# nLab Introduction to Cobordism and Complex Oriented Cohomology

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

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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.

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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

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cohomology

# Contents

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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

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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,

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

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.

### 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.

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#### Reduced cohomology

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

###### Definition
1. $\tilde E^\bullet \;\colon\; (Top^{\ast/}_{CW})^{op} \longrightarrow Ab^{\mathbb{Z}}$

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

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

$\sigma_E \;\colon\; \tilde E^\bullet(-) \overset{\simeq}{\longrightarrow} \tilde E^{\bullet +1}(\Sigma -)$

such that:

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

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

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

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

This is equivalent (prop. 1 below) to the following more succinct homotopy-theoretic definition:

###### Definition

A reduced generalized cohomology theory is a functor

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

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

$\sigma \;\colon\; \tilde E^{\bullet +1}(\Sigma -) \overset{\simeq}{\longrightarrow} \tilde E^\bullet(-)$

such that:

As a consequence (prop. 1 below), we find yet another equivalent definition:

###### Definition

A reduced generalized cohomology theory is a functor

$\tilde E^\bullet \;\colon\; (Top^{\ast/})^{op} \longrightarrow Ab^{\mathbb{Z}}$

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

$\sigma \;\colon\; \tilde E^{\bullet +1}(\Sigma -) \overset{\simeq}{\longrightarrow} \tilde E^\bullet(-)$

such that

###### Proposition

The three definitions

• def. 1

• def. 2

• def. 3

are indeed equivalent.

###### Proof

Regarding the equivalence of def. 1 with def. 2:

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. 1 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. 2 with def. 3:

This is the universal property of the classical homotopy category (thm.) which identifies it with the localization (def.) of $Top^{\ast/}$ 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 $F$ and $\tilde F$ in the following diagram (which is filled by a natural isomorphism itself):

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

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

Prop. 1 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:

###### Definition

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

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

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

$\sigma \; \colon \; \tilde E^\bullet \longrightarrow \tilde E^{\bullet+1} \circ \Sigma$

such that

Finally we need the following terminology:

###### Definition

Let $\tilde E^\bullet$ be a reduced cohomology theory according to either of def. 1, def. 2, def. 3 or def. 4.

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

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

$\tilde E^\bullet(\vee_{i \in I} X_i) \longrightarrow \prod_{i \in I} \tilde E^\bullet(X_i)$

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

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

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

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

A homomorphism of reduced cohomology theories

$\eta \;\colon\; \tilde E^\bullet \longrightarrow \tilde F^\bullet$

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

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

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

###### Definition

Given a generalized cohomology theory $(E^\bullet,\delta)$ on some $\mathcal{C}$ as in def. 4, and given a homotopy cofiber sequence in $\mathcal{C}$ (prop.),

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

then the corresponding connecting homomorphism is the composite

$\partial \;\colon\; E^\bullet(X) \stackrel{\sigma}{\longrightarrow} E^{\bullet+1}(\Sigma X) \stackrel{coker(g)^\ast}{\longrightarrow} E^{\bullet+1}(Z) \,.$
###### Proposition

The connecting homomorphisms of def. 6 are parts of long exact sequences

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

By the defining exactness of $E^\bullet$, def. 4, and the way this appears in def. 6, using that $\sigma$ is by definition an isomorphism.

#### Unreduced cohomology

Given a reduced generalized cohomology theory as in def. 1, we may “un-reduce” it and evaluate it on unpointed topological spaces $X$ simply by evaluating it on $X_+$ (def.). It is conventional to further generalize to relative cohomology and evaluate on unpointed subspace inclusions $i \colon A \hookrightarrow X$, taken as placeholders for their mapping cones $Cone(i_+)$ (prop.).

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

###### Definition

A cohomology theory (unreduced, relative) is

1. $E^\bullet : (Top_{CW}^{\hookrightarrow})^{op} \to Ab^{\mathbb{Z}}$

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

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

such that:

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

$E^\bullet(f) \;\colon\; E^\bullet(X_2,A_2) \stackrel{\simeq}{\longrightarrow} E^\bullet(X_1,A_1)$

is an isomorphism;

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

$\cdots \to E^n(X, A) \longrightarrow E^n(X) \longrightarrow E^n(A) \stackrel{\delta}{\longrightarrow} E^{n+1}(X, A) \to \cdots$
3. (excision) For $U \hookrightarrow A \hookrightarrow X$ such that $\overline{U} \subset Int(A)$, then the natural inclusion of the pair $i \colon (X-U, A-U) \hookrightarrow (X, A)$ induces an isomorphism

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

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

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

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

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

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

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

A homomorphism of unreduced cohomology theories

$\eta \;\colon\; E^\bullet \longrightarrow F^\bullet$

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

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

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

###### Lemma

The excision axiom in def. 7 is equivalent to the following statement:

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

$i \colon (A, A \cap B) \longrightarrow (X,B)$

induces an isomorphism,

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

(e.g Switzer 75, 7.2)

###### Proof

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

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

and that

$(X-U, B-U) = (A, A \cap B) \,.$

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

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

\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

$(X-U, (X-U) \cap A) = (X-U, A - U) \,.$

Hence

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

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.

###### Lemma

Let $E^\bullet$ be an cohomology theory, def. 7, and let $A \hookrightarrow X$. Then there is an isomorphism

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

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

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

$E^\bullet(p) \;\colon\; E^\bullet(X/A,\ast) \longrightarrow E^\bullet(X,A) \,.$
###### Proof

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

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

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:

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

If $A \hookrightarrow X$ is a cofibration, then this is a homotopy equivalence since $Cone(A)$ is contractible and since by the dual factorization lemma (lem.) and by the invariance of homotopy fibers under weak equivalences (lem.), $X \cup Cone(A)\to X/A$ 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

$E^\bullet(X/A,\ast) \overset{\simeq}{\longrightarrow} E^\bullet( X\cup Cone(A), Cone(A) ) \overset{\simeq}{\longrightarrow} E^\bullet(X,A) \,.$
###### Example

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

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

is an isomorphism.

###### Proposition

(exact sequence of a triple)

For $E^\bullet$ an unreduced generalized cohomology theory, def. 7, then every inclusion of two consecutive subspaces

$Z \hookrightarrow Y \hookrightarrow X$

induces a long exact sequence of cohomology groups of the form

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

where

$\bar \delta \;\colon \; E^{q-1}(Y,Z) \longrightarrow E^{q-1}(Y) \stackrel{\delta}{\longrightarrow} E^{q}(X,Y) \,.$
###### Proof

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

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

See here for details.

###### Remark

The exact sequence of a triple in prop. 3 is what gives rise to the Cartan-Eilenberg spectral sequence for $E$-cohomology of a CW-complex $X$.

###### Example

For $(X,x)$ a pointed topological space and $Cone(X) = (X \wedge (I_+))/X$ its reduced cone, the long exact sequence of the triple $(\{x\}, X, Cone(X))$, prop. 3,

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

exhibits the connecting homomorphism $\bar \delta$ here as an isomorphism

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

This is the suspension isomorphism extracted from the unreduced cohomology theory, see def. 9 below.

###### Proposition

(Mayer-Vietoris sequence)

Given $E^\bullet$ an unreduced cohomology theory, def. 7. Given a topological space covered by the interior of two spaces as $X = Int(A) \cup Int(B)$, then for each $C \subset A \cap B$ there is a long exact sequence of cohomology groups of the form

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

#### Relation between unreduced and reduced cohomology

###### Definition

(unreduced to reduced cohomology)

Let $E^\bullet$ be an unreduced cohomology theory, def. 7. Define a reduced cohomology theory, def. 1 $(\tilde E^\bullet, \sigma)$ as follows.

For $x \colon \ast \to X$ a pointed topological space, set

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

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

$\sigma \;\colon\; \tilde E^{\bullet+1}(\Sigma X) = E^{\bullet+1}(\Sigma X, \{x\}) \overset{E^\bullet(p)}{\longrightarrow} E^{\bullet+1}(Cone(X),X) \overset{\bar \delta^{-1}}{\longrightarrow} E^\bullet(X,\{x\}) = \tilde E^{\bullet}(X)$

of the isomorphism $E^\bullet(p)$ from example 1 and the inverse of the isomorphism $\bar \delta$ from example 2.

###### Proposition

The construction in def. 9 indeed gives a reduced cohomology theory.

(e.g Switzer 75, 7.34)

###### Proof

We need to check the exactness axiom given any $A\hookrightarrow X$. By lemma 1 we have an isomorphism

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

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

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

where the vertical sequence on the right is exact by prop. 3. Hence the left vertical sequence is exact.

###### Definition

(reduced to unreduced cohomology)

Let $(\tilde E^\bullet, \sigma)$ be a reduced cohomology theory, def. 1. Define an unreduced cohomolog theory $E^\bullet$, def. 7, by

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

and let the connecting homomorphism be as in def. 6.

###### Proposition

The construction in def. 10 indeed yields an unreduced cohomology theory.

e.g. (Switzer 75, 7.35)

###### Proof

Exactness holds by prop. 2. For excision, it is sufficient to consider the alternative formulation of lemma 8. For CW-inclusions, this follows immediately with lemma 1.

###### Theorem

The constructions of def. 10 and def. 9 constitute a pair of functors between then categories of reduced cohomology theories, def. 1 and unreduced cohomology theories, def. 7 which exhbit an equivalence of categories.

###### Proof

(…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

\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 $A \hookrightarrow X$ 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 1.

The other composite is

\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 $X$, and so now the natural isomorphism follows with homotopy invariance.

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

###### Proposition

Let $E^\bullet$ be an unreduced cohomology theory, and $\tilde E^\bullet$ its reduced cohomology theory from def. 9. For $(X,\ast)$ a pointed topological space, then there is an identification

$E^\bullet(X) \simeq \tilde E^\bullet(X) \oplus E^\bullet(\ast)$

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

###### Proof

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

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

which by the exactness clause in def. 7 is exact.

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

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

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:

###### Definition

A reduced homology theory is a functor

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

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

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

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

$\sigma \;\colon\; \tilde E_\bullet(-) \overset{\simeq}{\longrightarrow} \tilde E_{\bullet +1}(\Sigma -)$

such that:

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

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

$\tilde E_\bullet(A) \overset{i_\ast}{\longrightarrow} \tilde E_\bullet(X) \overset{j_\ast}{\longrightarrow} \tilde E_\bullet(Cone(i)) \,.$

We say $\tilde E_\bullet$ is additive if in addition

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

$\oplus_{i \in I} \tilde E_\bullet(X_i) \longrightarrow \tilde E^\bullet(\vee_{i \in I} X_i)$

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

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

A homomorphism of reduced cohomology theories

$\eta \;\colon\; \tilde E_\bullet \longrightarrow \tilde F_\bullet$

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

$\array{ \tilde E_\bullet(X) &\overset{\eta_X}{\longrightarrow}& \tilde F_\bullet(X) \\ {}^{\mathllap{\sigma_E}}\downarrow && \downarrow^{\mathrlap{\sigma_F}} \\ \tilde E_{\bullet + 1}(\Sigma X) &\overset{\eta_{\Sigma X}}{\longrightarrow}& \tilde F_{\bullet + 1}(\Sigma X) } \,.$
###### Definition

A homology theory (unreduced, relative) is a functor

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

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

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

such that:

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

$E_\bullet(f) \;\colon\; E_\bullet(X_1,A_1) \stackrel{\simeq}{\longrightarrow} E_\bullet(X_2,A_2)$

is an isomorphism;

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

$\cdots \to E_{n+1}(X, A) \stackrel{\delta}{\longrightarrow} E_n(A) \longrightarrow E_n(X) \longrightarrow E_n(X, A) \to \cdots$
3. (excision) For $U \hookrightarrow A \hookrightarrow X$ such that $\overline{U} \subset Int(A)$, then the natural inclusion of the pair $i \colon (X-U, A-U) \hookrightarrow (X, A)$ induces an isomorphism

$E_\bullet(i) \;\colon\; E_n(X-U, A-U) \overset{\simeq}{\longrightarrow} E_n(X, A)$

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

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

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

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

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

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

A homomorphism of unreduced homology theories

$\eta \;\colon\; E_\bullet \longrightarrow F_\bullet$

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

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

#### 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:

###### Definition

Let $E_1, E_2, E_3$ be three unreduced generalized cohomology theories (def.). A pairing of cohomology theories

$\mu \;\colon\; E_1 \Box E_2 \longrightarrow E_3$

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

$\mu_{n_1,n_2} \;\colon\; E_1^{n_1}(X,A) \otimes E_2^{n_2}(Y,B) \longrightarrow E_3^{n_1 + n_2}(X\times Y \;,\; A\times Y \cup X \times B)$

such that this is compatible with the connecting homomorphisms $\delta_i$ of $E_i$, in that the following are commuting squares

$\array{ E_1^{n_1}(A) \otimes E_2^{n_2}(Y,B) &\overset{\delta_1 \otimes id_2}{\longrightarrow}& E_1^{n_1+1}(X,A) \otimes E_2^{n_2}(Y,B) \\ {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1+1, n_2}}} \\ \underoverset {E_3^{n_1 + n_2}(A \times Y \cup X \times B , X \times B)} {E_3^{n_1 + n_2}(A \times Y, A \times B)} {\simeq} &\overset{\delta_3}{\longrightarrow}& E_3^{n_1 + n_2+ 1}(X \times Y, A \times B) }$

and

$\array{ E_1^{n_1}(X,A) \otimes E_2^{n_2}(B) &\overset{(-1)^{n_1} id_1 \otimes \delta_2}{\longrightarrow}& E_1^{n_1+1}(X,A) \otimes E_2^{n_2}(Y,B) \\ {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1, n_2 + 1}}} \\ \underoverset {E_3^{n_1 + n_2}(A \times Y \cup X \times B , A \times Y)} {E_3^{n_1 + n_2}(X \times B, A \times B)} {\simeq} &\overset{\delta_3}{\longrightarrow}& E_3^{n_1 + n_2+ 1}(X \times Y, A \times B) } \,,$

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

###### Definition

An (unreduced) multiplicative cohomology theory is an unreduced generalized cohomology theory theory $E$ (def. 7) equipped with

1. (external multiplication) a pairing (def. 13) of the form $\mu \;\colon\; E \Box E \longrightarrow E$;

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

such that

1. (associativity) $\mu \circ (id \otimes \mu) = \mu \circ (\mu \otimes id)$;

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

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

$\array{ E^{n_1}(X,A) \otimes E^{n_2}(Y,B) &\overset{(u \otimes v) \mapsto (-1)^{n_1 n_2} (v \otimes u) }{\longrightarrow}& E^{n_2}(Y,B) \otimes E^{n_1}_{X,A} \\ {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_2,n_1}}} \\ E^{n_1 + n_2}( X \times Y , A \times Y \cup X \times B) &\underset{(switch_{(X,A), (Y,B)})^\ast}{\longrightarrow}& E^{n_1 + n_2}( Y \times X , B \times X \cup Y \times A) } \,.$

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

$(-) \cup (-) \;\colon\; E^{n_1}(X,A) \otimes E^{n_2}(X,A) \overset{\mu_{n_1,n_2}}{\longrightarrow} E^{n_1 + n_2}( X \times X, \; A \times X \cup X \times A) \overset{\Delta^\ast_{(X,A)}}{\longrightarrow} E^{n_1 + n_2}(X, \; A \cup B) \,.$

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

###### Proposition

Let $(E,\mu,1)$ be a multiplicative cohomology theory, def. 14. Then

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

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

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

###### Proof

Regarding the third point:

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

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

For connecting homomorphisms this is the (graded) commutativity of the squares in def. 14:

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

### 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.

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

###### Definition

A Brown functor is a functor

$F\;\colon \; Ho(Top_{\geq 1}^{\ast/})^{op} \longrightarrow Set^{\ast/}$

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

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

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

###### Proposition

For every additive reduced cohomology theory $\tilde E^\bullet(-) \colon Ho(Top^{\ast/})^{op}\to Set^{\ast/}$ (def. 2) and for each degree $n \in \mathbb{N}$, the restriction of $\tilde E^n(-)$ to connected spaces is a Brown functor (def. 15).

###### Proof

Under the relation between reduced and unreduced cohomology above, this follows from the exactness of the Mayer-Vietoris sequence of prop. 4.

###### Theorem

(Brown representability)

Every Brown functor $F$ (def. 15) is representable, hence there exists $X \in Top_{\geq 1}^{\ast/}$ and a natural isomorphism

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

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

(e.g. AGP 02, theorem 12.2.22)

###### Remark

A key subtlety in theorem 2 is the restriction to connected pointed topological spaces in def. 15. This comes about since the proof of the theorem requires that continuous functions $f \colon X \longrightarrow Y$ that induce isomorphisms on pointed homotopy classes

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

for all $n$ are weak homotopy equivalences (For instance in AGP 02 this is used in the proof of theorem 12.2.19 there). But $[S^n,X]_\ast = \pi_n(X,x)$ gives the $n$th homotopy group of $X$ 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 $X\to Y$ making all the $[S^n,X]_\ast \longrightarrow [S^n,Y]_\ast$ into isomorphisms.

The representability result applied degreewise to an additive reduced cohomology theory will yield (prop. 10 below) the following concept.

###### Definition

An Omega-spectrum $X$ (def.) is

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

2. weak homotopy equivalences

$\tilde \sigma_n \;\colon\; X_n \underoverset{\in W_{cl}}{\tilde \sigma_n}{\longrightarrow} \Omega X_{n+1}$

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

###### Proposition

Every additive reduced cohomology theory $\tilde E^\bullet(-) \colon (Top_{CW}^\ast)^{op} \longrightarrow Ab^{\mathbb{Z}}$ according to def. 2, is represented by an Omega-spectrum $E$ (def. 16) in that in each degree $n \in \mathbb{N}$

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

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

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

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

###### Proof

If it were not for the connectedness clause in def. 15 (remark 2), then theorem 2 with prop. 9 would immediately give the existence of the $\{E_n\}_{n \in \mathbb{N}}$ 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 $X$ a pointed topological space, write $X^{(0)}$ for the connected component of its basepoint. Observe that the loop space of a pointed topological space only depends on this connected component:

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

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

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

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

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

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

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

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

Conversely:

###### Proposition

Every Omega-spectrum $E$, def. 16, represents an additive reduced cohomology theory def. 1 $\tilde E^\bullet$ by

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

with suspension isomorphism given by

$\sigma_n \;\colon\; \tilde E^n(X) = [X,E_n]_\ast \overset{[X,\tilde \sigma_n]}{\longrightarrow} [X, \Omega E_{n+1}]_\ast \overset{\simeq}{\to} [\Sigma X, E_{n+1}] = \tilde E^{n+1}(\Sigma X) \,.$
###### Proof

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

###### Remark

If we consider the stable homotopy category $Ho(Spectra)$ of spectra (def.) and consider any topological space $X$ in terms of its suspension spectrum $\Sigma^\infty X \in Ho(Spectra)$ (exmpl.), then the statement of prop. 11 is more succinctly summarized by saying that the graded reduced cohomology groups of a topological space $X$ represented by an Omega-spectrum $E$ are the hom-groups

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

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

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

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

to be the graded reduced generalized $E$-cohomology groups of the spectrum $X$.

#### Application to ordinary cohomology

###### Example

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

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

Hence singular cohomology is a generalized cohomology theory which is “ordinary cohomology” in the sense of def. 5.

Applying the Brown representability theorem as in prop. 10 hence produces an Omega-spectrum (def. 16) whose $n$th component space is characterized as having homotopy groups concentrated in degree $n$ on $A$. These are called Eilenberg-MacLane spaces $K(A,n)$

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

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

Such spectra are called Eilenberg-MacLane spectra $H A$:

$(H A)_n \simeq K(A,n) \,.$

As a consequence of example 3 one obtains the uniqueness result of Eilenberg-Steenrod:

###### Proposition

Let $\tilde E_1$ and $\tilde E_2$ be ordinary (def. 5) generalized (Eilenberg-Steenrod) cohomology theories. If there is an isomorphism

$\tilde E_1(S^0) \simeq \tilde E_2(S^0)$

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

$\tilde E_1 \overset{\simeq}{\longrightarrow} \tilde E_2 \,.$

#### 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.

###### Definition

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

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

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

is called a Brown functor if

1. it sends small coproducts to products;

2. it sends homotopy pushouts in $\mathcal{C}\to Ho(\mathcal{C})$ to weak pullbacks in Set (see remark 4).

###### 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. 17 says that for a homotopy pushout square

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

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

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

into the actual pullback is an epimorphism.

###### Definition

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

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

$\{S_i \in \mathcal{C}\}_{i \in I}$

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

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

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

###### Example

(suspensions are H-cogroup objects)

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

Then the fold map

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

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

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

$Ho(\mathcal{C})(\Sigma X, -) \;\colon\; Y \mapsto Ho(\mathcal{C})(\Sigma X, Y) \simeq Ho(\mathcal{C})(X, \Omega Y) \simeq \pi_1 Ho(\mathcal{C})(X, Y) \,.$
###### Example

In bare pointed homotopy types $\mathcal{C} = Top^{\ast/}_{Quillen}$, the (homotopy types of) n-spheres $S^n$ are cogroup objects for $n \geq 1$, but not for $n = 0$, by example 4. And of course they are compact objects.

So while $\{S^n\}_{n \in \mathbb{N}}$ generates all of the homotopy theory of $Top^{\ast/}$, the latter is not an example of def. 18 due to the failure of $S^0$ to have cogroup structure.

Removing that generator, the homotopy theory generated by $\{S^n\}_{{n \in \mathbb{N}} \atop {n \geq 1}}$ is $Top^{\ast/}_{\geq 1}$, 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 2 above.

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

###### Proposition

In a homotopy theory compactly generated by cogroup objects $\{S_i\}_{i \in I}$ closed under forming suspensions, according to def. 18, a morphism $f\colon X \longrightarrow Y$ is an equivalence precisely if for each $i \in I$ the induced function of maps in the homotopy category

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

is an isomorphism (a bijection).

###### Proof

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

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

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

Now the maps

$\mathcal{C}(S_i,f) \;\colon\; \mathcal{C}(S_i,X) \longrightarrow \mathcal{C}(S_i,Y)$

are weak equivalences in $Top_{Quillen}$ if they are weak homotopy equivalences, hence if they induce isomorphisms on all homotopy groups $\pi_n$ for all basepoints.

It is this last condition of testing on all basepoints that the assumed cogroup structure on the $S_i$ allows to do away with: this cogroup structure implies that $\mathcal{C}(S_i,-)$ has the structure of an $H$-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 $\pi_n$ based on the connected component of the zero morphism

$\pi_n\mathcal{C}(S_i,f) \;\colon\; \pi_n \mathcal{C}(S_i,X) \longrightarrow \pi_n\mathcal{C}(S_i,Y) \,.$

Now in this pointed situation we may use that

\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 $f$ is an equivalence in $\mathcal{C}$ precisely if the induced morphisms

$Ho(\mathcal{C})(\Sigma^n S_i, f) \;\colon\; Ho(\mathcal{C})(\Sigma^n S_i,X) \longrightarrow Ho(\mathcal{C})(\Sigma^n S_i,Y)$

are isomorphisms for all $i \in I$ and $n \in \mathbb{N}$.

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

###### Theorem

(Brown representability)

Let $\mathcal{C}$ be a model category compactly generated by cogroup objects closed under forming suspensions, according to def. 18. Then a functor

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

(from the opposite of the homotopy category of $\mathcal{C}$ to Set) is representable precisely if it is a Brown functor, def. 17.

###### Proof

Due to the version of the Whitehead theorem of prop. 13 we are essentially reduced to showing that Brown functors $F$ are representable on the $S_i$. To that end consider the following lemma. (In the following we notationally identify, via the Yoneda lemma, objects of $\mathcal{C}$, hence of $Ho(\mathcal{C})$, with the functors they represent.)

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

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

$\eta_0 \;\colon\; X \sqcup \left( \underset{{i \in I,} \atop {\gamma \colon S_i \stackrel{}{\to} F}}{\coprod} S_i \right) \longrightarrow F$

to be the canonical map. (Using that $F$, by assumption, turns coproducts into products, we may indeed treat the coproduct in $\mathcal{C}$ 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 $X_0$. To that end proceed by induction and suppose that $\eta_n \colon X_n \to F$ has been constructed. Then for $i \in I$ let

$K_i \coloneqq ker \left( Ho(\mathcal{C})(S_i, X_n) \stackrel{\eta_n \circ (-)}{\longrightarrow} F(S_i) \right)$

be the kernel of $\eta_n$ evaluated on $S_i$. These $K_i$ are the pieces that need to go away in order to make a bijection. Hence define $X_{n+1}$ to be their joint homotopy cofiber

$X_{n+1} \coloneqq coker\left( \left( \underset{{i \in I,} \atop {\gamma \in K_i}}{\sqcup} S_i \right) \overset{(\gamma)_{{i \in I} \atop {\gamma\in K_i}}}{\longrightarrow} X_n \right) \,.$

Then by the assumption that $F$ takes this homotopy cokernel to a weak fiber (as in remark 4), there exists an extension $\eta_{n+1}$ of $\eta_n$ along $X_n \to X_{n+1}$:

Then by the assumption that $F$ takes this homotopy cokernel to a weak fiber (as in remark 4), there exists an extension $\eta_{n+1}$ of $\eta_n$ along $X_n \to X_{n+1}$:

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

It is now clear that we want to take

$X' \coloneqq \underset{\rightarrow}{\lim}_n X_n$

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

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

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

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

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

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

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

$\array{ 0 \colon & S_i &\overset{\hat \gamma}{\longrightarrow}& X_{n(\gamma)} &\overset{\eta_n}{\longrightarrow}& F \\ & \downarrow && \downarrow & \\ & \ast &\longrightarrow& X_{n(\gamma)+1} \\ & && \downarrow \\ & && X' }$

This concludes the proof of Lemma ($\star$).

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

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

is a bijection.

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

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

Moreover, Lemma $(\star)$ gives that evaluated on all $S_i$, the two diagonal morphisms here become isomorphisms. But then prop. 13 implies that $X' \longrightarrow Z$ is in fact an equivalence. Hence the component map $Y \to Z \simeq Z$ is a lift of $\kappa$ through $\theta$.

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

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

along the codiagonal of $Y$. Using that $F$ sends this to a weak pullback by assumption, we obtain an extension $\bar \eta$ of $\eta'$ along $X' \to Z$. Applying Lemma $(\star)$ to this gives a further extension $\bar \eta' \colon Z' \to Z$ which now makes the following diagram

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

such that the diagonal maps become isomorphisms when evaluated on the $S_i$. As before, it follows via prop. 13 that the morphism $h \colon X' \longrightarrow Z'$ is an equivalence.

Since by this construction $h\circ f$ and $h\circ g$ are homotopic

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

it follows with $h$ being an equivalence that already $f$ and $g$ were homotopic, hence that they represented the same element.

###### Proposition

Given a reduced additive cohomology functor $H^\bullet \colon Ho(\mathcal{C})^{op}\to Ab^{\mathbb{Z}}$, def. 4, its underlying Set-valued functors $H^n \colon Ho(\mathcal{C})^{op}\to Ab\to Set$ are Brown functors, def. 17.

###### Proof

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

$\array{ X_1 &\stackrel{f_1}{\longrightarrow}& Y_1 \\ \downarrow^{} && \downarrow \\ X_2 &\stackrel{f_2}{\longrightarrow}& Y_2 }$

in $\mathcal{C}$, consider the induced morphism of the long exact sequences given by prop. 2

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

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.

###### Corollary

Let $\mathcal{C}$ be a model category which satisfies the conditions of theorem 3, and let $(H^\bullet, \delta)$ be a reduced additive generalized cohomology functor on $\mathcal{C}$, def. 4. Then there exists a spectrum object $E \in Stab(\mathcal{C})$ such that

1. $H\bullet$ is degreewise represented by $E$:

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

$\delta \colon H^n(-) \simeq Ho(\mathcal{C})(-,E_n) \stackrel{Ho(\mathcal{C})(-,\tilde\sigma_n) }{\longrightarrow} Ho(\mathcal{C})(-,\Omega E_{n+1}) \simeq Ho(\mathcal{C})(\Sigma (-), E_{n+1}) \simeq H^{n+1}(\Sigma(-)) \,.$
###### Proof

Via prop. 14, theorem 3 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 $B U(1)$ plays a key role in the complex oriented cohomology-theory discussed below, and via the equivalence $B U(1) \simeq \mathbb{C}P^\infty$ to the homotopy type of the infinite complex projective space (def. 43), which is the direct limit of finite dimensional projective spaces $\mathbb{C}P^n$, this is an inverse limit of the generalized cohomology groups of the $\mathbb{C}P^n$s. 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.

#### $Lim^1$

###### Definition

Given a tower $A_\bullet$ of abelian groups

$\cdots \to A_3 \stackrel{f_2}{\to} A_2 \stackrel{f_1}{\to} A_1 \stackrel{f_0}{\to} A_0$

write

$\partial \;\colon\; \underset{n}{\prod} A_n \longrightarrow \underset{n}{\prod} A_n$

for the homomorphism given by

$\partial \;\colon\; (a_n)_{n \in \mathbb{N}} \mapsto (a_n - f_n(a_{n+1}))_{n \in \mathbb{N}}.$
###### Remark

The limit of a sequence as in def. 19 – hence the group $\underset{\longleftarrow}{\lim}_n A_n$ universally equipped with morphisms $\underset{\longleftarrow}{\lim}_n A_n \overset{p_n}{\to} A_n$ such that all

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

commute – is equivalently the kernel of the morphism $\partial$ in def. 19.

###### Definition

Given a tower $A_\bullet$ of abelian groups

$\cdots \to A_3 \stackrel{f_2}{\to} A_2 \stackrel{f_1}{\to} A_1 \stackrel{f_0}{\to} A_0$

then $\underset{\longleftarrow}{\lim}^1 A_\bullet$ is the cokernel of the map $\partial$ in def. 19, hence the group that makes a long exact sequence of the form

$0 \to \underset{\longleftarrow}{\lim}_n A_n \longrightarrow \underset{n}{\prod} A_n \stackrel{\partial}{\longrightarrow} \underset{n}{\prod} A_n \longrightarrow \underset{\longleftarrow}{\lim}^1_n A_n \to 0 \,,$
###### Proposition

The functor $\underset{\longleftarrow}{\lim}^1 \colon Ab^{(\mathbb{N}, \geq)} \longrightarrow Ab$ (def. 20) satisfies

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

$0 \to \underset{\longleftarrow}{\lim}_n A_n \to \underset{\longleftarrow}{\lim}_n B_n \to \underset{\longleftarrow}{\lim}_n C_n \to \underset{\longleftarrow}{\lim}_n^1 A_n \to \underset{\longleftarrow}{\lim}_n^1 B_n \to \underset{\longleftarrow}{\lim}_n^1 C_n \to 0$

is a long exact sequence of abelian groups;

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

###### Proof

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

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

for the homomorphism from def. 19 regarded as the single non-trivial differential in a cochain complex of abelian groups. Then by remark 5 and def. 20 we have $H^0(L(A_\bullet)) \simeq \underset{\longleftarrow}{\lim} A_\bullet$ and $H^1(L(A_\bullet)) \simeq \underset{\longleftarrow}{\lim}^1 A_\bullet$.

With this, then for a short exact sequence of towers $0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0$ the long exact sequence in question is the long exact sequence in homology of the corresponding short exact sequence of complexes

$0 \to L^\bullet(A_\bullet) \longrightarrow L^\bullet(B_\bullet) \longrightarrow L^\bullet(C_\bullet) \to 0 \,.$

For the second statement: If all the $f_k$ are surjective, then inspection shows that the homomorphism $\partial$ in def. 19 is surjective. Hence its cokernel vanishes.

###### Lemma

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

###### Proof

The functor $(-)_n \colon Ab^{(\mathbb{N}, \geq)} \to Ab$ that picks the $n$-th component of the tower has a right adjoint $r_n$, which sends an abelian group $A$ to the tower

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

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

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

$A_\bullet \overset{(\eta_n)_{n \in \mathbb{N}}}{\longrightarrow} \underset{n \in \mathbb{R}}{\prod} r_n A_n \hookrightarrow \underset{n \in \mathbb{N}}{\prod} r_n \tilde A_n$

is an injective resolution for $A_\bullet$.

###### Proposition

The functor $\underset{\longleftarrow}{\lim}^1 \colon Ab^{(\mathbb{N}, \geq)} \longrightarrow Ab$ (def. 20) is the first right derived functor of the limit functor $\underset{\longleftarrow}{\lim} \colon Ab^{(\mathbb{N},\geq)} \longrightarrow Ab$.

###### Proof

By lemma 2 there are enough injectives in $Ab^{(\mathbb{N}, \geq)}$. So for $A_\bullet \in Ab^{(\mathbb{N}, \geq)}$ the given tower of abelian groups, let

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

be an injective resolution. We need to show that

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

Since limits preserve kernels, this is equivalently

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

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

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

Therefore by the second item of prop. 15 the long exact sequence from the first item of prop. 15 applied to the short exact sequence

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

becomes

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

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

###### Proposition

The functor $\underset{\longleftarrow}{\lim}^1 \colon Ab^{(\mathbb{N}, \geq)} \longrightarrow Ab$ (def. 20) is in fact the unique functor, up to natural isomorphism, satisfying the conditions in prop. 17.

###### Proof

The proof of prop. 16 only used the conditions from prop. 15, hence any functor satisfying these conditions is the first right derived functor of $\underset{\longleftarrow}{\lim}$, up to natural isomorphism.

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

###### Lemma

Given a cotower

$A_\bullet = (A_0 \overset{f_0}{\to} A _1 \overset{f_1}{\to} A_2 \to \cdots)$

of abelian groups, then for every abelian group $B \in Ab$ there is a short exact sequence of the form

$0 \to \underset{\longleftarrow}{\lim}^1_n Hom(A_n, B) \longrightarrow Ext^1( \underset{\longrightarrow}{\lim}_n A_n, B ) \longrightarrow \underset{\longleftarrow}{\lim}_n Ext^1( A_n, B) \to 0 \,,$

where $Hom(-,-)$ denotes the hom-group, $Ext^1(-,-)$ denotes the first Ext-group (and so $Hom(-,-) = Ext^0(-,-)$).

###### Proof

Consider the homomorphism

$\tilde \partial \;\colon\; \underset{n}{\oplus} A_n \longrightarrow \underset{n}{\oplus} A_n$

which sends $a_n \in A_n$ to $a_n - f_n(a_n)$. Its cokernel is the colimit over the cotower, but its kernel is trivial (in contrast to the otherwise formally dual situation in remark 5). Hence (as opposed to the long exact sequence in def. 20) there is a short exact sequence of the form

$0 \to \underset{n}{\oplus} A_n \overset{\tilde \partial}{\longrightarrow} \underset{n}{\oplus} A_n \overset{}{\longrightarrow} \underset{\longrightarrow}{lim}_n A_n \to 0 \,.$

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

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

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. 19 corresponding to the tower

$Hom(A_\bullet,B) = ( \cdots \to Hom(A_2,B) \to Hom(A_1,B) \to Hom(A_0,B) ) \,.$

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

#### Mittag-Leffler condition

###### Definition

A tower $A_\bullet$ of abelian groups

$\cdots \to A_3 \to A_2 \to A_1 \to A_0$

is said to satify the Mittag-Leffler condition if for all $k$ there exists $i \geq k$ such that for all $j \geq i \geq k$ the image of the homomorphism $A_i \to A_k$ equals that of $A_j \to A_k$

$im(A_i \to A_k) \simeq im(A_j \to A_k) \,.$

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

###### Example

The Mittag-Leffler condition, def. 21, is satisfied in particular when all morphisms $A_{i+1}\to A_i$ are epimorphisms (hence surjections of the underlying sets).

###### Proposition

If a tower $A_\bullet$ satisfies the Mittag-Leffler condition, def. 21, then its $\underset{\leftarrow}{\lim}^1$ vanishes:

$\underset{\longleftarrow}{\lim}^1 A_\bullet = 0 \,.$
###### Proof idea

One needs to show that with the Mittag-Leffler condition, then the cokernel of $\partial$ in def. 19 vanishes, hence that $\partial$ is an epimorphism in this case, hence that every $(a_n)_{n \in \mathbb{N}} \in \underset{n}{\prod} A_n$ has a preimage under $\partial$. So use the Mittag-Leffler condition to find pre-images of $a_n$ by induction over $n$.

#### Mapping telescopes

Given a sequence

$X_\bullet = \left( X_0 \overset{f_0}{\longrightarrow} X_1 \overset{f_1}{\longrightarrow} X_2 \overset{f_2}{\longrightarrow} \cdots \right)$

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

###### Definition

For

$X_\bullet = \left( X_0 \overset{f_0}{\longrightarrow} X_1 \overset{f_1}{\longrightarrow} X_2 \overset{f_2}{\longrightarrow} \cdots \right)$

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

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

where the equivalence relation quotiented out is

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

for all $n\in \mathbb{N}$ and $x_n \in X_n$.

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

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

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

$Tel(X_\bullet) \longrightarrow X$

from the mapping telescope, def. 22, is a weak homotopy equivalence.

###### Proof

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

$Tel(X_n) \longrightarrow X_n$

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, so that elements of homotopy groups $\pi_q(Tel(X))$ are represented at some finite stage $\pi_q(Tel(X_n))$ it follows that

$\underset{\longrightarrow}{\lim}_n \pi_q(Tel(X_n)) \overset{\simeq}{\longrightarrow} \pi_q(Tel(X))$

are isomorphisms for all $q\in \mathbb{N}$ and all choices of basepoints (not shown).

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

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

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

#### Milnor exact sequences

###### Proposition

(Milnor exact sequence for homotopy groups)

Let

$\cdots \to X_3 \overset{p_2}{\longrightarrow} X_2 \overset{p_1}{\longrightarrow} X_1 \overset{p_0}{\longrightarrow} X_0$

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

$0 \to \underset{\longleftarrow}{\lim}^1_i \pi_{q+1}(X_i) \longrightarrow \pi_q(\underset{\longleftarrow}{\lim}_i X_i) \longrightarrow \underset{\longleftarrow}{\lim}_i \pi_q(X_i) \to 0 \,,$

for $\pi_\bullet$ the homotopy group-functor (exact as pointed sets for $i = 0$, as groups for $i \geq 1$) 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 $\underset{\longleftarrow}{\lim}^1$ (def. 20) 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.

###### Proof

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 $[(\mathbb{N},\geq), sSet]_{inj}$ ($[(\mathbb{N},\geq), Top]_{inj}$) (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

$\array{ holim X_\bullet &\longrightarrow& \underset{n}{\prod} Path(X_n) \\ \downarrow &(pb)& \downarrow \\ \underset{n}{\prod} X_n &\underset{(id,p_n)_n}{\longrightarrow}& \underset{n}{\prod} X_ n \times X_n } \,,$

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

$\underset{n}{\prod} \Omega X_n \longrightarrow holim X_\bullet \longrightarrow \underset{n}{\prod} X_n \,.$

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

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

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

###### Proposition

(Milnor exact sequence for generalized cohomology)

Let $X$ be a pointed CW-complex, $X = \underset{\longrightarrow}{\lim}_n X_n$ and let $\tilde E^\bullet$ an additive reduced cohomology theory, def. 1.

Then the canonical morphisms make a short exact sequence

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

saying that

1. the failure of the canonical comparison map $\tilde E^\bullet(X) \to \underset{\longleftarrow}{\lim} \tilde E^\bullet(X_n)$ 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 $\lim^1$ (def. 20) of the cohomology groups at the finite stages in one degree lower.

###### Proof

For

$X_\bullet = \left( X_0 \overset{i_0}{\hookrightarrow} X_1 \overset{i_1}{\hookrightarrow} X_2 \overset{i_1}{\hookrightarrow} \cdots \right)$

the sequence of stages of the (pointed) CW-complex $X = \underset{\longleftarrow}{\lim}_n X_n$, write

\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 $Tel(X_\bullet)$ (def.), which we denote by

$\array{ A_X && && B_X \\ & {}_{\mathllap{\iota_{A_x}}}\searrow && \swarrow_{\mathrlap{\iota_{B_x}}} \\ && Tel(X_\bullet) } \,.$

Observe that

1. $A_X \cup B_X \simeq Tel(X_\bullet)$;

2. $A_X \cap B_X \simeq \underset{n \in \mathbb{N}}{\sqcup} X_n$;

and that there are homotopy equivalences

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

2. $B_X \simeq \underset{n \in \mathbb{N}}{\sqcup} X_{2n}$

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

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

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

$\array{ \tilde E^{\bullet-1}(A_X)\oplus \tilde E^{\bullet-1}(B_X) &\longrightarrow& \tilde E^{\bullet-1}(A_X \cap B_X) &\longrightarrow& \tilde E^\bullet(X) &\overset{(\iota_{A_x})^\ast - (\iota_{B_x})^\ast}{\longrightarrow}& \tilde E^\bullet(A_X)\oplus \tilde E^\bullet(B_X) &\overset{}{\longrightarrow}& \tilde E^\bullet(A_X \cap B_X) \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{=}} && {}^{\mathllap{(id, -id)}}\downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ \underset{n}{\prod}\tilde E^{\bullet-1}(X_n) &\underset{\partial}{\longrightarrow}& \underset{n}{\prod}\tilde E^{\bullet-1}(X_n) &\longrightarrow& \tilde E^\bullet(X) &\overset{(i_n^\ast)_{n}}{\longrightarrow}& \underset{n}{\prod}\tilde E^\bullet(X_n) &\underset{\partial}{\longrightarrow}& \underset{n}{\prod}\tilde E^\bullet(X_n) } \,,$

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

To identify the morphism $\partial$, notice that it comes from pulling back $E$-cohomology classes along the inclusions $A \cap B \to A$ and $A\cap B \to B$. Comonentwise these are the inclusions of each $X_n$ 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 $i_n \colon X_n \hookrightarrow X_{n+1}$ into the cylinder over $X_{n+1}$. In conclusion, $\partial$ acts by

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

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

This is the morphism from def. 19 for the sequence

$\cdots \to \tilde E^\bullet(X_{n+1}) \overset{i_n^\ast}{\longrightarrow} \tilde E^\bullet(X_n) \overset{i_n^\ast}{\longrightarrow} \tilde E^{\bullet}(X_{n-1}) \to \cdots \,.$

Hence truncating the above long exact sequence by forming kernel and cokernel of $\partial$, the result follows via remark 5 and definition 20.

In contrast:

###### Proposition

Let $X$ be a pointed CW-complex, $X = \underset{\longleftarrow}{\lim}_n X_n$.

For $\tilde E_\bullet$ an additive reduced generalized homology theory, then

$\underset{\longrightarrow}{\lim}_n \tilde E_\bullet(X_n) \overset{\simeq}{\longrightarrow} \tilde E_\bullet(X)$

is an isomorphism.

There is also a version for cohomology of spectra:

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

\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 the stable model structure on orthogonal spectra, in that its localization at the stable weak homotopy equivalences is of the form

$\gamma \;\colon\; OrthSpec(Top_{cg})_{stable} \longrightarrow Ho(Spectra) \,.$

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

###### Proposition

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

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

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

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

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

###### Corollary

For $X,E \in Ho(Spectra)$ two spectra such that the tower $n \mapsto E^{n -1}(X_{n})$ satisfies the Mittag-Leffler condition (def. 21), then two morphisms of spectra $X \longrightarrow E$ are homotopic already if all their morphisms of component spaces $X_n \to E_n$ are.

###### Proof

By prop. 18 the assumption implies that the $lim^1$-term in prop. 22 vanishes, hence by exactness it follows that in this case there is an isomorphism

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

### 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 $E$, there is a spectral sequence known as the Atiyah-Hirzebruch spectral sequence (AHSS) which serves to compute $E$-cohomology of $F$-fiber bundles over a simplicial complex $X$ in terms of ordinary cohomology with coefficients in the generalized cohomology $E^\bullet(F)$ of the fiber. For $E =$ 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)

#### Converging spectral sequences

###### Definition

A cohomology spectral sequence $\{E_r^{p,q}, d_r\}$ is

1. a sequence $\{E_r^{\bullet,\bullet}\}$ (for $r \in \mathbb{N}$, $r \geq 1$) of bigraded abelian groups (the “pages”);

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

$\{d_r \;\colon\; E_r^{\bullet,\bullet} \longrightarrow E_r^{\bullet+r, \bullet-r+1}\}$

such that

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

Given a $\mathbb{Z}$-graded abelian group_ $C^\bullet$ equipped with a decreasing filtration

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

such that

$C^\bullet = \underset{s}{\cup} F^s C^\bullet \;\;\;\; and \;\;\;\; 0 = \underset{s}{\cap} F^s C^\bullet$

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

$E_2^{\bullet,\bullet} \Rightarrow C^\bullet$

if

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

$E_\infty^{s,t} \coloneqq E_{\gg 1}^{s,t}$;

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

$E_\infty^{s,t} \simeq F^s C^{s+t} / F^{s+1}C^{s+t}$.

The converging spectral sequence is called a multiplicative spectral sequence if

1. $\{E_2^{\bullet,\bullet}\}$ is equipped with the structure of a bigraded algebra;

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

such that

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

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

###### Remark

The point of spectral sequences is that by subdividing the data in any graded abelian group $C^\bullet$ 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 $C^\bullet$. 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.

###### Example

The archetypical example of (co-)homology spectral sequences as in def. 24 are induced from a 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 Interlude -- Spectral sequences the section For filtered complexes.

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

###### Definition

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

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

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

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

Given an exact couple, then its derived exact couple is

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

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

###### Proposition

(cohomological spectral sequence of an exact couple)

Given an exact couple, def. 24,

$\array{ D_1 && \stackrel{g_1}{\longrightarrow} && D_1 \\ & {}_{\mathllap{f_1}}\nwarrow && \swarrow_{\mathrlap{h_1}} \\ && E_1 }$

its derived exact couple

$\array{ D_2 && \stackrel{g_2}{\longrightarrow} && D_2 \\ & {}_{\mathllap{f_2}}\nwarrow && \swarrow_{\mathrlap{h_2}} \\ && E_2 }$

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

$\array{ D_r && \stackrel{g_r}{\longrightarrow} && D_r \\ & {}_{\mathllap{f_r}}\nwarrow && \swarrow_{\mathrlap{h_r}} \\ && E_r } \,.$

If the abelian groups $D$ and $E$ are equipped with bigrading such that

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

then $\{E_r^{\bullet,\bullet}, d_r\}$ with

\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. 24.

(As before in prop. 23, the notation $g^{-n}$ with $n \in \mathbb{N}$ denotes the function given by choosing, on representatives, a preimage under $g^n = \underset{n\;times}{\underbrace{g \circ \cdots \circ g \circ g}}$, with the implicit claim that all possible choices represent the same equivalence class.)

If for every bidegree $(s,t)$ there exists $R_{s,t} \gg 1$ such that for all $r \geq R_{s,t}$

1. $g \colon D^{s+R,t-R} \stackrel {\simeq}{\longrightarrow} D^{s+R -1, t-R-1}$;

2. $g\colon D^{s-R+1, t+R-2} \stackrel{0}{\longrightarrow} D^{s-R,t+R-1}$

then this spectral sequence converges to the inverse limit group

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

filtered by

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

(e.g. Kochmann 96, lemma 2.6.2)

###### Proof

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

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

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

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

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

By assumption there is for each $(s,t)$ an $R_{s,t}$ such that for all $r \geq R_{s,t}$ then $ker(d_{r-1})^{s,t}$ is independent of $r$.

Moreover, $im(d_{r-1})$ consists of the image under $h$ of those $x \in D^{s-1,t}$ such that $g^{r-2}(x)$ is in the image of $f$, hence (since $im(f) = ker(g)$ by exactness of the exact couple) such that $g^{r-2}(x)$ is in the kernel of $g$, hence such that $x$ is in the kernel of $g^{r-1}$. If $r \gt R$ then by assumption $g^{r-1}|_{D^{s-1,t}} = 0$ and so then $im(d_{r-1}) = im(h)$.

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

It follows that

\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 $E_\infty$-page is a sub-group of the image of the $E_1$-page under $f$.)

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

$E_\infty^{p,n-p} \simeq F^p G^n / F^{p+1} G^n \,.$
###### Remark

Given a spectral sequence (def. 24), 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 $F^\bullet H^\bullet$

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

Then by definition of convergence there are isomorphisms

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

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

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

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

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

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

But in general this need not be the case.

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

The following proposition requires, in general, to evaluate cohomology functors not just on CW-complexes, but on all topological spaces. Hence we invoke prop. 1 to regard a reduced cohomology theory as a contravariant functor on all pointed topological spaces, which sends weak homotopy equivalences to isomorphisms (def. 3).

###### Proposition

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

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

If at least one of the following two conditions is met

• $B$ is finite-dimensional as a CW-complex;

• $A^\bullet(F)$ is bounded below in degree and the sequences $\cdots \to A^p(X_{n+1}) \to A^p(X_n) \to \cdots$ satisfy the Mittag-Leffler condition (def. 21) for all $p$;

then there is a cohomology spectral sequence, def. 24, whose $E_2$-page is the ordinary cohomology $H^\bullet(B,A^\bullet(F))$ of $B$ with coefficients in the $A$-cohomology groups $A^\bullet(F)$ of the fiber, and which converges to the $A$-cohomology groups of the total space

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

with respect to the filtering given by

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

where $X_{p} \coloneqq \pi^{-1}(B_{p})$ is the fiber over the $p$th stage of the CW-complex $B = \underset{\longleftarrow}{\lim}_n B_n$.

###### Proof

The exactness axiom for $A$ gives an exact couple, def. 24, of the form

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

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

In order to determine the $E_2$-page, we analyze the $E_1$-page: By definition

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

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

$(\pi^{-1}(\sigma), \pi^{-1}(\partial \sigma)) \simeq (D^n, S^{n-1}) \times F_\sigma \,,$

where $F_\sigma$ is weakly homotopy equivalent to $F$ (exmpl.).

This implies that

\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 $\tilde A$, prop. 10 together with lemma 1, then the wedge axiom and the suspension isomorphism of the latter.

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

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

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

Consider the following diagram, which commutes due to the naturality of the connecting homomorphism $\delta$ of $A^\bullet$:

$\array{ \partial^\ast \colon & C^{s-1}_{cell}(X,A^t(\ast)) & =& \underset{i \in I_{s-1}}{\prod} A^t(\ast) && \longrightarrow && \underset{i \in I_s}{\prod} A^t(\ast) & = & C_{cell}^{s}(X,A^t(\ast)) \\ && & {}^{\mathllap{\simeq}}\downarrow && && \downarrow^{\mathrlap{\simeq}} \\ && & \underset{i \in I_{s-1}}{\prod} \tilde A^{s+t-1}(S^{s-1}) && && \underset{i \in I_s}{\prod} \tilde A^{s+t}(S^{s}) \\ && & {}^{\mathllap{\simeq}}\downarrow && && \downarrow^{\mathrlap{\simeq}} \\ && d_1 \colon & A^{s+t-1}(X_{s-1}, X_{s-2}) &\overset{}{\longrightarrow}& A^{s+t-1}(X_{s-1}) &\overset{\delta}{\longrightarrow}& A^{s+t}(X_s, X_{s-1}) \\ && & \downarrow && \downarrow && \downarrow \\ && & A^{s+t-1}(S^{s-1}, \emptyset) &\overset{}{\longrightarrow}& A^{s+t-1}(S^{s-1}) &\overset{\delta}{\longrightarrow}& A^{s+t}(D^s , S^{s-1}) } \,.$

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

The differential $d_1$ 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 $X$ with coefficients in $A^\bullet(\ast)$ (def.). This cellular cohomology coincides with singular cohomology of the CW-complex $X$ (thm.), hence computes the ordinary cohomology of $X$.

Now to see the convergence. If $B$ is finite dimensional then the convergence condition as stated in prop. 23 is met. Alternatively, if $A^\bullet(F)$ is bounded below in degree, then by the above analysis the $E_1$-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. 23 to obtain convergence.

By that proposition the convergence is to the inverse limit

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

If $X$ is finite dimensional or more generally if the sequences that this limit is over satisfy the Mittag-Leffler condition (def. 21), then this limit is $A^\bullet(X)$, by prop. 18.

#### Multiplicative structure

###### Proposition

For $E^\bullet$ a multiplicative cohomology theory (def. 14), then the Atiyah-Hirzebruch spectral sequences (prop. 24) for $E^\bullet(X)$ 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.

###### Proposition

Given a multiplicative cohomology theory $(A,\mu,1)$ (def. 14), then for every Serre fibration $X \to B$ (def.) all the differentials in the corresponding Atiyah-Hirzebruch spectral sequence of prop. 24

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

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

###### Proof

By the proof of prop. 24, the differentials are those induced by the exact couple

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

consisting of the pullback homomorphisms and the connecting homomorphisms of $A$.

By prop. 23 its differentials on page $r$ are the composites of one pullback homomorphism, the preimage of $(r-1)$ pullback homomorphisms, and one connecting homomorphism of $A$. Hence the statement follows with prop. 8.

###### Proposition

For $E$ a homotopy commutative ring spectrum (def.) and $X$ a finite CW-complex, then the Kronecker pairing

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

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

## 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).

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 $G$-Structure

Idea. Every manifold $X$ of dimension $n$ carries a canonical vector bundle of rank $n$: its tangent bundle. There is a universal vector bundle of rank $n$, 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 $B GL(n) \simeq B O(n)$ (for $O(n)$ 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 $X$ is classified by a map $X \longrightarrow B O(n)$, unique up to homotopy. For $G$ a subgroup of $O(n)$, then a lift of this map through the canonical map $B G \longrightarrow B O(n)$ of classifying spaces is a G-structure on $X$

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

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

###### Proposition

For $X$ a smooth manifold and $G$ a compact Lie group equipped with a free smooth action on $X$, then the quotient projection

$X \longrightarrow X/G$

is a $G$-principal bundle (hence in particular a Serre fibration).

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

###### Corollary

For $G$ a Lie group and $H \subset G$ a compact subgroup, then the coset quotient projection

$G \longrightarrow G/H$

is an $H$-principal bundle (hence in particular a Serre fibration).

###### Proposition

For $G$ a compact Lie group and $K \subset H \subset G$ closed subgroups, then the projection map on coset spaces

$p \;\colon\; G/K \longrightarrow G/H$

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

###### Proof

Observe that the projection map in question is equivalently

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

(where on the left we form the Cartesian product and then divide out the diagonal action by $H$). This exhibits it as the $H/K$-fiber bundle associated to the $H$-principal bundle of corollary 3.

#### Orthogonal and Unitary groups

###### Proposition

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

###### Proposition

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

###### Example

The n-spheres are coset spaces of orthogonal groups:

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

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

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

Regarding the first statement:

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

The second statement follows by the same kind of reasoning:

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

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

where $A \in U(n)$.

###### Proposition

For $n,k \in \mathbb{N}$, $n \leq k$, then the canonical inclusion of orthogonal groups

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

is an (n-1)-equivalence, hence induces an isomorphism on homotopy groups in degrees $\lt n-1$ and a surjection in degree $n-1$.

###### Proof

Consider the coset quotient projection

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

By prop. 30 and by corollary 3, the projection $O(n+1)\to O(n+1)/O(n)$ is a Serre fibration. Furthermore, example 8 identifies the coset with the n-sphere

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

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

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

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

$\pi_{\lt n-1}(O(n)) \overset{\simeq}{\longrightarrow} \pi_{\lt n-1}(O(n+1))$

is an isomorphism and that

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

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

Similarly:

###### Proposition

For $n,k \in \mathbb{N}$, $n \leq k$, then the canonical inclusion of unitary groups

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

is a 2n-equivalence, hence induces an isomorphism on homotopy groups in degrees $\lt 2n$ and a surjection in degree $2n$.

###### Proof

Consider the coset quotient projection

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

By prop. 31 and corollary 3, the projection $U(n+1)\to U(n+1)/U(n)$ is a Serre fibration. Furthermore, example 8 identifies the coset with the (2n+1)-sphere

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

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

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

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

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

is an isomorphism and that

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

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

#### Stiefel manifolds and Grassmannians

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

###### Definition

For $n, k \in \mathbb{N}$ and $n \leq k$, then the $n$th real Stiefel manifold of $\mathbb{R}^k$ is the coset topological space.

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

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

Similarly the $n$th complex Stiefel manifold of $\mathbb{C}^k$ is

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

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

###### Definition

For $n, k \in \mathbb{N}$ and $n \leq k$, then the $n$th real Grassmannian of $\mathbb{R}^k$ is the coset topological space.

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

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

Similarly the $n$th complex Grassmannian of $\mathbb{C}^k$ is the coset topological space.

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

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

###### Example
• $G_1(\mathbb{R}^{n+1}) \simeq \mathbb{R}P^n$ is real projective space of dimension $n$.

• $G_1(\mathbb{C}^{n+1}) \simeq \mathbb{C}P^n$ is complex projective space of dimension $n$ (def. 43).

###### Proposition

For all $n \leq k \in \mathbb{N}$, the canonical projection from the Stiefel manifold (def. 25) to the Grassmannian is a $O(n)$-principal bundle

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

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

$\array{ U(n) &\hookrightarrow& V_n(\mathbb{C}^k) \\ && \downarrow \\ && Gr_n(\mathbb{C}^k) } \,.$
###### Proof

By prop 3 and prop 29.

###### Proposition

The real Grassmannians $Gr_n(\mathbb{R}^k)$ and the complex Grassmannians $Gr_n(\mathbb{C}^k)$ of def. 26 admit the structure of CW-complexes. Moreover the canonical inclusions

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

are subcomplex incusion (hence relative cell complex inclusions).

Accordingly there is an induced CW-complex structure on the classifying space (def. 27).

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

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

###### Proposition

The Stiefel manifolds $V_n(\mathbb{R}^k)$ and $V_n(\mathbb{C}^k)$ from def. 25 admits the structure of a CW-complex.

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

###### Proposition

The real Stiefel manifold $V_n(\mathbb{R}^k)$ (def. 25) is (k-n-1)-connected.

###### Proof

Consider the coset quotient projection

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

By prop. 30 and by corollary 3, the projection $O(k)\to O(k)/O(k-n)$ is a Serre fibration. Therefore there is induced the long exact sequence of homotopy groups of this fiber sequence, and by prop. 32 it has the following form in degrees bounded by $n$:

$\cdots \to \pi_{\bullet \leq k-n-1}(O(k-n)) \overset{epi}{\longrightarrow} \pi_{\bullet \leq k-n-1}(O(k)) \overset{0}{\longrightarrow} \pi_{\bullet \leq k-n-1}(V_n(\mathbb{R}^k)) \overset{0}{\longrightarrow} \pi_{\bullet-1 \lt k-n-1}(O(k)) \overset{\simeq}{\longrightarrow} \pi_{\bullet-1 \lt k-n-1}(O(k-n)) \to \cdots \,.$

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

Similarly:

###### Proposition

The complex Stiefel manifold $V_n(\mathbb{C}^k)$ (def. 25) is 2(k-n)-connected.

###### Proof

Consider the coset quotient projection

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

By prop. 31 and by corollary 3 the projection $U(k)\to U(k)/U(k-n)$ is a Serre fibration. Therefore there is induced the long exact sequence of homotopy groups of this fiber sequence, and by prop. 33 it has the following form in degrees bounded by $n$:

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

This implies the claim.

#### Classifying spaces

###### Definition

By def. 26 there are canonical inclusions

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

and

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

for all $k \in \mathbb{N}$. The colimit (in Top, see there, or rather in $Top_{cg}$, see this cor.) over these inclusions is denoted

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

and

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

respectively.

Moreover, by def. 25 there are canonical inclusions

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

and

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

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

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

and

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

respectively.

The inclusions are in fact compatible with the bundle structure from prop. 34, so that there are induced projections

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

and

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

respectively. These are the standard models for the universal principal bundles for $O$ and $U$, respectively. The corresponding associated vector bundles

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

and

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

are the corresponding universal vector bundles.

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

\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 $E U(n)$.

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

###### Definition

There are canonical inclusions

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

and

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

given by adjoining one coordinate to the ambient space and to any subspace. Under the colimit of def. 27 these induce maps of classifying spaces

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

and

$B U(n) \longrightarrow B U(n+1) \,.$
###### Definition

There are canonical maps

$Gr_{n_1}(\mathbb{R}^{k_1}) \times Gr_{n_2}(\mathbb{R}^{k_2}) \longrightarrow Gr_{n_1 + n_2}(\mathbb{R}^{k_1 + k_2})$

and

$Gr_{n_1}(\mathbb{C}^{k_1}) \times Gr_{n_2}(\mathbb{C}^{k_2}) \longrightarrow Gr_{n_1 + n_2}(\mathbb{C}^{k_1 + k_2})$

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

Under the colimit of def. 27 these induce maps of classifying spaces

$B O(n_1) \times B O(n_2) \longrightarrow B O(n_1 + n_2)$

and

$B U(n_1) \times B U(n_2) \longrightarrow B U(n_1 + n_2)$
###### Proposition

The colimiting space $E O(n) = \underset{\longrightarrow}{\lim}_k V_n(\mathbb{R}^k)$ from def. 27 is weakly contractible.

The colimiting space $E U(n) = \underset{\longrightarrow}{\lim}_k V_n(\mathbb{C}^k)$ from def. 27 is weakly contractible.

###### Proof

By propositions 37, and 38, the Stiefel manifolds are more and more highly connected as $k$ increases. Since the inclusions are relative cell complex inclusions by prop. 36, the claim follows.

###### Proposition

The homotopy groups of the classifying spaces $B O(n)$ and $B U(n)$ (def. 27) are those of the orthogonal group $O(n)$ and of the unitary group $U(n)$, respectively, shifted up in degree: there are isomorphisms

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

and

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

(for homotopy groups based at the canonical basepoint).

###### Proof

Consider the sequence

$O(n) \longrightarrow E O(n) \longrightarrow B O(n)$

from def. 27, with $O(n)$ the fiber. Since (by prop. 29) 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

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

Since by cor. 39 $\pi_\bullet(E O(n))= 0$, exactness of the sequence implies that

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

is an isomorphism.

The same kind of argument applies to the complex case.

###### Proposition

For $n \in \mathbb{N}$ there are homotopy fiber sequence (def.)

$S^n \longrightarrow B O(n) \longrightarrow B O(n+1)$

and

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

exhibiting the n-sphere ($(2n+1)$-sphere) as the homotopy fiber of the canonical maps from def. 28.

This means (thm.), that there is a replacement of the canonical inclusion $B O(n) \hookrightarrow B O(n+1)$ (induced via def. 27) by a Serre fibration

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

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

###### Proof

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

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

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

By prop. 29 both bottom vertical maps are Serre fibrations and so both vertical sequences are fiber sequences. By prop. 40 part of the induced morphisms of long exact sequences of homotopy groups looks like this

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

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

That $B O(n)\to \tilde B O(n+1)$ is indeed a Serre fibration follows again with prop. 29, which gives the fiber sequence

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

The claim then follows with the identification

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

of example 8.

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

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

from example 8.

#### $G$-Structure on the Stable normal bundle

###### Definition

Given a smooth manifold $X$ of dimension $n$ and equipped with an embedding

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

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

$g_i \;\colon\; X \to Gr_{k-n}(\mathbb{R}^k) \hookrightarrow B O(k-n)$

which sends $x \in X$ to the normal of the tangent space

$N_x X = (T_x X)^{\perp} \hookrightarrow \mathbb{R}^k$

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

The normal bundle of $i$ itself is the subbundle of the tangent bundle

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

consisting of those vectors which are orthogonal to the tangent vectors of $X$:

$N_i \coloneqq \left\{ x\in X, v \in T_{i(x)}\mathbb{R}^k \;\vert\; v \,\perp\, i_\ast T_x X \subset T_{i(x)}\mathbb{R}^k \right\} \,.$
###### Definition

A $(B,f)$-structure is

1. for each $n\in \mathbb{N}$ a pointed CW-complex $B_n \in Top_{CW}^{\ast/}$

2. equipped with a pointed Serre fibration

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

to the classifying space $B O(n)$ (def.);

3. for all $n_1 \leq n_2$ a pointed continuous function

$g_{n_1, n_2} \;\colon\; B_{n_1} \longrightarrow B_{n_2}$

which is the identity for $n_1 = n_2$;

such that for all $n_1 \leq n_2 \in \mathbb{N}$ these squares commute

$\array{ B_{n_1} &\overset{g_{n_1,n_2}}{\longrightarrow}& B_{n_2} \\ {}^{\mathllap{f_{n_1}}}\downarrow && \downarrow^{\mathrlap{f_{n_2}}} \\ B O(n_1) &\longrightarrow& B O(n_2) } \,,$

where the bottom map is the canonical one from def. 28.

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

$\array{ B_{n_1} \times B_{n_2} &\overset{\mu_{n_1, n_2}}{\longrightarrow}& B_{n_1 + n_2} \\ {}^{\mathllap{f_{n_1} \times f_{n_2}}}\downarrow && \downarrow^{\mathrlap{f_{n_1 + n_2}}} \\ B O(n_1) \times B O(n_2) &\longrightarrow& B O(n_1 + n_2) }$

and which satisfy the evident associativity and unitality, for $B_0 = \ast$ the unit, and, finally, which commute with the maps $g$ in that all $n_1,n_2, n_3 \in \mathbb{N}$ these squares commute:

$\array{ B_{n_1} \times B_{n_2} &\overset{id \times g_{n_2,n_2+n_3}}{\longrightarrow}& B_{n_1} \times B_{n_2 + n_3} \\ {}^{\mathllap{\mu_{n_1, n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1,n_2 + n_3}}} \\ B_{n_1 + n_2} &\underset{g_{n_1+n_2, n_1 + n_2 + n_3}}{\longrightarrow}& B_{n_1 + n_2 + n_3} }$

and

$\array{ B_{n_1} \times B_{n_2} &\overset{g_{n_1,n_1+n_3} \times id}{\longrightarrow}& B_{n_1+n_3} \times B_{n_2 } \\ {}^{\mathllap{\mu_{n_1, n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1 + n_3 , n_2}}} \\ B_{n_1 + n_2} &\underset{g_{n_1+n_2, n_1 + n_2 + n_3}}{\longrightarrow}& B_{n_1 + n_2 + n_3} } \,.$

Similarly, an $S^2$-$(B,f)$-structure is a compatible system

$f_{2n} \colon B_{2n} \longrightarrow B O(2n)$

indexed only on the even natural numbers.

Generally, an $S^k$-$(B,f)$-structure for $k \in \mathbb{N}$, $k \geq 1$ is a compatible system

$f_{k n} \colon B_{k n} \longrightarrow B O(k n)$

for all $n \in \mathbb{N}$, hence for all $k n \in k \mathbb{N}$.

###### Example

Examples of $(B,f)$-structures (def. 31) include the following:

1. $B_n = B O(n)$ and $f_n = id$ is orthogonal structure (or “no structure”);

2. $B_n = E O(n)$ and $f_n$ the universal principal bundle-projection is framing-structure;

3. $B_n = B SO(n) = E O(n)/SO(n)$ the classifying space of the special orthogonal group and $f_n$ the canonical projection is orientation structure;

4. $B_n = B Spin(n) = E O(n)/Spin(n)$ the classifying space of the spin group and $f_n$ the canonical projection is spin structure.

Examples of $S^2$-$(B,f)$-structures (def. 31) include

1. $B_{2n} = B U(n) = E O(2n)/U(n)$ the classifying space of the unitary group, and $f_{2n}$ the canonical projection is almost complex structure (or rather: almost Hermitian structure).

2. $B_{2n} = B Sp(2n) = E O(2n)/Sp(2n)$ the classifying space of the symplectic group, and $f_{2n}$ the canonical projection is almost symplectic structure.

Examples of $S^4$-$(B,f)$-structures (def. 31) include

1. $B_{4n} = B U_{\mathbb{H}}(n) = E O(4n)/U_{\mathbb{H}}(n)$ the classifying space of the quaternionic unitary group, and $f_{4n}$ the canonical projection is almost quaternionic structure.
###### Definition

Given a smooth manifold $X$ of dimension $n$, and given a $(B,f)$-structure as in def. 31, then a $(B,f)$-structure on the stable normal bundle of the manifold is an equivalence class of the following structure:

1. an embedding $i_X \; \colon \; X \hookrightarrow \mathbb{R}^k$ for some $k \in \mathbb{N}$;

2. a homotopy class of a lift $\hat g$ of the classifying map $g$ of the normal bundle (def. 30)

$\array{ && B_{k-n} \\ &{}^{\mathllap{\hat g}}\nearrow& \downarrow^{\mathrlap{f_{k-n}}} \\ X &\overset{g}{\longrightarrow}& B O(k-n) } \,.$

The equivalence relation on such structures is to be that generated by the relation $((i_{X})_1, \hat g_1) \sim ((i_{X})_,\hat g_2)$ if

1. $k_2 \geq k_1$

2. the second inclusion factors through the first as

$(i_X)_2 \;\colon\; X \overset{(i_X)_1}{\hookrightarrow} \mathbb{R}^{k_1} \hookrightarrow \mathbb{R}^{k_2}$
3. the lift of the classifying map factors accordingly (as homotopy classes)

$\hat g_2 \;\colon\; X \overset{\hat g_1}{\longrightarrow} B_{k_1-n} \overset{g_{k_1-n, k_2-n}}{\longrightarrow} B_{k_2-n} \,.$

### Thom spectra

Idea. Given a vector bundle $V$ of rank $n$ over a compact topological space, then its one-point compactification is equivalently the result of forming the bundle $D(V) \hookrightarrow V$ of unit n-balls, and identifying with one single point all the boundary unit n-spheres $S(V)\hookrightarrow V$. Generally, this construction $Th(C) \coloneqq D(V)/S(V)$ is called the Thom space of $V$.

Thom spaces occur notably as codomains for would-be left inverses of embeddings of manifolds $X \hookrightarrow Y$. The Pontrjagin-Thom collapse map $Y \to Th(N X)$ of such an embedding is a continuous function going the other way around, but landing not quite in $X$ but in the Thom space of the normal bundle of $X$ in $Y$. 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 $B O(k)$, denoted $M O(k)$. In particular in the case that $Y = S^n$ is an n-sphere (and every manifold embeds into a large enough $n$-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 $M O(k)$.

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

###### Definition

Let $X$ be a topological space and let $V \to X$ be a vector bundle over $X$ of rank $n$, which is associated to an O(n)-principal bundle. Equivalently this means that $V \to X$ is the pullback of the universal vector bundle $E_n \to B O(n)$ (def. 27) over the classifying space. Since $O(n)$ preserves the metric on $\mathbb{R}^n$, by definition, such $V$ inherits the structure of a metric space-fiber bundle. With respect to this structure:

1. the unit disk bundle $D(V) \to X$ is the subbundle of elements of norm $\leq 1$;

2. the unit sphere bundle $S(V)\to X$ is the subbundle of elements of norm $= 1$;

$S(V) \overset{i_V}{\hookrightarrow} D(V) \hookrightarrow V$;

3. the Thom space $Th(V)$ is the cofiber (formed in Top (prop.)) of $i_V$

$Th(V) \coloneqq cofib(i_V)$

canonically regarded as a pointed topological space.

$\array{ S(V) &\overset{i_V}{\longrightarrow}& D(V) \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& Th(V) } \,.$

If $V \to X$ is a general real vector bundle, then there exists an isomorphism to an $O(n)$-associated bundle and the Thom space of $V$ is, up to based homeomorphism, that of this orthogonal bundle.

###### Remark

If the rank of $V$ is positive, then $S(V)$ is non-empty and then the Thom space (def. 33) is the quotient topological space

$Th(V) \simeq D(V)/S(V) \,.$

However, in the degenerate case that the rank of $V$ vanishes, hence the case that $V = X\times \mathbb{R}^0 \simeq X$, then $D(V) \simeq V \simeq X$, but $S(V) = \emptyset$. Hence now the pushout defining the cofiber is

$\array{ \emptyset &\overset{i_V}{\longrightarrow}& X \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& Th(V) \simeq X_* } \,,$

which exhibits $Th(V)$ as the coproduct of $X$ with the point, hence as $X$ with a basepoint freely adjoined.

$Th(X \times \mathbb{R}^0) = Th(X) \simeq X_+ \,.$
###### Proposition

Let $V \to X$ be a vector bundle over a CW-complex $X$. Then the Thom space $Th(V)$ (def. 33) is equivalently the homotopy cofiber (def.) of the inclusion $S(V) \longrightarrow D(V)$ of the sphere bundle into the disk bundle.

###### Proof

The Thom space is defined as the ordinary cofiber of $S(V)\to D(V)$. 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:

###### Proposition

Let $V \to X$ be a vector bundle over a CW-complex $X$. Write $V-X$ for the complement of its 0-section. Then the Thom space $Th(V)$ (def. 33) is homotopy equivalent to the mapping cone of the inclusion $(V-X) \hookrightarrow V$ (hence to the pair $(V,V-X)$ in the language of generalized (Eilenberg-Steenrod) cohomology).

###### Proof

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:

$\array{ V-X &\longrightarrow& V \\ {}^{\mathllap{\in W_{cl}}}\downarrow && \downarrow^{\mathrlap{\in W_{cl}}} \\ S(V) &\hookrightarrow& D(V) } \,.$
###### Proposition

Let $V_1,V_2 \to X$ be two real vector bundles. Then the Thom space (def. 33) of the direct sum of vector bundles $V_1 \oplus V_2 \to X$ is expressed in terms of the Thom space of the pullbacks $V_2|_{D(V_1)}$ and $V_2|_{S(V_1)}$ of $V_2$ to the disk/sphere bundle of $V_1$ as

$Th(V_1 \oplus V_2) \simeq Th(V_2|_{D(V_1)})/Th(V_2|_{S(V_1)}) \,.$
###### Proof

Notice that

1. $D(V_1 \oplus V_2) \simeq D(V_2|_{Int D(V_1)}) \cup S(V_1)$;

2. $S(V_1 \oplus V_2) \simeq S(V_2|_{Int D(V_1)}) \cup Int D(V_2|_{S(V_1)})$.

(Since a point at radius $r$ in $V_1 \oplus V_2$ is a point of radius $r_1 \leq r$ in $V_2$ and a point of radius $\sqrt{r^2 - r_1^2}$ in $V_1$.)

###### Proposition

For $V$ a vector bundle then the Thom space (def. 33) of $\mathbb{R}^n \oplus V$, the direct sum of vector bundles with the trivial rank $n$ vector bundle, is homeomorphic to the smash product of the Thom space of $V$ with the $n$-sphere (the $n$-fold reduced suspension).

$Th(\mathbb{R}^n \oplus V) \simeq S^n \wedge Th(V) = \Sigma^n Th(V) \,.$
###### Proof

Apply prop. 44 with $V_1 = \mathbb{R}^n$ and $V_2 = V$. Since $V_1$ is a trivial bundle, then

$V_2|_{D(V_1)} \simeq V_2\times D^n$

(as a bundle over $X\times D^n$) and similarly

$V_2|_{S(V_1)} \simeq V_2\times S^n \,.$
###### Example

By prop. 45 and remark 7 the Thom space (def. 33) of a trivial vector bundle of rank $n$ is the $n$-fold suspension of the base space

\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. 54 below) for a given Thom space is a twisted version of the suspension isomorphism (above).

###### Proposition

For $V_1 \to X_1$ and $V_2 \to X_2$ to vector bundles, let $V_1 \boxtimes V_2 \to X_1 \times X_2$ be the direct sum of vector bundles of their pullbacks to $X_1 \times X_2$. The corresponding Thom space (def. 33) is the smash product of the individual Thom spaces:

$Th(V_1 \boxtimes V_2) \simeq Th(V_1) \wedge Th(V_2) \,.$
###### Remark

Given a vector bundle $V \to X$ of rank $n$, then the reduced ordinary cohomology of its Thom space $Th(V)$ (def. 33) vanishes in degrees $\lt n$:

$\tilde H^{\bullet \lt n}(Th(V)) \simeq H^{\bullet \lt n}(D(V), S(V)) \simeq 0 \,.$
###### Proof

Consider the long exact sequence of relative cohomology (from above)

$\cdots \to H^{\bullet-1}(D(V)) \overset{i^\ast}{\longrightarrow} H^{\bullet-1}(S(V)) \longrightarrow H^\bullet(D(V), S(V)) \longrightarrow H^{\bullet}(D(V)) \overset{i^\ast}{\longrightarrow} H^{\bullet}(S(V)) \to \cdots \,.$

Since the cohomology in degree $k$ only depends on the $k$-skeleton, and since for $k \lt n$ the $k$-skeleton of $S(V)$ equals that of $X$, and since $D(V)$ is even homotopy equivalent to $X$, the morhism $i^\ast$ is an isomorphism in degrees lower than $n$. Hence by exactness of the sequence it follows that $H^{\bullet \lt n}(D(V),S(V)) = 0$.

#### Universal Thom spectra $M G$

###### Proposition

For each $n \in \mathbb{N}$ the pullback of the rank-$(n+1)$ universal vector bundle to the classifying space of rank $n$ vector bundles is the direct sum of vector bundles of the rank $n$ universal vector bundle with the trivial rank-1 bundle: there is a pullback diagram of topological spaces of the form

$\array{ \mathbb{R}\oplus (E O(n)\underset{O(n)}{\times} \mathbb{R}^n) &\longrightarrow& E O(n+1) \underset{O(n+1)}{\times} \mathbb{R}^{n+1} \\ \downarrow &(pb)& \downarrow \\ B O(n) &\longrightarrow& B O(n+1) } \,,$

where the bottom morphism is the canonical one (def.).

(e.g. Kochmann 96, p. 25)

###### Proof

For each $k \in \mathbb{N}$, $k \geq n$ there is such a pullback of the canonical vector bundles over Grassmannians

$\array{ \left\{ {V_{n}\subset \mathbb{R}^k, } \atop {v \in V_n, v_{n+1} \in \mathbb{R}} \right\} &\longrightarrow& \left\{ {V_{n+1} \subset \mathbb{R}^{k+1}}, \atop v \in V_{n+1} \right\} \\ \downarrow && \downarrow \\ Gr_n(\mathbb{R}^k) &\longrightarrow& Gr_{n+1}(\mathbb{R}^{k+1}) }$

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 $Top_{cg}$ 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. 35. 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.

###### Definition

The universal real Thom spectrum $M O$ is the spectrum, which is represented by the sequential prespectrum (def.) whose $n$th component space is the Thom space (def. 33)

$(M O)_n \coloneqq Th(E O(n)\underset{O(n)}{\times}\mathbb{R}^n)$

of the rank-$n$ universal vector bundle, and whose structure maps are the image under the Thom space functor $Th(-)$ of the top morphisms in prop. 47, via the homeomorphisms of prop. 45:

$\sigma_n \;\colon\; \Sigma (M O)_n \simeq Th(\mathbb{R}\oplus (E O(n)\underset{O(n)}{\times} \mathbb{R}^n)) \stackrel{}{\longrightarrow} Th(E O(n+1) \underset{O(n+1)}{\times} \mathbb{R}^{n+1}) = (M O)_{n+1} \,.$

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 $SO(n)$, or the spin groups $Spin(n)$, or the string 2-group $String(n)$, or the fivebrane 6-group $Fivebrane(n)$,…, or any level in the Whitehead tower of $O(n)$. To any of these groups there corresponds a Thom spectrum (denoted, respectively, $M SO$, MSpin, $M String$, $M Fivebrane$, etc.), which is in turn related to oriented cobordism, spin cobordism, string cobordism, et cetera.:

###### Definition

Given a (B,f)-structure $\mathcal{B}$ (def. 31), write $V^\mathcal{B}_n$ for the pullback of the universal vector bundle (def. 27) to the corresponding space of the $(B,f)$-structure and with

$\array{ V^{\mathcal{B}} &\overset{}{\longrightarrow}& V O(n) \underset{O(n)}{\times} \mathbb{R}^n \\ \downarrow &(pb)& \downarrow \\ B_n &\underset{f_n}{\longrightarrow}& B O(n) }$

and we write $e_{n_1,n_2}$ for the maps of total space of vector bundles over the $g_{n_1,n_2}$:

$\array{ V^{\mathcal{B}}_{n_1} &\overset{e_{n_1,n_2}}{\longrightarrow}& V^{\mathcal{B}}_{n_2} \\ \downarrow &(pb)& \downarrow \\ B_{n_1} &\underset{g_{n_1,n_2}}{\longrightarrow}& B_{n_2} } \,.$

Observe that the analog of prop. 47 still holds:

###### Proposiiton

Given a (B,f)-structure $\mathcal{B}$ (def. 31), then the pullback of its rank-$(n+1)$ vector bundle $V^{\mathcal{B}}_{n+1}$ (def. 35) along the map $g_{n,n+1} \colon B_n \to B_{n+1}$ is the direct sum of vector bundles of the rank-$n$ bundle $V^{\mathcal{B}}_n$ with the trivial rank-1-bundle: there is a pullback square

$\array{ \mathbb{R} \oplus V^{\mathcal{B}}_n &\overset{e_{n,n+1}}{\longrightarrow}& V^{\mathcal{B}}_{n+1} \\ \downarrow &(pb)& \downarrow \\ B_n &\underset{g_{n,n+1}}{\longrightarrow}& B_{n+1} } \,.$
###### Proof

Unwinding the definitions, the pullback in question is

\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. 47.

###### Definition

Given a (B,f)-structure $\mathcal{B}$ (def. 31), its universal Thom spectrum $M \mathcal{B}$ is, as a sequential prespectrum, given by component spaces being the Thom spaces (def. 33) of the $\mathcal{B}$-associated vector bundles of def. 35

$(M \mathcal{B})_n \coloneqq Th(V^{\mathcal{B}}_n)$

and with structure maps given via prop. 45 by the top maps in prop. 48:

$\sigma_n \;\colon\; \Sigma (M \mathcal{B})_n = \Sigma Th(V^{\mathcal{E}}_n) \simeq Th(\mathbb{R}\oplus V^{\mathcal{E}}_n) \overset{Th(e_{n,n+1})}{\longrightarrow} Th(V^{\mathcal{B}}_{n+1}) = (M \mathcal{B})_{n+1} \,.$

Similarly for an $S^k-(B,f)$-structure indexed on every $k$th natural number (such as almost complex structure, almost quaternionic structure, example 10), there is the corresponding Thom spectrum as a sequential $S^k$ spectrum (def.).

If $B_n = B G_n$ for some natural system of groups $G_n \to O(n)$, then one usually writes $M G$ for $M \mathcal{B}$. For instance $M SO$, MSpin, MU, MSp etc.

If the $(B,f)$-structure is multiplicative (def. 31), then the Thom spectrum $M \mathcal{B}$ canonical becomes a ring spectrum (for more on this see Part 1-2 the section on orthogonal Thom spectra ): the multiplication maps $B_{n_1} \times B_{n_2}\to B_{n_1 + n_2}$ are covered by maps of vector bundles

$V^{\mathcal{B}}_{n_1} \boxtimes V^{\mathcal{B}}_{n_2} \longrightarrow V^{\mathcal{B}}_{n_1 + n_2}$

and under forming Thom spaces this yields (via prop. 46) maps

$(M \mathcal{B})_{n_1} \wedge (M \mathcal{B})_{n_2} \longrightarrow (M \mathcal{B})_{n_1 + n_2}$

which are associative by the associativity condition in a multiplicative $(B,f)$-structure. The unit is

$(M \mathcal{B})_0 = Th(V^{\mathcal{B}}_0) \simeq Th(\ast) \simeq S^0 \,,$

by remark 7.

###### Example

The universal Thom spectrum (def. 36) for framing structure (exmpl.) is equivalently the sphere spectrum (def.)

$M 1 \simeq \mathbb{S} \,.$

Because in this case $B_n \simeq \ast$ and so $E^{\mathcal{B}}_n \simeq \mathbb{R}^n$, whence $Th(E^{\mathcal{B}}_n) \simeq S^n$.

#### Pontrjagin-Thom construction

###### Definition

For $X$ a smooth manifold and $i \colon X \hookrightarrow \mathbb{R}^k$ an embedding, then a tubular neighbourhood of $X$ is a subset of the form

$\tau_i X \coloneqq \left\{ x \in \mathbb{R}^k \;\vert\; d(x,i(X)) \lt \epsilon \right\}$

for some $\epsilon \in \mathbb{R}$, $\epsilon \gt 0$, small enough such that the map

$N_i X \longrightarrow \tau_i X$

from the normal bundle (def. 30) given by

$(i(x),v) \mapsto (i(x), \epsilon (1-e^{- {\vert v\vert}}) v )$

is a diffeomorphism.

###### Proposition

(tubular neighbourhood theorem)

For every embedding of smooth manifolds, there exists a tubular neighbourhood according to def. 37.

###### Remark

Given an embedding $i \colon X \hookrightarrow \mathbb{R}^k$ with a tubuluar neighbourhood $\tau_i X \hookrigtharrow \mathbb{R}^k$ (def. 37) then by construction:

1. the Thom space (def. 33) of the normal bundle (def. 30) is homeomorphic to the quotient topological space of the topological closure of the tubular neighbourhood by its boundary:

$Th(N_i(X)) \simeq \overline{ \tau_i(X)}/\partial \overline{\tau_i(X)}$;

2. there exists a continous function

$\mathbb{R}^k \longrightarrow \overline{ \tau_i(X)}/\partial \overline{\tau_i(X)}$

which is the identity on $\tau_i(X)\subset \mathbb{R}^k$ and is constant on the basepoint of the quotient on all other points.

###### Definition

For $X$ a smooth manifold of dimension $n$ and for $i \colon X \hookrightarrow \mathbb{R}^k$ an embedding, then the Pontrjagin-Thom collapse map is, for any choice of tubular neighbourhood $\tau_i(X)\subset \mathbb{R}^k$ (def. 37) the composite map of pointed topological spaces

$S^k \overset{\simeq}{\to} (\mathbb{R}^k)^\ast \longrightarrow \overline{ \tau_i(X)}/\partial \overline{\tau_i(X)} \overset{\simeq}{\to} Th(N_i X)$

where the first map identifies the k-sphere as the one-point compactification of $\mathbb{R}^k$; and where the second and third maps are those of remark 9.

The Pontrjagin-Thom construction is the further composite

$\xi_i \;\colon\; S^k \longrightarrow Th(N_i X) \overset{Th(e_i)}{\longrightarrow} Th( E O(k-n) \underset{O(k-n)}{\times} \mathbb{R}^{k-n} ) \simeq (M O)_{k-n}$

with the image under the Thom space construction of the morphism of vector bundles

$\array{ \nu &\overset{e_i}{\longrightarrow}& E O(k-n)\underset{O(k-n)}{\times} \mathbb{R}^{k-n} \\ \downarrow &(pb)& \downarrow \\ X &\underset{g_i}{\longrightarrow}& B O(k-n) }$

induced by the classifying map $g_i$ of the normal bundle (def. 30).

This defines an element

$[S^{n+(k-n)} \overset{\xi_i}{\to} (M O)_{k-n}] \in \pi_{n} M O$

in the $n$th stable homotopy group (def.) of the Thom spectrum $M O$ (def. 34).

More generally, for $X$ a smooth manifold with normal (B,f)-structure $(X,i,\hat g_i)$ according to def. 32, then its Pontrjagin-Thom construction is the composite

$\xi_i \;\colon\; S^k \longrightarrow Th(N_i X) \overset{Th(\hat e_i)}{\longrightarrow} Th( V^{\mathcal{B}}_{k-n} ) \simeq (M \mathcal{B})_{k-n}$

with

$\array{ \nu &\overset{\hat e_i}{\longrightarrow}& V^{\mathcal{B}}_{k-n} \\ \downarrow &(pb)& \downarrow \\ X &\underset{\hat g_i}{\longrightarrow}& B O(k-n) } \,.$
###### Proposition

The Pontrjagin-Thom construction (def. 38) respects the equivalence classes entering the definition of manifolds with stable normal $\mathcal{B}$-structure (def. 32) hence descends to a function (of sets)

$\xi \;\colon\; \left\{ {n\text{-}manifolds\;with\;stable} \atop {normal\;\mathcal{B}\text{-}structure} \right\} \longrightarrow \pi_n(M\mathcal{B}) \,.$
###### Proof

It is clear that the homotopies of classifying maps of $\mathcal{B}$-structures that are devided out in def. 32 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 $X \overset{i}{\hookrightarrow}\mathbb{R}^{k_1}$ with normal $\mathcal{B}$-structure

$\array{ && B_{k_1-n} \\ & {}^{\mathllap{\hat g_i}}\nearrow & \downarrow^{\mathrlap{f_{k-n}}} \\ X &\underset{g_i}{\longrightarrow}& B O(k_1-n) }$

write

$\alpha \;\colon\; S^{n+(k_1-n)} \overset{}{\longrightarrow} Th(E^{\mathcal{B}_{k_1-n}})$

for its image under the Pontrjagin-Thom construction (def. 38). Now given $k_2 \in \mathbb{N}$, consider the induced embedding $X \overset{i}{\hookrightarrow} \mathbb{R}^{k_1}\hookrightarrow \mathbb{R}^{k_1 + k_2}$ with normal $\mathcal{B}$-structure given by the composite

$\array{ && B_{k_1-n} &\overset{g_{k_1-n, k_1+ k_2 -n}}{\longrightarrow}& B_{k_1 + k_2-n} \\ & {}^{\mathllap{\hat g_i}}\nearrow & \downarrow^{\mathrlap{f_{k_1 - n} \times f_{k_2}}} && \downarrow^{\mathrlap{f_{k_1 + k_2-n}}} \\ X &\underset{g_i}{\longrightarrow}& B O(k_1-n) &\longrightarrow& B O(k_1 + k_2-n) } \,.$

By prop. 48 and using the pasting law for pullbacks, the classifying map $\hat g'_i$ for the enlarged normal bundle sits in a diagram of the form

$\array{ (\nu_i \oplus \mathbb{R}^{k_2}) &\overset{(\hat e_i \oplus id)}{\longrightarrow}& (V^{\mathcal{B}}_{k_1-n} \oplus \mathbb{R}^{k_2}) &\overset{e_{k_1-n,k_1+k_2-n}}{\longrightarrow}& V^{\mathcal{B}}_{k_1 + k_2 - n} \\ \downarrow &(pb)& \downarrow &(pb)& \downarrow \\ X &\underset{\hat g_i}{\longrightarrow}& B_{k_1-n} &\underset{g_{k_1-n, k_1 + k_2 - n}}{\longrightarrow}& B_{k_1 +k_2 - n} } \,.$

Hence the Pontrjagin-Thom construction for the enlarged embedding space is (using prop. 45) the composite

$\alpha_{k_2} \;\colon\; S^{n + (k_1+ k_2 - n)} \simeq Th(\mathbb{R}^{k_2}) \wedge S^{n + (k_1 - n)} \overset{}{\longrightarrow} Th(\mathbb{R}^{k_2}) \wedge Th(\nu_i) \overset{Th(id)\wedge Th(\hat e_i)}{\longrightarrow} Th(\mathbb{R}^{k_2}) \wedge Th(E^{\mathcal{B}}_{k_1-n})) \overset{Th(e_{k_1-n, k_1 + k_2 - n})}{\longrightarrow} Th(V^{\mathcal{B}}_{k_1 + k_2 - n}) \,.$

The composite of the first two morphisms here is $S^{k_k}\wedge \alpha$, while last morphism $Th(\hat e_{k_1-n,k_1+k_2-n})$ is the structure map in the Thom spectrum (by def. 36):

$\alpha_{k_2} \;\colon\; S^{k_2} \wedge S^{n + (k_1 - n)} \overset{S^{k_2} \wedge \alpha}{\longrightarrow} S^{k_2} \wedge Th(E^{\mathcal{B}}_{k_1 + k_2 - n}) \overset{\sigma^{M \mathcal{B}}_{k_1-n,k_1 + k_2 - n} }{\longrightarrow} Th(V^{\mathcal{B}}_{k_1+k_2 - n})$

This manifestly identifies $\alpha_{k_2}$ as being the image of $\alpha$ under the component map in the sequential colimit that defines the stable homotopy groups (def.). Therefore $\alpha$ and $\alpha_{k_2}$, for all $k_2 \in \mathbb{N}$, represent the same element in $\pi_{\bullet}(M \mathcal{B})$.

### Bordism and Thom’s theorem

Idea. By the Pontryagin-Thom collapse construction above, there is an assignment

$n Manifolds \longrightarrow \pi_n(M O)$

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 $\pi_\bullet(M O)$ of the universal Thom spectrum precisely if they are connected by a bordism. The bordism-classes $\Omega_\bullet^O$ of manifolds form a commutative ring under disjoint union and Cartesian product, called the bordism ring, and Pontrjagin-Thom collapse produces a ring homomorphism

$\Omega_\bullet^O \longrightarrow \pi_\bullet(M O) \,.$

Thom's theorem states that this homomorphism is an isomorphism.

More generally, for $\mathcal{B}$ a multiplicative (B,f)-structure, def. 31, there is such an identification

$\Omega_\bullet^{\mathcal{B}} \simeq \pi_\bullet(M \mathcal{B})$

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)

#### Bordism

Throughout, let $\mathcal{B}$ be a multiplicative (B,f)-structure (def. 31).

###### Definition

Write $I \coloneqq [0,1]$ for the standard interval, regarded as a smooth manifold with boundary. For $c \in \mathbb{R}_+$ Consider its embedding

$e \;\colon\; I \hookrightarrow \mathbb{R}\oplus \mathbb{R}_{\geq 0}$

as the arc

$e \;\colon\; t \mapsto \cos(\pi t) \cdot e_1 + \sin(\pi t) \cdot e_2 \,,$

where $(e_1, e_2)$ denotes the canonical linear basis of $\mathbb{R}^2$, and equipped with the structure of a manifold with normal framing structure (example 10) by equipping it with the canonical framing

$fr \;\colon\; t \mapsto \cos(\pi t) \cdot e_1 + \sin(\pi t) \cdot e_2$

of its normal bundle.

Let now $\mathcal{B}$ be a (B,f)-structure (def. 31). Then for $X \overset{i}{\hookrightarrow}\mathbb{R}^k$ any embedded manifold with $\mathcal{B}$-structure $\hat g \colon X \to B_{k-n}$ on its normal bundle (def. 32), define its negative or orientation reversal $-(X,i,\hat g)$ of $(X,i, \hat g)$ to be the restriction of the structured manifold

$(X \times I \overset{(i,e)}{\hookrightarrow} \mathbb{R}^{k+2}, \hat g \times fr)$

to $t = 1$.

###### Definition

Two closed manifolds of dimension $n$ equipped with normal $\mathcal{B}$-structure $(X_1, i_1, \hat g_1)$ and $(X_2,i_2,\hat g_2)$ (def.) are called bordant if there exists a manifold with boundary $W$ of dimension $n+1$ equipped with $\mathcal{B}$-strcuture $(W,i_W, \hat g_W)$ if its boundary with $\mathcal{B}$-structure restricted to that boundary is the disjoint union of $X_1$ with the negative of $X_2$, according to def. 39

$\partial(W,i_W,\hat g_W) \simeq (X_1, i_1, \hat g_1) \sqcup -(X_2, i_2, \hat g_2) \,.$
###### Proposition

The relation of $\mathcal{B}$-bordism (def. 40) is an equivalence relation.

Write $\Omega^\mathcal{B}_{\bullet}$ for the $\mathbb{N}$-graded set of $\mathcal{B}$-bordism classes of $\mathcal{B}$-manifolds.

###### Proposition

Under disjoint union of manifolds, then the set of $\mathcal{B}$-bordism equivalence classes of def. 51 becomes an $\mathbb{Z}$-graded abelian group

$\Omega^{\mathcal{B}}_\bullet \in Ab^{\mathbb{Z}}$

(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. 31), 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.

$\Omega^{\mathcal{B}}_\bullet \in CRing^{\mathbb{Z}} \,.$

e.g. (Kochmann 96, prop. 1.5.3)

#### Thom’s theorem

Recall that the Pontrjagin-Thom construction (def. 38) associates to an embbeded manifold $(X,i,\hat g)$ with normal $\mathcal{B}$-structure (def. 32) an element in the stable homotopy group $\pi_{dim(X)}(M \mathcal{B})$ of the universal $\mathcal{B}$-Thom spectrum in degree the dimension of that manifold.

###### Lemma

For $\mathcal{B}$ be a multiplicative (B,f)-structure (def. 31), the $\mathcal{B}$-Pontrjagin-Thom construction (def. 38) is compatible with all the relations involved to yield a graded ring homomorphism

$\xi \;\colon\; \Omega^{\mathcal{B}}_\bullet \longrightarrow \pi_\bullet(M \mathcal{B})$

from the $\mathcal{B}$-bordism ring (def. 52) 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. 36).

###### Proof

By prop. 50 the underlying function of sets is well-defined before dividing out the bordism relation (def. 40). 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 $\xi$ 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 $n$-manifolds with $\mathcal{B}$-structure, we may consider an embedding of their disjoint union into some $\mathbb{R}^{k}$ such that the tubular neighbourhoods of the two direct summands do not intersect. There is then a map from two copies of the k-cube, glued at one face

$\Box^k \underset{\Box^{k-1}}{\sqcup} \Box^k \longrightarrow \mathbb{R}^k$

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 $\xi$ 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 $\xi$ 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 $\mathbb{R}^0$, by the identity map. The corresponding normal bundle is of rank 0 and hence (by remark 7) its Thom space is $S^0$, the 0-sphere. Also $V^{\mathcal{B}}_0$ is the rank-0 vector bundle over the point, and hence $(M \mathcal{B})_0 \simeq S^0$ (by def. 36) and so $\xi(\ast) \colon (S^0 \overset{\simeq}{\to} S^0)$ indeed represents the unit element in $\pi_\bullet(M\mathcal{B})$.

Finally regarding respect for the ring product structure: for two manifolds with stable normal $\mathcal{B}$-structure, represented by embeddings into $\mathbb{R}^{k_i}$, 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. 46 there is a diagram of the form

$\array{ \nu_1 \boxtimes \nu_2 &\overset{\hat e_1 \boxtimes \hat e_2}{\longrightarrow}& V^{\mathcal{B}}_{n_1} \boxtimes V^{\mathcal{B}}_{n_2} &\overset{\kappa_{n_1,n_2}}{\longrightarrow}& V^{\mathcal{B}}_{n_1 + n_2} \\ \downarrow &(pb)& \downarrow &(pb)& \downarrow \\ X_1 \times X_2 &\underset{\hat g_1 \times \hat g_2}{\longrightarrow}& B_{k_1-n_1} \times B_{k_2-n_2} &\underset{\mu_{k_1-n_1,k_2-n_2}}{\longrightarrow}& B_{k_1 + k_2 - n_1 - n_2} } \,.$

To the Pontrjagin-Thom construction of the product manifold is by definition the top composite in the diagram

$\begin{array}{ccccccc}{S}^{{n}_{1}+{n}_{2}+\left({k}_{1}+{k}_{2}-{n}_{1}-{n}_{2}\right)}& \stackrel{}{⟶}& \mathrm{Th}\left({\nu }_{1}⊠{\nu }_{2}\right)& \stackrel{\mathrm{Th}\left({\stackrel{^}{e}}_{1}⊠{\stackrel{^}{e}}_{2}\right)}{⟶}& \mathrm{Th}\left({V}_{{k}_{1}-{n}_{1}}^{ℬ}⊠{V}_{{k}_{2}-{n}_{2}}^{ℬ}\right)& \stackrel{{\kappa }_{{k}_{1}-{n}_{1},{k}_{2}-{n}_{2}}}{⟶}& \mathrm{Th}\left({V}_{{k}_{1}+{k}_{2}-{n}_{1}-{n}_{2}}^{ℬ}\end{array}$