on their homotopy fibers (prop.). But we have such a weak homotopy equivalence, given by contracting away the fibers of the vector bundle:

VX V W cl W cl S(V) D(V). \array{ V-X &\longrightarrow& V \\ {}^{\mathllap{\in W_{cl}}}\downarrow && \downarrow^{\mathrlap{\in W_{cl}}} \\ S(V) &\hookrightarrow& D(V) } \,.

Let V 1,V 2XV_1,V_2 \to X be two real [[vector bundles]]. Then the Thom space (def. ) of the [[direct sum of vector bundles]] V 1V 2XV_1 \oplus V_2 \to X is expressed in terms of the Thom space of the [[pullbacks]] V 2| D(V 1)V_2|_{D(V_1)} and V 2| S(V 1)V_2|_{S(V_1)} of V 2V_2 to the disk/sphere bundle of V 1V_1 as

Th(V 1V 2)Th(V 2| D(V 1))/Th(V 2| S(V 1)). Th(V_1 \oplus V_2) \simeq Th(V_2|_{D(V_1)})/Th(V_2|_{S(V_1)}) \,.

Notice that

  1. D(V 1V 2)D(V 2| IntD(V 1))S(V 1)D(V_1 \oplus V_2) \simeq D(V_2|_{Int D(V_1)}) \cup S(V_1);

  2. S(V 1V 2)S(V 2| IntD(V 1))IntD(V 2| S(V 1))S(V_1 \oplus V_2) \simeq S(V_2|_{Int D(V_1)}) \cup Int D(V_2|_{S(V_1)}).

(Since a point at radius rr in V 1V 2V_1 \oplus V_2 is a point of radius r 1rr_1 \leq r in V 2V_2 and a point of radius r 2r 1 2\sqrt{r^2 - r_1^2} in V 1V_1.)


For VV a [[vector bundle]] then the Thom space (def. ) of nV\mathbb{R}^n \oplus V, the [[direct sum of vector bundles]] with the trivial rank nn vector bundle, is [[homeomorphism|homeomorphic]] to the [[smash product]] of the Thom space of VV with the nn-[[sphere]] (the nn-fold [[reduced suspension]]).

Th( nV)S nTh(V)=Σ nTh(V). Th(\mathbb{R}^n \oplus V) \simeq S^n \wedge Th(V) = \Sigma^n Th(V) \,.

Apply prop. with V 1= nV_1 = \mathbb{R}^n and V 2=VV_2 = V. Since V 1V_1 is a trivial bundle, then

V 2| D(V 1)V 2×D n V_2|_{D(V_1)} \simeq V_2\times D^n

(as a bundle over X×D nX\times D^n) and similarly

V 2| S(V 1)V 2×S n. V_2|_{S(V_1)} \simeq V_2\times S^n \,.

By prop. and remark the Thom space (def. ) of a trivial vector bundle of rank nn is the nn-fold [[suspension]] of the base space

Th(X× n) S nTh(X× 0) S n(X +). \begin{aligned} Th(X \times \mathbb{R}^n) & \simeq S^n \wedge Th(X\times \mathbb{R}^0) \\ & \simeq S^n \wedge (X_+) \end{aligned} \,.

Therefore a general Thom space may be thought of as a “twisted suspension”, with twist encoded by a vector bundle (or rather by its underlying [[spherical fibration]]). See at Thom spectrum – For infinity-module bundles for more on this.

Correspondingly the [[Thom isomorphism]] (prop. below) for a given Thom space is a twisted version of the [[suspension isomorphism]] (above).


For V 1X 1V_1 \to X_1 and V 2X 2V_2 \to X_2 to vector bundles, let V 1V 2X 1×X 2V_1 \boxtimes V_2 \to X_1 \times X_2 be the [[direct sum of vector bundles]] of their [[pullbacks]] to X 1×X 2X_1 \times X_2. The corresponding Thom space (def. ) is the [[smash product]] of the individual Thom spaces:

Th(V 1V 2)Th(V 1)Th(V 2). Th(V_1 \boxtimes V_2) \simeq Th(V_1) \wedge Th(V_2) \,.

Given a [[vector bundle]] VXV \to X of [[rank]] nn, then the [[reduced cohomology|reduced]] [[ordinary cohomology]] of its [[Thom space]] Th(V)Th(V) (def. ) vanishes in degrees <n\lt n:

H˜ <n(Th(V))H <n(D(V),S(V))0. \tilde H^{\bullet \lt n}(Th(V)) \simeq H^{\bullet \lt n}(D(V), S(V)) \simeq 0 \,.

Consider the [[long exact sequence]] of [[relative cohomology]] (from above)

H 1(D(V))i *H 1(S(V))H (D(V),S(V))H (D(V))i *H (S(V)). \cdots \to H^{\bullet-1}(D(V)) \overset{i^\ast}{\longrightarrow} H^{\bullet-1}(S(V)) \longrightarrow H^\bullet(D(V), S(V)) \longrightarrow H^{\bullet}(D(V)) \overset{i^\ast}{\longrightarrow} H^{\bullet}(S(V)) \to \cdots \,.

Since the cohomology in degree kk only depends on the kk-skeleton, and since for k<nk \lt n the kk-skeleton of S(V)S(V) equals that of XX, and since D(V)D(V) is even homotopy equivalent to XX, the morhism i *i^\ast is an isomorphism in degrees lower than nn. Hence by exactness of the sequence it follows that H <n(D(V),S(V))=0H^{\bullet \lt n}(D(V),S(V)) = 0.

Universal Thom spectra MGM G


For each nn \in \mathbb{N} the [[pullback]] of the [[rank]]-(n+1)(n+1) [[universal vector bundle]] to the [[classifying space]] of rank nn vector bundles is the [[direct sum of vector bundles]] of the rank nn universal vector bundle with the trivial rank-1 bundle: there is a [[pullback]] [[diagram]] of [[topological spaces]] of the form

(EO(n)×O(n) n) EO(n+1)×O(n+1) n+1 (pb) BO(n) BO(n+1), \array{ \mathbb{R}\oplus (E O(n)\underset{O(n)}{\times} \mathbb{R}^n) &\longrightarrow& E O(n+1) \underset{O(n+1)}{\times} \mathbb{R}^{n+1} \\ \downarrow &(pb)& \downarrow \\ B O(n) &\longrightarrow& B O(n+1) } \,,

where the bottom morphism is the canonical one (def.).

(e.g. Kochmann 96, p. 25)


For each kk \in \mathbb{N}, knk \geq n there is such a pullback of the canonical vector bundles over [[Grassmannians]]

{V n k,vV n,v n+1} {V n+1 k+1,vV n+1} Gr n( k) Gr n+1( k+1) \array{ \left\{ {V_{n}\subset \mathbb{R}^k, } \atop {v \in V_n, v_{n+1} \in \mathbb{R}} \right\} &\longrightarrow& \left\{ {V_{n+1} \subset \mathbb{R}^{k+1}}, \atop v \in V_{n+1} \right\} \\ \downarrow && \downarrow \\ Gr_n(\mathbb{R}^k) &\longrightarrow& Gr_{n+1}(\mathbb{R}^{k+1}) }

where the bottom morphism is the canonical inclusion (def.).

Now we claim that taking the [[colimit]] in each of the four corners of this system of pullback diagrams yields again a pullback diagram, and this proves the claim.

To see this, remember that we work in the category Top cgTop_{cg} of [[compactly generated topological spaces]] (def.). By their nature, we may test the [[universal property]] of a would-be [[pullback]] space already by mapping [[compact topological spaces]] into it. Now observe that all the inclusion maps in the four corners of this system of diagrams are [[relative cell complex]] inclusions, by prop. . Together this implies (via this lemma) that we may test the universal property of the colimiting square at finite stages. And so this implies the claim by the above fact that at each finite stage there is a pullback diagram.


The universal real [[Thom spectrum]] MOM O is the [[spectrum]], which is represented by the [[sequential prespectrum]] (def.) whose nnth component space is the [[Thom space]] (def. )

(MO) nTh(EO(n)×O(n) n) (M O)_n \coloneqq Th(E O(n)\underset{O(n)}{\times}\mathbb{R}^n)

of the rank-nn [[universal vector bundle]], and whose structure maps are the image under the [[Thom space]] functor Th()Th(-) of the top morphisms in prop. , via the homeomorphisms of prop. :

σ n:Σ(MO) nTh((EO(n)×O(n) n))Th(EO(n+1)×O(n+1) n+1)=(MO) n+1. \sigma_n \;\colon\; \Sigma (M O)_n \simeq Th(\mathbb{R}\oplus (E O(n)\underset{O(n)}{\times} \mathbb{R}^n)) \stackrel{}{\longrightarrow} Th(E O(n+1) \underset{O(n+1)}{\times} \mathbb{R}^{n+1}) = (M O)_{n+1} \,.

More generally, there are universal Thom spectra associated with any other tangent structure (“[[(B,f)-structure]]”), notably for the orthogonal group replaced by the [[special orthogonal groups]] SO(n)SO(n), or the [[spin groups]] Spin(n)Spin(n), or the [[string 2-group]] String(n)String(n), or the [[fivebrane 6-group]] Fivebrane(n)Fivebrane(n),…, or any level in the [[Whitehead tower]] of O(n)O(n). To any of these groups there corresponds a Thom spectrum (denoted, respectively, MSOM SO, [[MSpin]], MStringM String, MFivebraneM Fivebrane, etc.), which is in turn related to oriented cobordism, spin cobordism, string cobordism, et cetera.:


Given a [[(B,f)-structure]] \mathcal{B} (def. ), write V n V^\mathcal{B}_n for the [[pullback]] of the [[universal vector bundle]] (def. ) to the corresponding space of the (B,f)(B,f)-structure and with

V VO(n)×O(n) n (pb) B n f n BO(n) \array{ V^{\mathcal{B}} &\overset{}{\longrightarrow}& V O(n) \underset{O(n)}{\times} \mathbb{R}^n \\ \downarrow &(pb)& \downarrow \\ B_n &\underset{f_n}{\longrightarrow}& B O(n) }

and we write e n 1,n 2e_{n_1,n_2} for the maps of total space of vector bundles over the g n 1,n 2g_{n_1,n_2}:

V n 1 e n 1,n 2 V n 2 (pb) B n 1 g n 1,n 2 B n 2. \array{ V^{\mathcal{B}}_{n_1} &\overset{e_{n_1,n_2}}{\longrightarrow}& V^{\mathcal{B}}_{n_2} \\ \downarrow &(pb)& \downarrow \\ B_{n_1} &\underset{g_{n_1,n_2}}{\longrightarrow}& B_{n_2} } \,.

Observe that the analog of prop. still holds:


Given a [[(B,f)-structure]] \mathcal{B} (def. ), then the pullback of its rank-(n+1)(n+1) vector bundle V n+1 V^{\mathcal{B}}_{n+1} (def. ) along the map g n,n+1:B nB n+1g_{n,n+1} \colon B_n \to B_{n+1} is the [[direct sum of vector bundles]] of the rank-nn bundle V n V^{\mathcal{B}}_n with the trivial rank-1-bundle: there is a pullback square

V n e n,n+1 V n+1 (pb) B n g n,n+1 B n+1. \array{ \mathbb{R} \oplus V^{\mathcal{B}}_n &\overset{e_{n,n+1}}{\longrightarrow}& V^{\mathcal{B}}_{n+1} \\ \downarrow &(pb)& \downarrow \\ B_n &\underset{g_{n,n+1}}{\longrightarrow}& B_{n+1} } \,.

Unwinding the definitions, the pullback in question is

(g n,n+1) *V n+1 =(g n,n+1) *f n+1 *(EO(n+1)×O(n+1) n+1) (g n,n+1f n+1) *(EO(n+1)×O(n+1) n+1) (f ni n) *(EO(n+1)×O(n+1) n+1) f n *i n *(EO(n+1)×O(n+1) n+1) f n *((EO(n)×O(n) n)) V n, \begin{aligned} (g_{n,n+1})^\ast V^{\mathcal{B}}_{n+1} & = (g_{n,n+1})^\ast f_{n+1}^\ast (E O(n+1)\underset{O(n+1)}{\times} \mathbb{R}^{n+1}) \\ & \simeq (g_{n,n+1} \circ f_{n+1})^\ast (E O(n+1)\underset{O(n+1)}{\times} \mathbb{R}^{n+1}) \\ & \simeq ( f_n \circ i_n )^\ast (E O(n+1)\underset{O(n+1)}{\times} \mathbb{R}^{n+1}) \\ & \simeq f_n^\ast i_n^\ast (E O(n+1)\underset{O(n+1)}{\times} \mathbb{R}^{n+1}) \\ & \simeq f_n^\ast (\mathbb{R} \oplus (E O(n)\underset{O(n)}{\times} \mathbb{R}^{n})) \\ &\simeq \mathbb{R} \oplus V^{\mathcal{B}_n} \,, \end{aligned}

where the second but last step is due to prop. .


Given a [[(B,f)-structure]] \mathcal{B} (def. ), its universal Thom spectrum MM \mathcal{B} is, as a [[sequential prespectrum]], given by component spaces being the [[Thom spaces]] (def. ) of the \mathcal{B}-associated vector bundles of def.

(M) nTh(V n ) (M \mathcal{B})_n \coloneqq Th(V^{\mathcal{B}}_n)

and with structure maps given via prop. by the top maps in prop. :

σ n:Σ(M) n=ΣTh(V n )Th(V n )Th(e n,n+1)Th(V n+1 )=(M) n+1. \sigma_n \;\colon\; \Sigma (M \mathcal{B})_n = \Sigma Th(V^{\mathcal{E}}_n) \simeq Th(\mathbb{R}\oplus V^{\mathcal{E}}_n) \overset{Th(e_{n,n+1})}{\longrightarrow} Th(V^{\mathcal{B}}_{n+1}) = (M \mathcal{B})_{n+1} \,.

Similarly for an S k(B,f)S^k-(B,f)-structure indexed on every kkth natural number (such as [[almost complex structure]], [[almost quaternionic structure]], example ), there is the corresponding Thom spectrum as a sequential S kS^k spectrum (def.).

If B n=BG nB_n = B G_n for some natural system of groups G nO(n)G_n \to O(n), then one usually writes MGM G for MM \mathcal{B}. For instance MSOM SO, [[MSpin]], [[MU]], [[MSp]] etc.

If the (B,f)(B,f)-structure is multiplicative (def. ), then the Thom spectrum MM \mathcal{B} canonical becomes a [[ring spectrum]] (for more on this see [[Introduction to Stable homotopy theory – 1-2|Part 1-2]] the section on orthogonal Thom spectra ): the multiplication maps B n 1×B n 2B n 1+n 2B_{n_1} \times B_{n_2}\to B_{n_1 + n_2} are covered by maps of vector bundles

V n 1 V n 2 V n 1+n 2 V^{\mathcal{B}}_{n_1} \boxtimes V^{\mathcal{B}}_{n_2} \longrightarrow V^{\mathcal{B}}_{n_1 + n_2}

and under forming [[Thom spaces]] this yields (via prop. ) maps

(M) n 1(M) n 2(M) n 1+n 2 (M \mathcal{B})_{n_1} \wedge (M \mathcal{B})_{n_2} \longrightarrow (M \mathcal{B})_{n_1 + n_2}

which are [[associativity|associative]] by the associativity condition in a multiplicative (B,f)(B,f)-structure. The unit is

(M) 0=Th(V 0 )Th(*)S 0, (M \mathcal{B})_0 = Th(V^{\mathcal{B}}_0) \simeq Th(\ast) \simeq S^0 \,,

by remark .


The universal [[Thom spectrum]] (def. ) for [[framing]] structure (exmpl.) is equivalently the [[sphere spectrum]] (def.)

M1𝕊. M 1 \simeq \mathbb{S} \,.

Because in this case B n*B_n \simeq \ast and so E n nE^{\mathcal{B}}_n \simeq \mathbb{R}^n, whence Th(E n )S nTh(E^{\mathcal{B}}_n) \simeq S^n.

Pontrjagin-Thom construction


For XX a [[smooth manifold]] and i:X ki \colon X \hookrightarrow \mathbb{R}^k an [[embedding]], then a [[tubular neighbourhood]] of XX is a subset of the form

τ iX{x k|d(x,i(X))<ϵ} \tau_i X \coloneqq \left\{ x \in \mathbb{R}^k \;\vert\; d(x,i(X)) \lt \epsilon \right\}

for some ϵ\epsilon \in \mathbb{R}, ϵ>0\epsilon \gt 0, small enough such that the map

N iXτ iX N_i X \longrightarrow \tau_i X

from the [[normal bundle]] (def. ) given by

(i(x),v)(i(x),ϵ(1e |v|)v) (i(x),v) \mapsto (i(x), \epsilon (1-e^{- {\vert v\vert}}) v )

is a [[diffeomorphism]].


([[tubular neighbourhood theorem]])

For every [[embedding]] of [[smooth manifolds]], there exists a [[tubular neighbourhood]] according to def. .


Given an embedding i:X ki \colon X \hookrightarrow \mathbb{R}^k with a tubuluar neighbourhood τ iXhookrigtharrow k\tau_i X \hookrigtharrow \mathbb{R}^k (def. ) then by construction:

  1. the [[Thom space]] (def. ) of the [[normal bundle]] (def. ) is [[homeomorphism|homeomorphic]] to the [[quotient topological space]] of the [[topological closure]] of the tubular neighbourhood by its [[boundary]]:

    Th(N i(X))τ i(X)¯/τ i(X)¯Th(N_i(X)) \simeq \overline{ \tau_i(X)}/\partial \overline{\tau_i(X)};

  2. there exists a continous function

    kτ i(X)¯/τ i(X)¯ \mathbb{R}^k \longrightarrow \overline{ \tau_i(X)}/\partial \overline{\tau_i(X)}

    which is the identity on τ i(X) k\tau_i(X)\subset \mathbb{R}^k and is constant on the basepoint of the quotient on all other points.


For XX a [[smooth manifold]] of [[dimension]] nn and for i:X ki \colon X \hookrightarrow \mathbb{R}^k an [[embedding]], then the [[Pontrjagin-Thom collapse map]] is, for any choice of [[tubular neighbourhood]] τ i(X) k\tau_i(X)\subset \mathbb{R}^k (def. ) the composite map of [[pointed topological spaces]]

S k( k) *τ i(X)¯/τ i(X)¯Th(N iX) S^k \overset{\simeq}{\to} (\mathbb{R}^k)^\ast \longrightarrow \overline{ \tau_i(X)}/\partial \overline{\tau_i(X)} \overset{\simeq}{\to} Th(N_i X)

where the first map identifies the [[n-sphere|k-sphere]] as the [[one-point compactification]] of k\mathbb{R}^k; and where the second and third maps are those of remark .

The Pontrjagin-Thom construction is the further composite

ξ i:S kTh(N iX)Th(e i)Th(EO(kn)×O(kn) kn)(MO) kn \xi_i \;\colon\; S^k \longrightarrow Th(N_i X) \overset{Th(e_i)}{\longrightarrow} Th( E O(k-n) \underset{O(k-n)}{\times} \mathbb{R}^{k-n} ) \simeq (M O)_{k-n}

with the image under the [[Thom space]] construction of the morphism of vector bundles

ν e i EO(kn)×O(kn) kn (pb) X g i BO(kn) \array{ \nu &\overset{e_i}{\longrightarrow}& E O(k-n)\underset{O(k-n)}{\times} \mathbb{R}^{k-n} \\ \downarrow &(pb)& \downarrow \\ X &\underset{g_i}{\longrightarrow}& B O(k-n) }

induced by the classifying map g ig_i of the normal bundle (def. ).

This defines an element

[S n+(kn)ξ i(MO) kn]π nMO [S^{n+(k-n)} \overset{\xi_i}{\to} (M O)_{k-n}] \in \pi_{n} M O

in the nnth [[stable homotopy group]] (def.) of the [[Thom spectrum]] MOM O (def. ).

More generally, for XX a smooth manifold with normal [[(B,f)-structure]] (X,i,g^ i)(X,i,\hat g_i) according to def. , then its Pontrjagin-Thom construction is the composite

ξ i:S kTh(N iX)Th(e^ i)Th(V kn )(M) kn \xi_i \;\colon\; S^k \longrightarrow Th(N_i X) \overset{Th(\hat e_i)}{\longrightarrow} Th( V^{\mathcal{B}}_{k-n} ) \simeq (M \mathcal{B})_{k-n}


ν e^ i V kn (pb) X g^ i BO(kn). \array{ \nu &\overset{\hat e_i}{\longrightarrow}& V^{\mathcal{B}}_{k-n} \\ \downarrow &(pb)& \downarrow \\ X &\underset{\hat g_i}{\longrightarrow}& B O(k-n) } \,.

The [[Pontrjagin-Thom construction]] (def. ) respects the equivalence classes entering the definition of manifolds with stable normal \mathcal{B}-structure (def. ) hence descends to a [[function]] (of [[sets]])

ξ:{n-manifoldswithstablenormal-structure}π n(M). \xi \;\colon\; \left\{ {n\text{-}manifolds\;with\;stable} \atop {normal\;\mathcal{B}\text{-}structure} \right\} \longrightarrow \pi_n(M\mathcal{B}) \,.

It is clear that the homotopies of classifying maps of \mathcal{B}-structures that are devided out in def. map to homotopies of representatives of stable homotopy groups. What needs to be shown is that the construction respects the enlargement of the embedding spaces.

Given a embedded manifold Xi k 1X \overset{i}{\hookrightarrow}\mathbb{R}^{k_1} with normal \mathcal{B}-structure

B k 1n g^ i f kn X g i BO(k 1n) \array{ && B_{k_1-n} \\ & {}^{\mathllap{\hat g_i}}\nearrow & \downarrow^{\mathrlap{f_{k-n}}} \\ X &\underset{g_i}{\longrightarrow}& B O(k_1-n) }


α:S n+(k 1n)Th(E k 1n) \alpha \;\colon\; S^{n+(k_1-n)} \overset{}{\longrightarrow} Th(E^{\mathcal{B}_{k_1-n}})

for its image under the [[Pontrjagin-Thom construction]] (def. ). Now given k 2k_2 \in \mathbb{N}, consider the induced embedding Xi k 1 k 1+k 2X \overset{i}{\hookrightarrow} \mathbb{R}^{k_1}\hookrightarrow \mathbb{R}^{k_1 + k_2} with normal \mathcal{B}-structure given by the composite

B k 1n g k 1n,k 1+k 2n B k 1+k 2n g^ i f k 1n×f k 2 f k 1+k 2n X g i BO(k 1n) BO(k 1+k 2n). \array{ && B_{k_1-n} &\overset{g_{k_1-n, k_1+ k_2 -n}}{\longrightarrow}& B_{k_1 + k_2-n} \\ & {}^{\mathllap{\hat g_i}}\nearrow & \downarrow^{\mathrlap{f_{k_1 - n} \times f_{k_2}}} && \downarrow^{\mathrlap{f_{k_1 + k_2-n}}} \\ X &\underset{g_i}{\longrightarrow}& B O(k_1-n) &\longrightarrow& B O(k_1 + k_2-n) } \,.

By prop. and using the [[pasting law]] for [[pullbacks]], the classifying map g^ i\hat g'_i for the enlarged normal bundle sits in a diagram of the form

(ν i k 2) (e^ iid) (V k 1n k 2) e k 1n,k 1+k 2n V k 1+k 2n (pb) (pb) X g^ i B k 1n g k 1n,k 1+k 2n B k 1+k 2n. \array{ (\nu_i \oplus \mathbb{R}^{k_2}) &\overset{(\hat e_i \oplus id)}{\longrightarrow}& (V^{\mathcal{B}}_{k_1-n} \oplus \mathbb{R}^{k_2}) &\overset{e_{k_1-n,k_1+k_2-n}}{\longrightarrow}& V^{\mathcal{B}}_{k_1 + k_2 - n} \\ \downarrow &(pb)& \downarrow &(pb)& \downarrow \\ X &\underset{\hat g_i}{\longrightarrow}& B_{k_1-n} &\underset{g_{k_1-n, k_1 + k_2 - n}}{\longrightarrow}& B_{k_1 +k_2 - n} } \,.

Hence the Pontrjagin-Thom construction for the enlarged embedding space is (using prop. ) the composite

α k 2:S n+(k 1+k 2n)Th( k 2)S n+(k 1n)Th( k 2)Th(ν i)Th(id)Th(e^ i)Th( k 2)Th(E k 1n ))Th(e k 1n,k 1+k 2n)Th(V k 1+k 2n ). \alpha_{k_2} \;\colon\; S^{n + (k_1+ k_2 - n)} \simeq Th(\mathbb{R}^{k_2}) \wedge S^{n + (k_1 - n)} \overset{}{\longrightarrow} Th(\mathbb{R}^{k_2}) \wedge Th(\nu_i) \overset{Th(id)\wedge Th(\hat e_i)}{\longrightarrow} Th(\mathbb{R}^{k_2}) \wedge Th(E^{\mathcal{B}}_{k_1-n})) \overset{Th(e_{k_1-n, k_1 + k_2 - n})}{\longrightarrow} Th(V^{\mathcal{B}}_{k_1 + k_2 - n}) \,.

The composite of the first two morphisms here is S k kαS^{k_k}\wedge \alpha, while last morphism Th(e^ k 1n,k 1+k 2n)Th(\hat e_{k_1-n,k_1+k_2-n}) is the structure map in the Thom spectrum (by def. ):

α k 2:S k 2S n+(k 1n)S k 2αS k 2Th(E k 1+k 2n )σ k 1n,k 1+k 2n MTh(V k 1+k 2n ) \alpha_{k_2} \;\colon\; S^{k_2} \wedge S^{n + (k_1 - n)} \overset{S^{k_2} \wedge \alpha}{\longrightarrow} S^{k_2} \wedge Th(E^{\mathcal{B}}_{k_1 + k_2 - n}) \overset{\sigma^{M \mathcal{B}}_{k_1-n,k_1 + k_2 - n} }{\longrightarrow} Th(V^{\mathcal{B}}_{k_1+k_2 - n})

This manifestly identifies α k 2\alpha_{k_2} as being the image of α\alpha under the component map in the sequential colimit that defines the stable homotopy groups (def.). Therefore α\alpha and α k 2\alpha_{k_2}, for all k 2k_2 \in \mathbb{N}, represent the same element in π (M)\pi_{\bullet}(M \mathcal{B}).

Bordism and Thom’s theorem

Idea. By the [[Pontryagin-Thom collapse]] construction above, there is an assignment

nManifoldsπ n(MO) n Manifolds \longrightarrow \pi_n(M O)

which sends [[disjoint union]] and [[Cartesian product]] of manifolds to sum and product in the [[ring]] of [[stable homotopy groups]] of the [[Thom spectrum]]. One finds then that two manifolds map to the same element in the [[stable homotopy groups]] π (MO)\pi_\bullet(M O) of the universal [[Thom spectrum]] precisely if they are connected by a [[bordism]]. The [[bordism]]-classes Ω O\Omega_\bullet^O of manifolds form a [[commutative ring]] under [[disjoint union]] and [[Cartesian product]], called the [[bordism ring]], and Pontrjagin-Thom collapse produces a ring [[homomorphism]]

Ω Oπ (MO). \Omega_\bullet^O \longrightarrow \pi_\bullet(M O) \,.

[[Thom’s theorem]] states that this homomorphism is an [[isomorphism]].

More generally, for \mathcal{B} a multiplicative [[(B,f)-structure]], def. , there is such an identification

Ω π (M) \Omega_\bullet^{\mathcal{B}} \simeq \pi_\bullet(M \mathcal{B})

between the ring of \mathcal{B}-cobordism classes of manifolds with \mathcal{B}-structure and the [[stable homotopy groups]] of the universal \mathcal{B}-[[Thom spectrum]].

Literature. (Kochman 96, 1.5)


Throughout, let \mathcal{B} be a multiplicative [[(B,f)-structure]] (def. ).


Write I[0,1]I \coloneqq [0,1] for the standard interval, regarded as a [[smooth manifold]] [[manifold with boundary|with boundary]]. For c +c \in \mathbb{R}_+ Consider its embedding

e:I 0 e \;\colon\; I \hookrightarrow \mathbb{R}\oplus \mathbb{R}_{\geq 0}

as the arc

e:tcos(πt)e 1+sin(πt)e 2, e \;\colon\; t \mapsto \cos(\pi t) \cdot e_1 + \sin(\pi t) \cdot e_2 \,,

where (e 1,e 2)(e_1, e_2) denotes the canonical [[linear basis]] of 2\mathbb{R}^2, and equipped with the structure of a manifold with normal [[framing]] structure (example ) by equipping it with the canonical framing

fr:tcos(πt)e 1+sin(πt)e 2 fr \;\colon\; t \mapsto \cos(\pi t) \cdot e_1 + \sin(\pi t) \cdot e_2

of its [[normal bundle]].

Let now \mathcal{B} be a [[(B,f)-structure]] (def. ). Then for Xi kX \overset{i}{\hookrightarrow}\mathbb{R}^k any embedded manifold with \mathcal{B}-structure g^:XB kn\hat g \colon X \to B_{k-n} on its [[normal bundle]] (def. ), define its negative or orientation reversal (X,i,g^)-(X,i,\hat g) of (X,i,g^)(X,i, \hat g) to be the restriction of the structured manifold

(X×I(i,e) k+2,g^×fr) (X \times I \overset{(i,e)}{\hookrightarrow} \mathbb{R}^{k+2}, \hat g \times fr)

to t=1t = 1.


Two closed manifolds of [[dimension]] nn equipped with normal \mathcal{B}-structure (X 1,i 1,g^ 1)(X_1, i_1, \hat g_1) and (X 2,i 2,g^ 2)(X_2,i_2,\hat g_2) (def.) are called bordant if there exists a [[manifold with boundary]] WW of dimension n+1n+1 equipped with \mathcal{B}-strcuture (W,i W,g^ W)(W,i_W, \hat g_W) if its [[boundary]] with \mathcal{B}-structure restricted to that boundary is the [[disjoint union]] of X 1X_1 with the negative of X 2X_2, according to def.

(W,i W,g^ W)(X 1,i 1,g^ 1)(X 2,i 2,g^ 2). \partial(W,i_W,\hat g_W) \simeq (X_1, i_1, \hat g_1) \sqcup -(X_2, i_2, \hat g_2) \,.

The [[relation]] of \mathcal{B}-[[bordism]] (def. ) is an [[equivalence relation]].

Write Ω \Omega^\mathcal{B}_{\bullet} for the \mathbb{N}-graded set of \mathcal{B}-bordism classes of \mathcal{B}-manifolds.


Under [[disjoint union]] of manifolds, then the set of \mathcal{B}-bordism equivalence classes of def. becomes an \mathbb{Z}-graded [[abelian group]]

Ω Ab \Omega^{\mathcal{B}}_\bullet \in Ab^{\mathbb{Z}}

(that happens to be concentrated in non-negative degrees). This is called the \mathcal{B}-bordism group.

Moreover, if the [[(B,f)-structure]] \mathcal{B} is multiplicative (def. ), then [[Cartesian product]] of manifolds followed by the multiplicative composition operation of \mathcal{B}-structures makes the \mathcal{B}-bordism ring into a [[commutative ring]], called the \mathcal{B}-bordism ring.

Ω CRing . \Omega^{\mathcal{B}}_\bullet \in CRing^{\mathbb{Z}} \,.

e.g. (Kochmann 96, prop. 1.5.3)

Thom’s theorem

Recall that the [[Pontrjagin-Thom construction]] (def. ) associates to an embbeded manifold (X,i,g^)(X,i,\hat g) with normal \mathcal{B}-structure (def. ) an element in the [[stable homotopy group]] π dim(X)(M)\pi_{dim(X)}(M \mathcal{B}) of the universal \mathcal{B}-[[Thom spectrum]] in degree the dimension of that manifold.


For \mathcal{B} be a multiplicative [[(B,f)-structure]] (def. ), the \mathcal{B}-[[Pontrjagin-Thom construction]] (def. ) is compatible with all the relations involved to yield a graded [[ring]] [[homomorphism]]

ξ:Ω π (M) \xi \;\colon\; \Omega^{\mathcal{B}}_\bullet \longrightarrow \pi_\bullet(M \mathcal{B})

from the \mathcal{B}-[[bordism ring]] (def. ) to the [[stable homotopy groups]] of the universal \mathcal{B}-[[Thom spectrum]] equipped with the ring structure induced from the canonical [[ring spectrum]] structure (def. ).


By prop. the underlying function of sets is well-defined before dividing out the bordism relation (def. ). To descend this further to a function out of the set underlying the bordism ring, we need to see that the Pontrjagin-Thom construction respects the bordism relation. But the definition of bordism is just so as to exhibit under ξ\xi a [[left homotopy]] of representatives of homotopy groups.

Next we need to show that it is

  1. a group homomorphism;

  2. a ring homomorphism.

Regarding the first point:

The element 0 in the [[cobordism group]] is represented by the empty manifold. It is clear that the Pontrjagin-Thom construction takes this to the trivial stable homotopy now.

Given two nn-manifolds with \mathcal{B}-structure, we may consider an embedding of their [[disjoint union]] into some k\mathbb{R}^{k} such that the [[tubular neighbourhoods]] of the two direct summands do not intersect. There is then a map from two copies of the [[n-cube|k-cube]], glued at one face

k k1 k k \Box^k \underset{\Box^{k-1}}{\sqcup} \Box^k \longrightarrow \mathbb{R}^k

such that the first manifold with its tubular neighbourhood sits inside the image of the first cube, while the second manifold with its tubular neighbourhood sits indide the second cube. After applying the Pontryagin-Thom construction to this setup, each cube separately maps to the image under ξ\xi of the respective manifold, while the union of the two cubes manifestly maps to the sum of the resulting elements of homotopy groups, by the very definition of the group operation in the homotopy groups (def.). This shows that ξ\xi is a group homomorphism.

Regarding the second point:

The element 1 in the [[cobordism ring]] is represented by the manifold which is the point. Without restriction we may consoder this as embedded into 0\mathbb{R}^0, by the identity map. The corresponding [[normal bundle]] is of [[rank]] 0 and hence (by remark ) its [[Thom space]] is S 0S^0, the [[0-sphere]]. Also V 0 V^{\mathcal{B}}_0 is the rank-0 vector bundle over the point, and hence (M) 0S 0(M \mathcal{B})_0 \simeq S^0 (by def. ) and so ξ(*):(S 0S 0)\xi(\ast) \colon (S^0 \overset{\simeq}{\to} S^0) indeed represents the unit element in π (M)\pi_\bullet(M\mathcal{B}).

Finally regarding respect for the ring product structure: for two manifolds with stable normal \mathcal{B}-structure, represented by embeddings into k i\mathbb{R}^{k_i}, then the normal bundle of the embedding of their [[Cartesian product]] is the [[direct sum of vector bundles]] of the separate normal bundles bulled back to the product manifold. In the notation of prop. there is a diagram of the form

ν 1ν 2 e^ 1e^ 2 V n 1 V n 2 κ n 1,n 2 V n 1+n 2 (pb) (pb) X 1×X 2 g^ 1×g^ 2 B k 1n 1×B k 2n 2 μ k 1n 1,k 2n 2 B k 1+k 2n 1n 2. \array{ \nu_1 \boxtimes \nu_2 &\overset{\hat e_1 \boxtimes \hat e_2}{\longrightarrow}& V^{\mathcal{B}}_{n_1} \boxtimes V^{\mathcal{B}}_{n_2} &\overset{\kappa_{n_1,n_2}}{\longrightarrow}& V^{\mathcal{B}}_{n_1 + n_2} \\ \downarrow &(pb)& \downarrow &(pb)& \downarrow \\ X_1 \times X_2 &\underset{\hat g_1 \times \hat g_2}{\longrightarrow}& B_{k_1-n_1} \times B_{k_2-n_2} &\underset{\mu_{k_1-n_1,k_2-n_2}}{\longrightarrow}& B_{k_1 + k_2 - n_1 - n_2} } \,.

To the Pontrjagin-Thom construction of the product manifold is by definition the top composite in the diagram

S n 1+n 2+(k 1+k 2n 1n 2) Th(ν 1ν 2) Th(e^ 1e^ 2) Th(V k 1n 1 V k 2n 2 ) κ k 1n 1,k 2n 2 Th(V k 1+k 2n 1n 2 ) = S n 1+(k 1n 1)S n 2+(k 2n 2) Th(ν 1)Th(ν 2) Th(e^ 1)Th(e^ 2) Th(V 1 )Th(V 2 ) κ k 1n 1,k 2n 2 Th(V k 1+k 2n 1n 2 ), \array{ S^{n_1 +n_2 + (k_1 + k_2 - n_1 - n_2)} &\overset{}{\longrightarrow}& Th(\nu_1 \boxtimes \nu_2) &\overset{Th(\hat e_1 \boxtimes \hat e_2)}{\longrightarrow}& Th(V^{\mathcal{B}}_{k_1-n_1} \boxtimes V^{\mathcal{B}}_{k_2-n_2}) &\overset{\kappa_{k_1-n_1, k_2-n_2}}{\longrightarrow}& Th(V^{\mathcal{B}}_{k_1 + k_2 - n_1 - n_2}) \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{=}} \\ S^{n_1 + (k_1 - n_1)} \wedge S^{n_2 + (k_2 - n_2)} &\overset{}{\longrightarrow}& Th(\nu_1) \wedge Th(\nu_2) &\overset{Th(\hat e_1)\wedge Th(\hat e_2)}{\longrightarrow}& Th(V^{\mathcal{B}}_1) \wedge Th(V^{\mathcal{B}}_2) &\overset{\kappa_{k_1-n_1, k_2-n_2}}{\longrightarrow}& Th(V^{\mathcal{B}}_{k_1 + k_2 - n_1 - n_2}) } \,,

which hence is equivalently the bottom composite, which in turn manifestly represents the product of the separate PT constructions in π (M)\pi_\bullet(M\mathcal{B}).


The ring homomorphsim in lemma is an [[isomorphism]].

Due to (Thom 54, Pontrjagin 55). See for instance (Kochmann 96, theorem 1.5.10).

Proof idea

Observe that given the result α:S n+(kn)Th(V kn)\alpha \colon S^{n+(k-n)} \to Th(V_{k-n}) of the Pontrjagin-Thom construction map, the original manifold Xi kX \overset{i}{\hookrightarrow} \mathbb{R}^k may be recovered as this [[pullback]]:

X i S n+(kn) g i (pb) α BO(kn) Th(V kn BO). \array{ X &\overset{i}{\longrightarrow}& S^{n + (k-n)} \\ {}^{\mathllap{g_i}}\downarrow &(pb)& \downarrow^{\mathrlap{\alpha}} \\ B O(k-n) &\longrightarrow& Th(V^{B O}_{k-n}) } \,.

To see this more explicitly, break it up into pieces:

X X + S n+(kn) (pb) (pb) X X +Th(X) Th(0) Th(ν i) (pb) (pb) B kn (B kn) +Th(B kn) Th(0) Th(V kn ) (pb) (pb) BO(kn) (BO(kn)) +Th(BO(kn)) Th(V kn BO). \array{ X &\longrightarrow& X_+ &\hookrightarrow& S^{n + (k-n)} \\ \downarrow &(pb)& \downarrow &(pb)& \downarrow \\ X &\longrightarrow& X_+ \simeq Th(X) &\overset{Th(0)}{\longrightarrow}& Th(\nu_i) \\ \downarrow &(pb)& \downarrow &(pb)& \downarrow \\ B_{k-n} &\longrightarrow& (B_{k-n})_+ \simeq Th(B_{k-n}) &\underset{Th(0)}{\longrightarrow}& Th(V^{\mathcal{B}}_{k-n}) \\ \downarrow &(pb)& \downarrow &(pb)& \downarrow \\ B O(k-n) &\longrightarrow& (B O(k-n))_+ \simeq Th(B O(k-n)) &\longrightarrow& Th(V^{B O}_{k-n}) } \,.

Moreover, since the [[n-spheres]] are [[compact topological spaces]], and since the [[classifying space]] BO(n)B O(n), and hence its universal Thom space, is a [[sequential colimit]] over [[relative cell complex]] inclusions, the right vertical map factors through some finite stage (by this lemma), the manifold XX is equivalently recovered as a pullback of the form

X S n+(kn) g i (pb) Gr kn( k) i Th(V kn( k)×O(kn) kn). \array{ X &\longrightarrow& S^{n + (k-n)} \\ {}^{\mathllap{g_i}}\downarrow &(pb)& \downarrow \\ Gr_{k-n}(\mathbb{R}^k) &\overset{i}{\longrightarrow}& Th( V_{k-n}(\mathbb{R}^k) \underset{O(k-n)}{\times} \mathbb{R}^{k-n}) } \,.

(Recall that V kn V^{\mathcal{B}}_{k-n} is our notation for the [[universal vector bundle]] with \mathcal{B}-structure, while V kn( k)V_{k-n}(\mathbb{R}^k) denotes a [[Stiefel manifold]].)

The idea of the proof now is to use this property as the blueprint of the construction of an [[inverse]] ζ\zeta to ξ\xi: given an element in π n(M)\pi_{n}(M \mathcal{B}) represented by a map as on the right of the above diagram, try to define XX and the structure map g ig_i of its normal bundle as the pullback on the left.

The technical problem to be overcome is that for a general continuous function as on the right, the pullback has no reason to be a smooth manifold, and for two reasons:

  1. the map S n+(kn)Th(V kn)S^{n+(k-n)} \to Th(V_{k-n}) may not be smooth around the image of ii;

  2. even if it is smooth around the image of ii, it may not be [[transversal map|transversal]] to ii, and the intersection of two non-transversal smooth functions is in general still not a smooth manifold.

The heart of the proof is in showing that for any α\alpha there are small homotopies relating it to an α\alpha' that is both smooth around the image of ii and transversal to ii.

The first condition is guaranteed by [[Sard’s theorem]], the second by [[Thom’s transversality theorem]].


Thom isomorphism

Idea. If a [[vector bundle]] EpXE \stackrel{p}{\longrightarrow} X of [[rank]] nn carries a cohomology class ωH n(Th(E),R)\omega \in H^n(Th(E),R) that looks fiberwise like a [[volume form]] – a [[Thom class]] – then the operation of pulling back from base space and then forming the [[cup product]] with this [[Thom class]] is an [[isomorphism]] on (reduced) cohomology

(()ω)p *:H (X,R)H˜ +n(Th(E),R). ( (-) \cup \omega) \circ p^\ast \;\colon\; H^\bullet(X,R) \stackrel{\simeq}{\longrightarrow} \tilde H^{\bullet+n}(Th(E),R) \,.

This is the [[Thom isomorphism]]. It follows from the [[Serre spectral sequence]] (or else from the [[Leray-Hirsch theorem]]). A closely related statement gives the [[Thom-Gysin sequence]].

In the special case that the vector bundle is trivial of rank nn, then its [[Thom space]] coincides with the nn-fold [[suspension]] of the base space (example ) and the Thom isomorphism coincides with the [[suspension isomorphism]]. In this sense the Thom isomorphism may be regarded as a twisted suspension isomorphism.

We need this below to compute (co)homology of universal Thom spectra MUM U in terms of that of the [[classifying spaces]] BUB U.

Composed with pullback along the [[Pontryagin-Thom collapse map]], the Thom isomorphism produces maps in cohomology that covariantly follow the underlying maps of spaces. These “[[Umkehr maps]]” have the interpretation of [[fiber integration]] against the Thom class.

Literature. (Kochman 96, 2.6)

Thom-Gysin sequence

The [[Thom-Gysin sequence]] is a type of [[long exact sequence in cohomology]] induced by a [[spherical fibration]] and expressing the [[cohomology groups]] of the total space in terms of those of the base plus correction. The sequence may be obtained as a corollary of the [[Serre spectral sequence]] for the given fibration. It induces, and is induced by, the [[Thom isomorphism]].


Let RR be a [[commutative ring]] and let

S n E π B \array{ S^n &\longrightarrow& E \\ && \downarrow^{\mathrlap{\pi}} \\ && B }

be a [[Serre fibration]] over a [[simply connected topological space|simply connected]] [[CW-complex]] with typical [[fiber]] (exmpl.) the [[n-sphere]].

Then there exists an element cH n+1(E;R)c \in H^{n+1}(E; R) (in the [[ordinary cohomology]] of the total space with [[coefficients]] in RR, called the Euler class of π\pi) such that the [[cup product]] operation c()c \cup (-) sits in a [[long exact sequence]] of [[cohomology groups]] of the form

H k(B;R)π *H k(E;R)H kn(B;R)c()H k+1(B;R). \cdots \to H^k(B; R) \stackrel{\pi^\ast}{\longrightarrow} H^k(E; R) \stackrel{}{\longrightarrow} H^{k-n}(B;R) \stackrel{c \cup (-)}{\longrightarrow} H^{k+1}(B; R) \to \cdots \,.

(e.g. Switzer 75, section 15.30, Kochman 96, corollary 2.2.6)


Under the given assumptions there is the corresponding [[Serre spectral sequence]]

E 2 s,t=H s(B;H t(S n;R))H s+t(E;R). E_2^{s,t} \;=\; H^s(B; H^t(S^n;R)) \;\Rightarrow\; H^{s+t}(E; R) \,.

Since the [[ordinary cohomology]] of the [[n-sphere]] [[fiber]] is concentrated in just two degees

H t(S n;R)={R fort=0andt=n 0 otherwise H^t(S^n; R) = \left\{ \array{ R & for \; t= 0 \; and \; t = n \\ 0 & otherwise } \right.

the only possibly non-vanishing terms on the E 2E_2 page of this spectral sequence, and hence on all the further pages, are in bidegrees (,0)(\bullet,0) and (,n)(\bullet,n):

E 2 ,0H (B;R),andE 2 ,nH (B;R). E^{\bullet,0}_2 \simeq H^\bullet(B; R) \,, \;\;\;\; and \;\;\; E^{\bullet,n}_2 \simeq H^\bullet(B; R) \,.

As a consequence, since the differentials d rd_r on the rrth page of the Serre spectral sequence have bidegree (r+1,r)(r+1,-r), the only possibly non-vanishing differentials are those on the (n+1)(n+1)-page of the form

E n+1 ,n H (B;R) d n+1 E n+1 +n+1,0 H +n+1(B;R). \array{ E_{n+1}^{\bullet,n} & \simeq & H^\bullet(B;R) \\ {}^{\mathllap{d_{n+1}}}\downarrow \\ E_{n+1}^{\bullet+n+1,0} & \simeq & H^{\bullet+n+1}(B;R) } \,.

Now since the [[coefficients]] RR is a [[ring]], the [[Serre spectral sequence]] is [[multiplicative spectral sequence|multiplicative]] under [[cup product]] and the [[differential]] is a [[derivation]] (of total degree 1) with respect to this product. (See at multiplicative spectral sequence – Examples – AHSS for multiplicative cohomology.)

To make use of this, write

ι1H 0(B;R)E n+1 0,n \iota \coloneqq 1 \in H^0(B;R) \stackrel{\simeq}{\longrightarrow} E_{n+1}^{0,n}

for the unit in the [[cohomology ring]] H (B;R)H^\bullet(B;R), but regarded as an element in bidegree (0,n)(0,n) on the (n+1)(n+1)-page of the spectral sequence. (In particular ι\iota does not denote the unit in bidegree (0,0)(0,0), and hence d n+1(ι)d_{n+1}(\iota) need not vanish; while by the [[derivation]] property, it does vanish on the actual unit 1H 0(B;R)E n+1 0,01 \in H^0(B;R) \simeq E_{n+1}^{0,0}.)


cd n+1(ι)E n+1 n+1,0H n+1(B;R) c \coloneqq d_{n+1}(\iota) \;\; \in E_{n+1}^{n+1,0} \stackrel{\simeq}{\longrightarrow} H^{n+1}(B; R)

for the image of this element under the differential. We will show that this is the Euler class in question.

To that end, notice that every element in E n+1 ,nE_{n+1}^{\bullet,n} is of the form ιb\iota \cdot b for bE n+1 ,0H (B;R)b\in E_{n+1}^{\bullet,0} \simeq H^\bullet(B;R).

(Because the [[multiplicative spectral sequence|multiplicative structure]] gives a group homomorphism ι():H (B;R)E n+1 0,0E n+1 0,nH (B;R)\iota \cdot(-) \colon H^\bullet(B;R) \simeq E_{n+1}^{0,0} \to E^{0,n}_{n+1} \simeq H^\bullet(B;R), which is an isomorphism because the product in the spectral sequence does come from the [[cup product]] in the [[cohomology ring]], see for instance (Kochman 96, first equation in the proof of prop. 4.2.9), and since hence ι\iota does act like the unit that it is in H (B;R)H^\bullet(B;R)).

Now since d n+1d_{n+1} is a graded [[derivation]] and vanishes on E n+1 ,0E_{n+1}^{\bullet,0} (by the above degree reasoning), it follows that its action on any element is uniquely fixed to be given by the product with cc:

d n+1(ιb) =d n+1(ι)b+(1) nιd n+1(b)=0 =cb. \begin{aligned} d_{n+1}(\iota \cdot b) & = d_{n+1}(\iota) \cdot b + (-1)^{n}\, \iota \cdot \underset{= 0}{\underbrace{d_{n+1}(b)}} \\ & = c \cdot b \end{aligned} \,.

This shows that d n+1d_{n+1} is identified with the cup product operation in question:

E n+1 s,n H s(B;R) d n+1 c() E n+1 s+n+1,0 H s+n+1(B;R). \array{ E_{n+1}^{s,n} & \simeq & H^s(B;R) \\ {}^{\mathllap{d_{n+1}}}\downarrow && \downarrow^{\mathrlap{c \cup (-)}} \\ E_{n+1}^{s+n+1, 0} & \simeq & H^{s+n+1}(B;R) } \,.

In summary, the non-vanishing entries of the E E_\infty-page of the spectral sequence sit in [[exact sequences]] like so

0 E s,n ker(d n+1) E n+1 s,n H s(B;R) d n+1 c() E n+1 s+n+1,0 H s+n+1(B;R) coker(d n+1) E s+n+1,0 0. \array{ 0 \\ \downarrow \\ E_\infty^{s,n} \\ {}^{\mathllap{ker(d_{n+1})}}\downarrow \\ E_{n+1}^{s,n} & \simeq & H^s(B;R) \\ {}^{\mathllap{d_{n+1}}}\downarrow && \downarrow^{\mathrlap{c \cup (-)}} \\ E_{n+1}^{s+n+1, 0} & \simeq & H^{s+n+1}(B;R) \\ {}^{\mathllap{coker(d_{n+1})}}\downarrow \\ E_\infty^{s+n+1,0} \\ \downarrow \\ 0 } \,.

Finally observe (lemma ) that due to the sparseness of the E E_\infty-page, there are also [[short exact sequences]] of the form

0E s,0H s(E;R)E sn,n0. 0 \to E_\infty^{s,0} \longrightarrow H^s(E; R) \longrightarrow E_\infty^{s-n,n} \to 0 \,.

Concatenating these with the above exact sequences yields the desired [[long exact sequence]].


Consider a cohomology [[spectral sequence]] converging to some [[filtered object|filtered]] [[graded abelian group]] F C F^\bullet C^\bullet such that

  1. F 0C =C F^0 C^\bullet = C^\bullet;

  2. F sC <s=0F^{s} C^{\lt s} = 0;

  3. E s,t=0E_\infty^{s,t} = 0 unless t=0t = 0 or t=nt = n,

for some nn \in \mathbb{N}, n1n \geq 1. Then there are [[short exact sequences]] of the form

0E s,0C sE sn,n0. 0 \to E_\infty^{s,0} \overset{}{\longrightarrow} C^s \longrightarrow E_\infty^{s-n,n} \to 0 \,.

(e.g. Switzer 75, p. 356)


By definition of convergence of a spectral sequence, the E s,tE_{\infty}^{s,t} sit in [[short exact sequences]] of the form

0F s+1C s+tiF sC s+tE s,t0. 0 \to F^{s+1}C^{s+t} \overset{i}{\longrightarrow} F^s C^{s+t} \longrightarrow E_\infty^{s,t} \to 0 \,.

So when E s,t=0E_\infty^{s,t} = 0 then the morphism ii above is an [[isomorphism]].

We may use this to either shift away the filtering degree

  • if tnt \geq n then F sC s+t=F (s1)+1C (s1)+(t+1)i s1F 0C (s1)+(t+1)=F 0C s+tC s+tF^s C^{s+t} = F^{(s-1)+1}C^{(s-1)+(t+1)} \underoverset{\simeq}{i^{s-1}}{\longrightarrow} F^0 C^{(s-1)+(t+1)} = F^0 C^{s+t} \simeq C^{s+t};

or to shift away the offset of the filtering to the total degree:

  • if 0t1n10 \leq t-1 \leq n-1 then F s+1C s+t=F s+1C (s+1)+(t1)i (t1)F s+tC (s+1)+(t1)=F s+tC s+tF^{s+1}C^{s+t} = F^{s+1}C^{(s+1)+(t-1)} \underoverset{\simeq}{i^{-(t-1)}}{\longrightarrow} F^{s+t}C^{(s+1)+(t-1)} = F^{s+t}C^{s+t}

Moreover, by the assumption that if t<0t \lt 0 then F sC s+t=0F^{s}C^{s+t} = 0, we also get

F sC sE s,0. F^{s}C^{s} \simeq E_\infty^{s,0} \,.

In summary this yields the vertical isomorphisms

0 F s+1C s+n F sC s+n E s,n 0 i (n1) i s1 = 0 F s+nC s+nE s+n,0 C s+n E s,n 0 \array{ 0 &\to& F^{s+1}C^{s+n} &\longrightarrow& F^{s}C^{s+n} &\longrightarrow& E_\infty^{s,n} &\to& 0 \\ && {}^{\mathllap{i^{-(n-1)}}}\downarrow^{\mathrlap{\simeq}} && {}^{\mathllap{i^{s-1}}}\downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{=}} \\ 0 &\to& F^{s+n}C^{s+n} \simeq E_\infty^{s+n,0} &\longrightarrow& C^{s+n} &\longrightarrow& E_\infty^{s,n} &\to& 0 }

and hence with the top sequence here being exact, so is the bottom sequence.

Thom isomorphism


Let VBV \to B be a topological [[vector bundle]] of [[rank]] n>0n \gt 0 over a [[simply connected topological space|simply connected]] [[CW-complex]] BB. Let RR be a [[commutative ring]].

There exists an element cH n(Th(V);R)c \in H^n(Th(V);R) (in the [[ordinary cohomology]], with [[coefficients]] in RR, of the [[Thom space]] of VV, called a [[Thom class]]) such that forming the [[cup product]] with cc induces an [[isomorphism]]

H (B;R)c()H˜ +n(Th(V);R) H^\bullet(B;R) \overset{c \cup (-)}{\longrightarrow} \tilde H^{\bullet + n}(Th(V);R)

of degree nn from the unreduced [[cohomology group]] of BB to the [[reduced cohomology]] of the [[Thom space]] of VV.


Choose an [[orthogonal structure]] on VV. Consider the fiberwise [[cofiber]]

ED(V)/ BS(V) E \coloneqq D(V)/_B S(V)

of the inclusion of the unit sphere bundle into the unit disk bundle of VV (def. ).

S n1 D n S n S(V) D(V) E p B = B = B \array{ S^{n-1} &\hookrightarrow& D^n &\longrightarrow& S^n \\ \downarrow && \downarrow && \downarrow \\ S(V) &\hookrightarrow& D(V) &\longrightarrow& E \\ \downarrow && \downarrow && \downarrow^{\mathrlap{p}} \\ B &=& B &=& B }

Observe that this has the following properties

  1. EpBE \overset{p}{\to} B is an [[n-sphere]] [[fiber bundle]], hence in particular a [[Serre fibration]];

  2. the [[Thom space]] Th(V)E/BTh(V)\simeq E/B is the quotient of EE by the base space, because of the [[pasting law]] applied to the following pasting diagram of [[pushout]] squares

    S(V) D(V) (po) B D(V)/ BS(V) (po) * Th(V) \array{ S(V) &\longrightarrow& D(V) \\ \downarrow &(po)& \downarrow \\ B &\longrightarrow& D(V)/_B S(V) \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& Th(V) }
  3. hence the [[reduced cohomology]] of the Thom space is (def.) the [[relative cohomology]] of EE relative BB

    H˜ (Th(V);R)H (E,B;R). \tilde H^\bullet(Th(V);R) \simeq H^\bullet(E,B;R) \,.
  4. EpBE \overset{p}{\to} B has a global [[section]] BsEB \overset{s}{\to} E (given over any point bBb \in B by the class of any point in the fiber of S(V)BS(V) \to B over bb; or abstractly: induced via the above pushout by the commutation of the projections from D(V)D(V) and from S(V)S(V), respectively).

In the following we write H ()H (;R)H^\bullet(-)\coloneqq H^\bullet(-;R), for short.

By the first point, there is the [[Thom-Gysin sequence]] (prop. ), an [[exact sequence]] running vertically in the following diagram

H (B) p * H˜ (Th(V)) H (E) s * H (B) H n(B). \array{ && H^\bullet(B) \\ && {}^{\mathllap{p^\ast}}\downarrow & \searrow^{\mathrlap{\simeq}} \\ \tilde H^\bullet(Th(V)) &\longrightarrow& H^\bullet(E) &\underset{s^\ast}{\longrightarrow}& H^\bullet(B) \\ && \downarrow \\ && H^{\bullet-n}(B) } \,.

By the second point above this is [[split exact sequence|split]], as shown by the diagonal isomorphism in the top right. By the third point above there is the horizontal exact sequence, as shown, which is the exact sequence in relative cohomology H (E,B)H (E)H (B)\cdots \to H^\bullet(E,B) \to H^\bullet(E) \to H^\bullet(B) \to \cdots induced from the section BEB \hookrightarrow E.

Hence using the splitting to decompose the term in the middle as a [[direct sum]], and then using horizontal and vertical exactness at that term yields

H (B) (0,id) H˜ (Th(V)) (id,0) H˜ (Th(V))H (B) (0,id) H (B) (id,0) H n(B) \array{ && H^\bullet(B) \\ && {}^{\mathllap{(0,id)}}\downarrow & \searrow^{\mathrlap{\simeq}} \\ \tilde H^\bullet(Th(V)) &\overset{(id,0)}{\hookrightarrow}& \tilde H^\bullet(Th(V)) \oplus H^\bullet(B) &\underset{(0,id)}{\longrightarrow}& H^\bullet(B) \\ && \downarrow^{\mathrlap{(id,0)}} \\ && H^{\bullet-n}(B) }

and hence an isomorphism

H˜ (Th(V))H n(B). \tilde H^\bullet(Th(V)) \overset{\simeq}{\longrightarrow} H^{\bullet-n}(B) \,.

To see that this is the inverse of a morphism of the form c()c \cup (-), inspect the proof of the Gysin sequence. This shows that H n(B)H^{\bullet-n}(B) here is identified with elements that on the second page of the corresponding [[Serre spectral sequence]] are cup products

ιb \iota \cup b

with ι\iota fiberwise the canonical class 1H n(S n)1 \in H^n(S^n) and with bH (B)b \in H^\bullet(B) any element. Since H (;R)H^\bullet(-;R) is a [[multiplicative cohomology theory]] (because the [[coefficients]] form a [[ring]] RR), cup producs are preserved as one passes to the E E_\infty-page of the spectral sequence, and the morphism H (E)B (B)H^\bullet(E) \to B^\bullet(B) above, hence also the isomorphism H˜ (Th(V))H (B)\tilde H^\bullet(Th(V)) \to H^\bullet(B), factors through the E E_\infty-page (see towards the end of the proof of the Gysin sequence). Hence the image of ι\iota on the E E_\infty-page is the Thom class in question.

Orientation in generalized cohomology

Idea. From the way the [[Thom isomorphism]] via a [[Thom class]] works in [[ordinary cohomology]] (as above), one sees what the general concept of [[orientation in generalized cohomology]] and of [[fiber integration in generalized cohomology]] is to be.

Specifically we are interested in [[complex oriented cohomology]] theories EE, characterized by an orientation class on infinity [[complex projective space]] P \mathbb{C}P^\infty (def. ), the [[classifying space]] for [[complex line bundles]], which restricts to a generator on S 2P S^2 \hookrightarrow \mathbb{C}P^\infty.

(Another important application is given by taking E=E = [[KU]] to be [[topological K-theory]]. Then [[orientation in generalized cohomology|orientation]] is [[spin^c structure]] and fiber integration with coefficients in EE is [[fiber integration in K-theory]]. This is classical [[index theory]].)

Literature. (Kochman 96, section 4.3, Adams 74, part III, section 10, Lurie 10, lecture 5)

  • [[Riccardo Pedrotti]], Complex oriented cohomology – Orientation in generalized cohomology, 2016 ([[PedrotticECohomology2016.pdf:file]])


Universal EE-orientation


Let EE be a [[multiplicative cohomology theory]] (def. ) and let VXV \to X be a topological [[vector bundle]] of [[rank]] nn. Then an EE-[[orientation in generalized cohomology|orientation]] or EE-[[Thom class]] on VV is an element of degree nn

uE˜ n(Th(V)) u \in \tilde E^n(Th(V))

in the [[reduced cohomology|reduced]] EE-[[cohomology ring]] of the [[Thom space]] (def. ) of VV, such that for every point xXx \in X its restriction i x *ui_x^* u along

i x:S nTh( n)Th(e x)Th(V) i_x \;\colon\; S^n \simeq Th(\mathbb{R}^n) \overset{Th(e_x)}{\longrightarrow} Th(V)

(for nfib xV\mathbb{R}^n \overset{fib_x}{\hookrightarrow} V the [[fiber]] of VV over xx) is a generator, in that it is of the form

i *u=ϵγ n i^\ast u = \epsilon \cdot \gamma_n


  • ϵE˜ 0(S 0)\epsilon \in \tilde E^0(S^0) a [[unit]] in E E^\bullet;

  • γ nE˜ n(S n)\gamma_n \in \tilde E^n(S^n) the image of the multiplicative unit under the [[suspension isomorphism]] E˜ 0(S 0)E˜ n(S n)\tilde E^0(S^0) \stackrel{\simeq}{\to}\tilde E^n(S^n).

(e.g. Kochmann 96, def. 4.3.4)


Recall that a [[(B,f)-structure]] \mathcal{B} (def. ) is a system of [[Serre fibrations]] B nf nBO(n)B_n \overset{f_n}{\longrightarrow} B O(n) over the [[classifying spaces]] for [[orthogonal structure]] equipped with maps

g n,n+1:B nB n+1 g_{n,n+1} \;\colon\; B_n \longrightarrow B_{n+1}

covering the canonical inclusions of classifying spaces. For instance for G nO(n)G_n \to O(n) a compatible system of [[topological group]] [[homomorphisms]], then the (B,f)(B,f)-structure given by the [[classifying spaces]] BG nB G_n (possibly suitably resolved for the maps BG nBO(n)B G_n \to B O(n) to become Serre fibrations) defines [[G-structure]].

Given a (B,f)(B,f)-structure, then there are the [[pullbacks]] V n f n *(EO(n)×O(n) n)V^{\mathcal{B}}_n \coloneqq f_n^\ast (E O(n)\underset{O(n)}{\times}\mathbb{R}^n) of the [[universal vector bundles]] over BO(n)B O(n), which are the universal vector bundles equipped with (B,f)(B,f)-structure

V n EO(n)×O(n) n (pb) B n f n BO(n). \array{ V^{\mathcal{B}}_n &\longrightarrow& E O(n)\underset{O(n)}{\times} \mathbb{R}^n \\ \downarrow &(pb)& \downarrow \\ B_n & \underset{f_n}{\longrightarrow} & B O(n) } \,.

Finally recall that there are canonical morphisms (prop.)

ϕ n:V n V n+1 \phi_n \;\colon\; \mathbb{R} \oplus V^{\mathcal{B}}_n \longrightarrow V^{\mathcal{B}}_{n+1}

Let EE be a [[multiplicative cohomology theory]] and let \mathcal{B} be a multiplicative [[(B,f)-structure]]. Then a universal EE-orientation for vector bundles with \mathcal{B}-structure is an EE-orientation, according to def. , for each rank-nn universal vector bundle with \mathcal{B}-structure:

ξ nE˜ n(Th(E n ))n \xi_n \in \tilde E^n(Th(E_n^{\mathcal{B}})) \;\;\;\; \forall n \in \mathbb{N}

such that these are compatible in that

  1. for all nn \in \mathbb{N} then

    ξ n=ϕ n *ξ n+1, \xi_n = \phi_n^\ast \xi_{n+1} \,,


    ξ nE˜ n(Th(V n))E˜ n+1(ΣTh(V n))E˜ n+1(Th(V n)) \xi_n \in \tilde E^n(Th(V_n)) \simeq \tilde E^{n+1}(\Sigma Th(V_n)) \simeq \tilde E^{n+1}(Th(\mathbb{R}\oplus V_n))

    (with the first isomorphism is the [[suspension isomorphism]] of EE and the second exhibiting the [[homeomorphism]] of Thom spaces Th(V)ΣTh(V)Th(\mathbb{R} \oplus V)\simeq \Sigma Th(V) (prop. ) and where

    ϕ n *:E˜ n+1(Th(V n+1))E˜ n+1(Th(V n)) \phi_n^\ast \;\colon\; \tilde E^{n+1}(Th(V_{n+1})) \longrightarrow \tilde E^{n+1}(Th(\mathbb{R}\oplus V_n))

    is pullback along the canonical ϕ n:V nV n+1\phi_n \colon \mathbb{R}\oplus V_n \to V_{n+1} (prop. ).

  2. for all n 1,n 2n_1, n_2 \in \mathbb{N} then

    ξ n+1ξ n+2=ξ n 1+n 2. \xi_{n+1} \cdot \xi_{n+2} = \xi_{n_1 + n_2} \,.

A universal EE-orientation, in the sense of def. , for vector bundles with [[(B,f)-structure]] \mathcal{B}, is equivalently (the homotopy class of) a homomorphism of [[ring spectra]]

ξ:ME \xi \;\colon\; M\mathcal{B} \longrightarrow E

from the universal \mathcal{B}-[[Thom spectrum]] to a spectrum which via the [[Brown representability theorem]] (theorem ) represents the given [[generalized (Eilenberg-Steenrod) cohomology theory]] EE (and which we denote by the same symbol).


The [[Thom spectrum]] MM\mathcal{B} has a standard structure of a [[CW-spectrum]]. Let now EE denote a [[sequential spectrum|sequential]] [[Omega-spectrum]] representing the multiplicative cohomology theory of the same name. Since, in the standard [[model structure on topological sequential spectra]], [[CW-spectra]] are cofibrant (prop.) and Omega-spectra are fibrant (thm.) we may represent all morphisms in the [[stable homotopy category]] (def.) by actual morphisms

ξ:ME \xi \;\colon\; M \mathcal{B} \longrightarrow E

of sequential spectra (due to this lemma).

Now by definition (def.) such a homomorphism is precissely a sequence of base-point preserving [[continuous functions]]

ξ n:(M) n=Th(V n )E n \xi_n \;\colon\; (M\mathcal{B})_n = Th(V_n^{\mathcal{B}}) \longrightarrow E_n

for nn \in \mathbb{N}, such that they are compatible with the structure maps σ n\sigma_n and equivalently with their (S 1()Maps(S 1,) *)(S^1 \wedge(-)\dashv Maps(S^1,-)_\ast)-[[adjuncts]] σ˜ n\tilde \sigma_n, in that these diagrams commute:

S 1Th(V n ) S 1ξ n S 1E n σ n M σ n E Th(V n+1 ) ξ n+1 E n+1Th(V n ) ξ n E n σ˜ n M σ˜ n E Maps(S 1,Th(V n+1 )) Maps(S 1,ξ n+1) * Maps(S 1,E n+1) * \array{ S^1 \wedge Th(V^{\mathcal{B}}_n) &\overset{S^1 \wedge \xi_n}{\longrightarrow}& S^1 \wedge E_n \\ {}^{\mathllap{\sigma^{M\mathcal{B}}_n}}\downarrow && \downarrow^{\mathrlap{\sigma^E_n}} \\ Th(V^{\mathcal{B}}_{n+1}) &\underset{\xi_{n+1}}{\longrightarrow}& E_{n+1} } \;\;\;\;\;\;\;\;\; \leftrightarrow \;\;\;\;\;\;\;\;\; \array{ Th(V^{\mathcal{B}}_n) &\overset{\xi_n}{\longrightarrow}& E_n \\ {}^{\mathllap{\tilde \sigma^{M\mathcal{B}}_n}}\downarrow && \downarrow^{\mathrlap{\tilde \sigma^E_n}} \\ Maps(S^1,Th(V^{\mathcal{B}}_{n+1})) &\underset{Maps(S^1,\xi_{n+1})_\ast}{\longrightarrow}& Maps(S^1, E_{n+1})_{\ast} }

for all nn \in \mathbb{N}.

First of all this means (via the identification given by the [[Brown representability theorem]], see prop. , that the components ξ n\xi_n are equivalently representatives of elements in the [[cohomology groups]]

ξ nE˜ n(Th(V n )) \xi_n \in \tilde E^n(Th(V^{\mathcal{B}}_n))

(which we denote by the same symbol, for brevity).

Now by the definition of universal [[Thom spectra]] (def. , def. ), the structure map σ n M\sigma_n^{M\mathcal{B}} is just the map ϕ n:Th(V n )Th(V n+1 )\phi_n \colon \mathbb{R}\oplus Th(V^{\mathcal{B}}_n)\to Th(V_{n+1}^{\mathcal{B}}) from above.

Moreover, by the [[Brown representability theorem]], the [[adjunct]] σ˜ n Eξ n\tilde \sigma_n^E \circ \xi_n