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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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group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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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).
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,
from (sufficiently nice) topological spaces to $\mathbb{Z}$-graded abelian groups, such that a few key properties of the homotopy types of topological spaces is preserved as one passes them from Ho(Top) to the much more tractable abelian category Ab.
Literature. (Aguilar-Gitler-Prieto 02, chapters 7,8 and 12, Kochman 96, 3.4, 4.2, Schwede 12, II.6)
Idea. A generalized (Eilenberg-Steenrod) cohomology theory is such a contravariant functor which satisfies the key properties exhibited by ordinary cohomology (as computed for instance by singular cohomology), notably homotopy invariance and excision, except that its value on the point is not required to be concentrated in degree 0. Dually for generalized homology. There are two versions of the axioms, one for reduced cohomology, and they are equivalent if properly set up.
An important example of a generalised cohomology theory other than ordinary cohomology is topological K-theory. The other two examples of key relevance below are cobordism cohomology and stable cohomotopy.
Literature. (Switzer 75, section 7, Aguilar-Gitler-Prieto 02, section 12 and section 9, Kochman 96, 3.4).
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The traditional formulation of reduced generalized cohomology in terms of point-set topology is this:
A reduced cohomology theory is
a functor
from the opposite of pointed topological spaces (CW-complexes) to $\mathbb{Z}$-graded abelian groups (“cohomology groups”), in components
equipped with a natural isomorphism of degree +1, to be called the suspension isomorphism, of the form
such that:
(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
(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
(e.g. AGP 02, def. 12.1.4)
This is equivalent (prop. below) to the following more succinct homotopy-theoretic definition:
A reduced generalized cohomology theory is a functor
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
such that:
As a consequence (prop. below), we find yet another equivalent definition:
A reduced generalized cohomology theory is a functor
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
such that
Regarding the equivalence of def. with def. :
By the existence of the classical model structure on topological spaces (thm.), the characterization of its homotopy category (cor.) and the existence of CW-approximations, the homotopy invariance axiom in def. is equivalent to the functor passing to the classical pointed homotopy category. In view of this and since on CW-complexes the standard topological mapping cone construction is a model for the homotopy cofiber (prop.), this gives the equivalence of the two versions of the exactness axiom.
Regarding the equivalence of def. with def. :
This is the universal property of the classical homotopy category (thm.) which identifies it with the localization (def.) of $Top^{\ast/}$ 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):
where $F$ sends weak homotopy equivalences to isomorphisms and where $(-)_\sim$ means identifying homotopic maps.
Prop. naturally suggests (e.g. Lurie 10, section 1.4) that the concept of generalized cohomology be formulated in the generality of any abstract homotopy theory (model category), not necessarily that of (pointed) topological spaces:
Let $\mathcal{C}$ be a model category (def.) with $\mathcal{C}^{\ast/}$ its pointed model category (prop.).
A reduced additive generalized cohomology theory on $\mathcal{C}$ is
a functor
a natural isomorphism (“suspension isomorphisms”) of degree +1
such that
Finally we need the following terminology:
Let $\tilde E^\bullet$ be a reduced cohomology theory according to either of def. , def. , def. or def. .
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
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:
If $\tilde E^\bullet$ is not ordinary, one also says that it is generalized or extraordinary.
A homomorphism of reduced cohomology theories
is a natural transformation between the underlying functors which is compatible with the suspension isomorphisms in that all the following squares commute
We now discuss some constructions and consequences implied by the concept of reduced cohomology theories:
Given a generalized cohomology theory $(E^\bullet,\delta)$ on some $\mathcal{C}$ as in def. , and given a homotopy cofiber sequence in $\mathcal{C}$ (prop.),
then the corresponding connecting homomorphism is the composite
The connecting homomorphisms of def. are parts of long exact sequences
By the defining exactness of $E^\bullet$, def. , and the way this appears in def. , using that $\sigma$ is by definition an isomorphism.
Given a reduced generalized cohomology theory as in def. , 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 $X \mapsto (X,\emptyset)$.
A cohomology theory (unreduced, relative) is
a functor
to the category of $\mathbb{Z}$-graded abelian groups,
a natural transformation of degree +1, to be called the connecting homomorphism, of the form
such that:
(homotopy invariance) For $f \colon (X_1,A_1) \to (X_2,A_2)$ a homotopy equivalence of pairs, then
is an isomorphism;
(exactness) For $A \hookrightarrow X$ the induced sequence
is a long exact sequence of abelian groups.
(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
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
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
A homomorphism of unreduced cohomology theories
is a natural transformation of the underlying functors that is compatible with the connecting homomorphisms, hence such that all these squares commute:
e.g. (AGP 02, def. 12.1.1).
The excision axiom in def. is equivalent to the following statement:
For all $A,B \hookrightarrow X$ with $X = Int(A) \cup Int(B)$, then the inclusion
induces an isomorphism,
(e.g Switzer 75, 7.2)
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
and that
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
and that
Hence
The following lemma shows that the dependence in pairs of spaces in a generalized cohomology theory is really a stand-in for evaluation on homotopy cofibers of inclusions.
Let $E^\bullet$ be an cohomology theory, def. , and let $A \hookrightarrow X$. Then there is an isomorphism
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.g AGP 02, corollary 12.1.10)
Consider $U \coloneqq (Cone(A)-A \times \{0\}) \hookrightarrow Cone(A)$, the cone on $A$ minus the base $A$. We have
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:
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
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
is an isomorphism.
(exact sequence of a triple)
For $E^\bullet$ an unreduced generalized cohomology theory, def. , then every inclusion of two consecutive subspaces
induces a long exact sequence of cohomology groups of the form
where
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.
The exact sequence of a triple in prop. is what gives rise to the Cartan-Eilenberg spectral sequence for $E$-cohomology of a CW-complex $X$.
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. ,
exhibits the connecting homomorphism $\bar \delta$ here as an isomorphism
This is the suspension isomorphism extracted from the unreduced cohomology theory, see def. below.
Given $E^\bullet$ an unreduced cohomology theory, def. . 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
e.g. (Switzer 75, theorem 7.19, Aguilar-Gitler-Prieto 02, theorem 12.1.22)
(unreduced to reduced cohomology)
Let $E^\bullet$ be an unreduced cohomology theory, def. . Define a reduced cohomology theory, def. $(\tilde E^\bullet, \sigma)$ as follows.
For $x \colon \ast \to X$ a pointed topological space, set
This is clearly functorial. Take the suspension isomorphism to be the composite
of the isomorphism $E^\bullet(p)$ from example and the inverse of the isomorphism $\bar \delta$ from example .
(e.g Switzer 75, 7.34)
We need to check the exactness axiom given any $A\hookrightarrow X$. By lemma we have an isomorphism
Unwinding the constructions shows that this makes the following diagram commute:
where the vertical sequence on the right is exact by prop. . Hence the left vertical sequence is exact.
(reduced to unreduced cohomology)
Let $(\tilde E^\bullet, \sigma)$ be a reduced cohomology theory, def. . Define an unreduced cohomolog theory $E^\bullet$, def. , by
e.g. (Switzer 75, 7.35)
Exactness holds by prop. . For excision, it is sufficient to consider the alternative formulation of lemma . For CW-inclusions, this follows immediately with lemma .
The constructions of def. and def. constitute a pair of functors between then categories of reduced cohomology theories, def. and unreduced cohomology theories, def. which exhbit an equivalence of categories.
(…careful with checking the respect for suspension iso and connecting homomorphism..)
To see that there are natural isomorphisms relating the two composites of these two functors to the identity:
One composite is
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 .
The other composite is
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:
Let $E^\bullet$ be an unreduced cohomology theory, and $\tilde E^\bullet$ its reduced cohomology theory from def. . For $(X,\ast)$ a pointed topological space, then there is an identification
of the unreduced cohomology of $X$ with the direct sum of the reduced cohomology of $X$ and the unreduced cohomology of the base point.
The pair $\ast \hookrightarrow X$ induces the sequence
which by the exactness clause in def. 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
with the right map an epimorphism. Hence this is a split exact sequence and the statement follows.
All of the above has a dual version with generalized cohomology replaced by generalized homology. For ease of reference, we record these dual definitions:
A reduced homology theory is a functor
from the category of pointed topological spaces (CW-complexes) to $\mathbb{Z}$-graded abelian groups (“homology groups”), in components
and equipped with a natural isomorphism of degree +1, to be called the suspension isomorphism, of the form
such that:
(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
(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
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
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:
A homomorphism of reduced cohomology theories
is a natural transformation between the underlying functors which is compatible with the suspension isomorphisms in that all the following squares commute
A homology theory (unreduced, relative) is a functor
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
such that:
(homotopy invariance) For $f \colon (X_1,A_1) \to (X_2,A_2)$ a homotopy equivalence of pairs, then
is an isomorphism;
(exactness) For $A \hookrightarrow X$ the induced sequence
is a long exact sequence of abelian groups.
(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
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
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
A homomorphism of unreduced homology theories
is a natural transformation of the underlying functors that is compatible with the connecting homomorphisms, hence such that all these squares commute:
The generalized cohomology theories considered above assign cohomology groups. It is familiar from ordinary cohomology with coefficients not just in a group but in a ring, that also the cohomology groups inherit compatible ring structure. The generalization of this phenomenon to generalized cohomology theories is captured by the concept of multiplicative cohomology theories:
Let $E_1, E_2, E_3$ be three unreduced generalized cohomology theories (def.). A pairing of cohomology theories
is a natural transformation (of functors on $(Top_{CW}^{\hookrightarrow}\times Top_{CW}^{\hookrightarrow})^{op}$) of the form
such that this is compatible with the connecting homomorphisms $\delta_i$ of $E_i$, in that the following are commuting squares
and
where the isomorphisms in the bottom left are the excision isomorphisms.
An (unreduced) multiplicative cohomology theory is an unreduced generalized cohomology theory theory $E$ (def. ) equipped with
(external multiplication) a pairing (def. ) of the form $\mu \;\colon\; E \Box E \longrightarrow E$;
(unit) an element $1 \in E^0(\ast)$
such that
(associativity) $\mu \circ (id \otimes \mu) = \mu \circ (\mu \otimes id)$;
(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
(graded commutativity)
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 $\Delta_{(X,A)} \colon (X,A) \longrightarrow (X\times X, A \times X \cup X \times A)$;
e.g. (Tamaki-Kono 06, II.6)
Let $(E,\mu,1)$ be a multiplicative cohomology theory, def. . Then
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.
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 $E^\bullet(\ast)$.
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 $(-1)^{n_1}$
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:
For connecting homomorphisms this is the (graded) commutativity of the squares in def. :
Idea. Given any functor such as the generalized (co)homology functor above, an important question to ask is whether it is a representable functor. Due to the $\mathbb{Z}$-grading and the suspension isomorphisms, if a generalized (co)homology functor is representable at all, it must be represented by a $\mathbb{Z}$-indexed sequence of pointed topological spaces such that the reduced suspension of one is comparable to the next one in the list. This is a spectrum or more specifically: a sequential spectrum .
Whitehead observed that indeed every spectrum represents a generalized (co)homology theory. The Brown representability theorem states that, conversely, every generalized (co)homology theory is represented by a spectrum, subject to conditions of additivity.
As a first application, Eilenberg-MacLane spectra representing ordinary cohomology may be characterized via Brown representability.
Literature. (Switzer 75, section 9, Aguilar-Gitler-Prieto 02, section 12, Kochman 96, 3.4)
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.
A Brown functor is a functor
(from the opposite of the classical homotopy category (def., def.) of connected pointed topological spaces) such that
(additivity) $F$ takes small coproducts (wedge sums) to products;
(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 $x_B = (x_X)|_B$.
For every additive reduced cohomology theory $\tilde E^\bullet(-) \colon Ho(Top^{\ast/})^{op}\to Set^{\ast/}$ (def. ) and for each degree $n \in \mathbb{N}$, the restriction of $\tilde E^n(-)$ to connected spaces is a Brown functor (def. ).
Under the relation between reduced and unreduced cohomology above, this follows from the exactness of the Mayer-Vietoris sequence of prop. .
(Brown representability)
Every Brown functor $F$ (def. ) is representable, hence there exists $X \in Top_{\geq 1}^{\ast/}$ and a natural isomorphism
(where $[-,-]_\ast$ denotes the hom-functor of $Ho(Top_{\geq 1}^{\ast/})$ (exmpl.)).
(e.g. AGP 02, theorem 12.2.22)
A key subtlety in theorem is the restriction to connected pointed topological spaces in def. . This comes about since the proof of the theorem requires that continuous functions $f \colon X \longrightarrow Y$ that induce isomorphisms on pointed homotopy classes
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. below) the following concept.
An Omega-spectrum $X$ (def.) is
a sequence $\{X_n\}_{n \in \mathbb{N}}$ of pointed topological spaces $X_n \in Top^{\ast/}$
for each $n \in \mathbb{N}$, form each space to the loop space of the following space.
Every additive reduced cohomology theory $\tilde E^\bullet(-) \colon (Top_{CW}^\ast)^{op} \longrightarrow Ab^{\mathbb{Z}}$ according to def. , is represented by an Omega-spectrum $E$ (def. ) in that in each degree $n \in \mathbb{N}$
$\tilde E^n(-)$ is represented by some $E_n \in Ho(Top^{\ast/})$;
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:
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.).
If it were not for the connectedness clause in def. (remark ), then theorem with prop. would immediately give the existence of the $\{E_n\}_{n \in \mathbb{N}}$ 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:
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 $\Omega E_{n+1}^{(0)}$:
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 $\Omega E_{n+1}^{(0)}$:
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:
Every Omega-spectrum $E$, def. , represents an additive reduced cohomology theory def. $\tilde E^\bullet$ by
with suspension isomorphism given by
The additivity is immediate from the construction. The exactnes follows from the long exact sequences of homotopy cofiber sequences given by this prop..
If we consider the stable homotopy category $Ho(Spectra)$ 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. 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
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
to be the graded reduced generalized $E$-cohomology groups of the spectrum $X$.
See also in part 1 this 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
Hence singular cohomology is a generalized cohomology theory which is “ordinary cohomology” in the sense of def. .
Applying the Brown representability theorem as in prop. hence produces an Omega-spectrum (def. ) whose $n$th component space is characterized as having homotopy groups concentrated in degree $n$ on $A$. These are called Eilenberg-MacLane spaces $K(A,n)$
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 $H A$:
As a consequence of example one obtains the uniqueness result of Eilenberg-Steenrod:
Let $\tilde E_1$ and $\tilde E_2$ be ordinary (def. ) generalized (Eilenberg-Steenrod) cohomology theories. If there is an isomorphism
of cohomology groups of the 0-sphere, then there is an isomorphism of cohomology theories
(e.g. Aguilar-Gitler-Prieto 02, theorem 12.3.6)
Using abstract homotopy theory in the guise of model category theory (see the lecture notes on classical homotopy theory), the traditional proof and further discussion of the Brown representability theorem above becomes more transparent (Lurie 10, section 1.4.1, for exposition see also Mathew 11).
This abstract homotopy-theoretic proof uses the general concept of homotopy colimits in model categories as well as the concept of derived hom-spaces (“∞-categories”). Even though in the accompanying Lecture notes on classical homotopy theory these concepts are only briefly indicated, the following is included for the interested reader.
Let $\mathcal{C}$ be a model category. A functor
(from the opposite of the homotopy category of $\mathcal{C}$ to Set)
is called a Brown functor if
it sends small coproducts to products;
it sends homotopy pushouts in $\mathcal{C}\to Ho(\mathcal{C})$ to weak pullbacks in Set (see remark ).
A weak pullback is a diagram that satisfies the existence clause of a pullback, but not necessarily the uniqueness condition. Hence the second clause in def. says that for a homotopy pushout square
in $\mathcal{C}$, then the induced universal morphism
into the actual pullback is an epimorphism.
Say that a model category $\mathcal{C}$ is compactly generated by cogroup objects closed under suspensions if
$\mathcal{C}$ is generated by a set
of compact objects (i.e. every object of $\mathcal{C}$ is a homotopy colimit of the objects $S_i$.)
each $S_i$ admits the structure of a cogroup object in the homotopy category $Ho(\mathcal{C})$;
the set $\{S_i\}$ is closed under forming reduced suspensions.
(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
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 $\Sigma X$:
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 . 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. 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 above.
See also (Lurie 10, example 1.4.1.4)
In homotopy theories compactly generated by cogroup objects closed under forming suspensions, the following strenghtening of the Whitehead theorem holds.
In a homotopy theory compactly generated by cogroup objects $\{S_i\}_{i \in I}$ closed under forming suspensions, according to def. , 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
is an isomorphism (a bijection).
(Lurie 10, p. 114, Lemma star)
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
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
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
Now in this pointed situation we may use that
to find that $f$ is an equivalence in $\mathcal{C}$ precisely if the induced morphisms
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.
(Brown representability)
Let $\mathcal{C}$ be a model category compactly generated by cogroup objects closed under forming suspensions, according to def. . Then a functor
(from the opposite of the homotopy category of $\mathcal{C}$ to Set) is representable precisely if it is a Brown functor, def. .
Due to the version of the Whitehead theorem of prop. 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
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
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
Then by the assumption that $F$ takes this homotopy cokernel to a weak fiber (as in remark ), 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 ), there exists an extension $\eta_{n+1}$ of $\eta_n$ along $X_n \to X_{n+1}$:
It is now clear that we want to take
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:
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 $\eta' \colon X' \to F$:
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:
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:
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
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:
Moreover, Lemma $(\star)$ gives that evaluated on all $S_i$, the two diagonal morphisms here become isomorphisms. But then prop. 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
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
such that the diagonal maps become isomorphisms when evaluated on the $S_i$. As before, it follows via prop. that the morphism $h \colon X' \longrightarrow Z'$ is an equivalence.
Since by this construction $h\circ f$ and $h\circ g$ are homotopic
it follows with $h$ being an equivalence that already $f$ and $g$ were homotopic, hence that they represented the same element.
Given a reduced additive cohomology functor $H^\bullet \colon Ho(\mathcal{C})^{op}\to Ab^{\mathbb{Z}}$, def. , its underlying Set-valued functors $H^n \colon Ho(\mathcal{C})^{op}\to Ab\to Set$ are Brown functors, def. .
The first condition on a Brown functor holds by definition of $H^\bullet$. For the second condition, given a homotopy pushout square
in $\mathcal{C}$, consider the induced morphism of the long exact sequences given by prop.
Here the outer vertical morphisms are isomorphisms, as shown, due to the pasting law (see also at fiberwise recognition of stable homotopy pushouts). This means that the four lemma applies to this diagram. Inspection shows that this implies the claim.
Let $\mathcal{C}$ be a model category which satisfies the conditions of theorem , and let $(H^\bullet, \delta)$ be a reduced additive generalized cohomology functor on $\mathcal{C}$, def. . Then there exists a spectrum object $E \in Stab(\mathcal{C})$ such that
$H\bullet$ is degreewise represented by $E$:
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
Via prop. , theorem gives the first clause. With this, the second clause follows by the Yoneda lemma.
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. ), 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.
Literature. (Switzer 75, section 7 from def. 7.57 on, Kochman 96, section 4.2, Goerss-Jardine 99, section VI.2, )
Given a tower $A_\bullet$ of abelian groups
write
for the homomorphism given by
The [[limit]] of a sequence as in def. – 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
[[commuting diagram|commute]] – is equivalently the [[kernel]] of the morphism $\partial$ in def. .
Given a [[tower]] $A_\bullet$ of [[abelian groups]]
then $\underset{\longleftarrow}{\lim}^1 A_\bullet$ is the [[cokernel]] of the map $\partial$ in def. , hence the group that makes a [[long exact sequence]] of the form
The [[functor]] $\underset{\longleftarrow}{\lim}^1 \colon Ab^{(\mathbb{N}, \geq)} \longrightarrow Ab$ (def. ) satisfies
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
is a [[long exact sequence]] of abelian groups;
if $A_\bullet$ is a tower such that all maps are [[surjections]], then $\underset{\longleftarrow}{\lim}^1_n A_n \simeq 0$.
(e.g. Switzer 75, prop. 7.63, Goerss-Jardine 96, section VI. lemma 2.11)
For the first property: Given $A_\bullet$ a tower of abelian groups, write
for the homomorphism from def. regarded as the single non-trivial differential in a [[cochain complex]] of abelian groups. Then by remark and def. we have $H^0(L(A_\bullet)) \simeq \underset{\longleftarrow}{\lim} A_\bullet$ 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
For the second statement: If all the $f_k$ are surjective, then inspection shows that the homomorphism $\partial$ in def. is surjective. Hence its [[cokernel]] vanishes.
The category $Ab^{(\mathbb{N}, \geq)}$ of [[towers]] of [[abelian groups]] has [[enough injectives]].
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
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
is an injective resolution for $A_\bullet$.
The [[functor]] $\underset{\longleftarrow}{\lim}^1 \colon Ab^{(\mathbb{N}, \geq)} \longrightarrow Ab$ (def. ) is the [[derived functor in homological algebra|first right derived functor]] of the [[limit]] functor $\underset{\longleftarrow}{\lim} \colon Ab^{(\mathbb{N},\geq)} \longrightarrow Ab$.
By lemma 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
be an [[injective resolution]]. We need to show that
Since limits preserve [[kernels]], this is equivalently
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
Therefore by the second item of prop. the long exact sequence from the first item of prop. applied to the [[short exact sequence]]
becomes
Exactness of this sequence gives the desired identification $\underset{\longleftarrow}{\lim}^1 A_\bullet \simeq (\underset{\longleftarrow}{\lim}(ker(j^2)_\bullet))/im(\underset{\longleftarrow}{\lim}(j^1)) \,.$
The [[functor]] $\underset{\longleftarrow}{\lim}^1 \colon Ab^{(\mathbb{N}, \geq)} \longrightarrow Ab$ (def. ) is in fact the unique functor, up to [[natural isomorphism]], satisfying the conditions in prop. .
The proof of prop. only used the conditions from prop. , hence any functor satisfying these conditions is the first right derived functor of $\underset{\longleftarrow}{\lim}$, up to natural isomorphism.
The following is a kind of double dual version of the $\lim^1$ construction which is sometimes useful:
Given a [[cotower]]
of [[abelian groups]], then for every abelian group $B \in Ab$ there is a [[short exact sequence]] of the form
where $Hom(-,-)$ denotes the [[hom-object|hom-group]], $Ext^1(-,-)$ denotes the first [[Ext]]-group (and so $Hom(-,-) = Ext^0(-,-)$).
Consider the homomorphism
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 [[formal dual|formally dual]] situation in remark ). Hence (as opposed to the long exact sequence in def. ) there is a [[short exact sequence]] of the form
Every short exact sequence gives rise to a [[long exact sequence]] of [[derived functors]] (prop.) which in the present case starts out as
where we used that [[direct sum]] is the [[coproduct]] in abelian groups, so that homs out of it yield a [[product]], and where the morphism $\partial$ is the one from def. corresponding to the [[tower]]
Hence truncating this long sequence by forming kernel and cokernel of $\partial$, respectively, it becomes the short exact sequence in question.
A tower $A_\bullet$ of [[abelian groups]]
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 $A_j \to A_k$
(e.g. Switzer 75, def. 7.74)
The Mittag-Leffler condition, def. , is satisfied in particular when all morphisms $A_{i+1}\to A_i$ are [[epimorphisms]] (hence [[surjections]] of the underlying [[sets]]).
If a tower $A_\bullet$ satisfies the [[Mittag-Leffler condition]], def. , then its $\underset{\leftarrow}{\lim}^1$ vanishes:
e.g. (Switzer 75, theorem 7.75, Kochmann 96, prop. 4.2.3, Weibel 94, prop. 3.5.7)
One needs to show that with the Mittag-Leffler condition, then the [[cokernel]] of $\partial$ in def. vanishes, hence that $\partial$ is an [[epimorphism]] in this case, hence that every $(a_n)_{n \in \mathbb{N}} \in \underset{n}{\prod} A_n$ has a preimage under $\partial$. So use the Mittag-Leffler condition to find pre-images of $a_n$ by [[induction]] over $n$.
Given a sequence
of ([[pointed topological space|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
For
a sequence in [[Top]], its mapping telescope is the [[quotient topological space]] of the [[disjoint union]] of [[product topological spaces]]
where the [[equivalence relation]] quotiented out is
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
For $X_\bullet$ the sequence of stages of a ([[pointed topological space|pointed]]) [[CW-complex]] $X = \underset{\longleftarrow}{\lim}_n X_n$, then the canonical map
from the [[mapping telescope]], def. , is a [[weak homotopy equivalence]].
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
is a [[homotopy equivalence]], hence (prop.) a weak homotopy equivalence. A concrete construction of a homotopy inverse is given for instance in (Switzer 75, proof of prop. 7.53).
Moreover, since spheres are [[compact object|compact]], so that elements of [[homotopy groups]] $\pi_q(Tel(X))$ are represented at some finite stage $\pi_q(Tel(X_n))$ it follows that
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.
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 sequence for homotopy groups)
Let
be a [[tower of fibrations]] ([[Serre fibrations]] (def.)). Then for each $q \in \mathbb{N}$ there is a [[short exact sequence]]
for $\pi_\bullet$ the [[homotopy group]]-functor (exact as [[pointed sets]] for $i = 0$, as [[groups]] for $i \geq 1$) which says that
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;
that kernel is the $\underset{\longleftarrow}{\lim}^1$ (def. ) of the homotopy groups of the stages.
An elementary but tedious proof is indicated in (Bousfield-Kan 72, chapter IX, theorem 3.1. The following is a neat [[model category]]-theoretic proof following (Goerss-Jardine 96, section VI. prop. 2.15), which however requires the concept of [[homotopy limit]] over towers.
With respect to the [[classical model structure on simplicial sets]] or the [[classical model structure on topological spaces]], a tower of fibrations as stated is a fibrant object in the injective [[model structure on functors]] $[(\mathbb{N},\geq), sSet]_{inj}$ ($[(\mathbb{N},\geq), Top]_{inj}$) (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
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]]
The [[long exact sequence of homotopy groups]] of this fiber sequence goes
Chopping that off by forming kernel and cokernel yields the claim for positive $q$. For $q = 0$ it follows by inspection.
(Milnor exact sequence for generalized cohomology)
Let $X$ be a [[pointed topological space|pointed]] [[CW-complex]], $X = \underset{\longrightarrow}{\lim}_n X_n$ and let $\tilde E^\bullet$ an additive [[reduced cohomology theory]], def. .
Then the canonical morphisms make a [[short exact sequence]]
saying that
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]];
this kernel is precisely the $\lim^1$ (def. ) of the cohomology groups at the finite stages in one degree lower.
e.g. (Switzer 75, prop. 7.66, Kochmann 96, prop. 4.2.2)
For
the sequence of stages of the ([[pointed topological space|pointed]]) [[CW-complex]] $X = \underset{\longleftarrow}{\lim}_n X_n$, write
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
Observe that
$A_X \cup B_X \simeq Tel(X_\bullet)$;
$A_X \cap B_X \simeq \underset{n \in \mathbb{N}}{\sqcup} X_n$;
and that there are [[homotopy equivalences]]
$A_X \simeq \underset{n \in \mathbb{N}}{\sqcup} X_{2n+1}$
$B_X \simeq \underset{n \in \mathbb{N}}{\sqcup} X_{2n}$
$Tel(X_\bullet) \simeq X$.
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:
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
(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. for the sequence
Hence truncating the above long exact sequence by forming kernel and cokernel of $\partial$, the result follows via remark and definition .
In contrast:
Let $X$ be a [[pointed topological space|pointed]] [[CW-complex]], $X = \underset{\longleftarrow}{\lim}_n X_n$.
For $\tilde E_\bullet$ an additive reduced [[generalized homology theory]], then
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.)
The [[stable homotopy category]] is, in particular, the [[homotopy category of a model category|homotopy category]] of the stable [[model structure on orthogonal spectra]], in that its [[localization]] at the [[stable weak homotopy equivalences]] is of the form
In the following when considering an [[orthogonal spectrum]] $X \in OrthSpec(Top_{cg})$, we use, for brevity, the same symbol for its image under $\gamma$.
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
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
(Schwede 12, chapter II prop. 6.5 (ii)) (using that symmetric spectra underlying orthogonal spectra are semistable (Schwede 12, p. 40))
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. ), then two morphisms of spectra $X \longrightarrow E$ are homotopic already if all their morphisms of component spaces $X_n \to E_n$ are.
By prop. the assumption implies that the $lim^1$-term in prop. vanishes, hence by exactness it follows that in this case there is an [[isomorphism]]
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)
See also the accompanying [[Introduction to Stable homotopy theory – I|lecture notes on spectral sequences]].
A cohomology [[spectral sequence]] $\{E_r^{p,q}, d_r\}$ is
a sequence $\{E_r^{\bullet,\bullet}\}$ (for $r \in \mathbb{N}$, $r \geq 1$) of [[bigraded object|bigraded]] [[abelian groups]] (the “pages”);
a sequence of [[linear maps]] (the “[[differentials]]”)
such that
Given a $\mathbb{Z}$-[[graded abelian group]]_ $C^\bullet$ equipped with a decreasing [[filtration]]
such that
then the spectral sequence is said to converge to $C^\bullet$, denoted,
if
in each bidegree $(s,t)$ the sequence $\{E_r^{s,t}\}_r$ eventually becomes constant on a group
$E_\infty^{s,t} \coloneqq E_{\gg 1}^{s,t}$;
$E_\infty^{\bullet,\bullet}$ 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
$\{E_2^{\bullet,\bullet}\}$ is equipped with the structure of a [[bigraded object|bigraded]] [[associative algebra|algebra]];
$F^\bullet C^\bullet$ 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
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;
the multiplication on $E_\infty^{\bullet,\bullet}$ is compatible with that on $C^\bullet$.
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.
The archetypical example of (co-)homology spectral sequences as in def. are induced from a [[filtered chain complex|filtering]] on a (co-)chain complex, converging to the (co-)[[chain homology]] of the chain complex by consecutively computing relative (co-)chain homologies, relative to decreasing (increasing) filtering degrees. For more on such [[spectral sequences of filtered complexes]] see at [[Introduction to Stable homotopy theory – I|Interlude – Spectral sequences]] the section For filtered complexes.
A useful way to generate spectral sequences is via [[exact couples]]:
An [[exact couple]] is three [[homomorphisms]] of [[abelian groups]] of the form
such that the [[image]] of one is the [[kernel]] of the next.
Given an exact couple, then its derived exact couple is
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.
(cohomological spectral sequence of an exact couple)
its derived exact couple
is itself an exact couple. Accordingly there is induced a sequence of exact couples
If the abelian groups $D$ and $E$ are equipped with [[bigraded object|bigrading]] such that
then $\{E_r^{\bullet,\bullet}, d_r\}$ with
is a cohomological spectral sequence, def. .
(As before in prop. , 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 $r \geq R_{s,t}$
$g \colon D^{s+R,t-R} \stackrel {\simeq}{\longrightarrow} D^{s+R -1, t-R-1}$;
$g\colon D^{s-R+1, t+R-2} \stackrel{0}{\longrightarrow} D^{s-R,t+R-1}$
then this spectral sequence converges to the [[inverse limit]] group
filtered by
(e.g. Kochmann 96, lemma 2.6.2)
We check the claimed form of the $E_\infty$-page:
Since $ker(h) = im(g)$ in the exact couple, the kernel
consists of those elements $x$ such that $g^{-r+2} (f(x)) = g(y)$, for some $y$, hence
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
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
Given a [[spectral sequence]] (def. ), then even if it converges strongly, computing its infinity-page still just gives the [[associated graded]] of the [[filtered object]] that it converges to, not the filtered object itself. The latter is in each filter stage an [[extension]] of the previous stage by the corresponding stage of the infinity-page, but there are in general several possible extensions (the trivial extension or some twisted extensions). The problem of determining these extensions and hence the problem of actually determining the filtered object from a spectral sequence converging to it is often referred to as the extension problem.
More in detail, consider, for definiteness, a cohomology spectral sequence converging to some [[filtered object|filtered]] $F^\bullet H^\bullet$
Then by definition of convergence there are isomorphisms
Equivalently this means that there are [[short exact sequences]] of the form
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]]
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 following proposition requires, in general, to evaluate cohomology functors not just on [[CW-complexes]], but on all topological spaces. Hence we invoke prop. to regard a [[reduced cohomology theory]] as a contravariant functor on all pointed topological spaces, which sends [[weak homotopy equivalences]] to isomorphisms (def. ).
(Serre-Cartan-Eilenberg-Whitehead-Atiyah-Hirzebruch spectral sequence)
Let $A^\bullet$ be a an additive unreduced [[generalized (Eilenberg-Steenrod) cohomology|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 topological space|weakly contractible]] or such that $B$ is [[simply connected topological space|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:
If at least one of the following two conditions is met
$B$ is [[finite number|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. ) for all $p$;
then there is a cohomology [[spectral sequence]], def. , 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
with respect to the filtering given by
where $X_{p} \coloneqq \pi^{-1}(B_{p})$ is the fiber over the $p$th stage of the [[CW-complex]] $B = \underset{\longleftarrow}{\lim}_n B_n$.
The exactness axiom for $A$ gives an [[exact couple]], def. , of the form
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
Let $C(s)$ be the set of $s$-dimensional cells of $B$, and notice that for $\sigma \in C(s)$ then
where $F_\sigma$ is [[weak homotopy equivalence|weakly homotopy equivalent]] to $F$ (exmpl.).
This implies that
where we used the relation to [[reduced cohomology]] $\tilde A$, prop. together with lemma , 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 cohomology|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 [[commuting diagram|commutes]] due to the [[natural transformation|naturality]] of the connecting homomorphism $\delta$ of $A^\bullet$:
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. 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. to obtain convergence.
By that proposition the convergence is to the [[inverse limit]]
If $X$ is finite dimensional or more generally if the sequences that this limit is over satisfy the [[Mittag-Leffler condition]] (def. ), then this limit is $A^\bullet(X)$, by prop. .
For $E^\bullet$ a [[multiplicative cohomology theory]] (def. ), then the Atiyah-Hirzebruch spectral sequences (prop. ) 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.
Given a multiplicative cohomology theory $(A,\mu,1)$ (def. ), then for every [[Serre fibration]] $X \to B$ (def.) all the differentials in the corresponding [[Atiyah-Hirzebruch spectral sequence]] of prop.
are linear over $A^\bullet(\ast)$.
By the proof of prop. , the differentials are those induced by the [[exact couple]]
consisting of the pullback homomorphisms and the connecting homomorphisms of $A$.
By prop. 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. .
For $E$ a [[homotopy commutative ring spectrum]] (def.) and $X$ a finite [[CW-complex]], then the [[Kronecker pairing]]
extends to a compatible pairing of [[Atiyah-Hirzebruch spectral sequences]].
Idea. As one passes from [[abelian groups]] to [[spectra]], a miracle happens: even though the latter are just the proper embodiment of [[linear algebra]] in the context of [[homotopy theory]] (“[[higher algebra]]”) their inspection reveals that spectra natively know about deep phenomena of [[differential topology]], [[index theory]] and in fact [[string theory]] (for instance via a close relation between [[genera and partition functions - table|genera and partition functions]]).
A strong manifestation of this phenomenon comes about in [[complex oriented cohomology theory]]/[[chromatic homotopy theory]] that we eventually come to below. It turns out to be higher algebra over the complex Thom spectrum [[MU]].
Here we first concentrate on its real avatar, the [[Thom spectrum]] [[MO]]. The seminal result of [[Thom’s theorem]] says that the [[stable homotopy groups]] of [[MO]] form the [[cobordism ring]] of [[cobordism]]-[[equivalence classes]] of [[manifolds]]. In the course of discussing this [[cobordism theory]] one encounters various phenomena whose complex version also governs the complex oriented cohomology theory that we are interested in below.
Literature. (Kochman 96, chapter I and sections II.2, II6). A quick efficient account is in (Malkiewich 11). See also (Aguilar-Gitler-Prieto 02, section 11).
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 $X$
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)
For $X$ a [[smooth manifold]] and $G$ a [[compact Lie group]] equipped with a [[free action|free]] smooth [[action]] on $X$, then the [[quotient]] [[projection]]
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)
For $G$ a [[Lie group]] and $H \subset G$ a [[compact Lie group|compact]] [[subgroup]], then the [[coset]] [[quotient]] [[projection]]
is an $H$-[[principal bundle]] (hence in particular a [[Serre fibration]]).
For $G$ a [[compact Lie group]] and $K \subset H \subset G$ [[closed subspace|closed]] [[subgroups]], then the [[projection]] map on [[coset spaces]]
is a locally trivial $H/K$-[[fiber bundle]] (hence in particular a [[Serre fibration]]).
Observe that the projection map in question is equivalently
(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 bundle|associated]] to the $H$-[[principal bundle]] of corollary .
The orthogonal group $O(n)$ is [[compact topological space]], hence in particular a [[compact Lie group]].
The unitary group $U(n)$ is [[compact topological space]], hence in particular a [[compact Lie group]].
The [[n-spheres]] are [[coset]] spaces of [[orthogonal groups]]:
The odd-dimensional spheres are also coset spaces of [[unitary groups]]:
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 [[stabilizer group|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 $U(n+1)$ [[transitive action|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
where $A \in U(n)$.
For $n,k \in \mathbb{N}$, $n \leq k$, then the canonical inclusion of [[orthogonal groups]]
is an [[n-equivalence|(n-1)-equivalence]], hence induces an [[isomorphism]] on [[homotopy groups]] in degrees $\lt n-1$ and a [[surjection]] in degree $n-1$.
Consider the [[coset]] [[quotient]] [[projection]]
By prop. and by corollary , the projection $O(n+1)\to O(n+1)/O(n)$ is a [[Serre fibration]]. Furthermore, example identifies the [[coset]] with the [[n-sphere]]
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
Since $\pi_{\lt n}(S^n) = 0$, this implies that
is an isomorphism and that
is surjective. Hence now the statement follows by [[induction]] over $k-n$.
Similarly:
For $n,k \in \mathbb{N}$, $n \leq k$, then the canonical inclusion of [[unitary groups]]
is a [[n-equivalence|2n-equivalence]], hence induces an [[isomorphism]] on [[homotopy groups]] in degrees $\lt 2n$ and a [[surjection]] in degree $2n$.
Consider the [[coset]] [[quotient]] [[projection]]
By prop. and corollary , the projection $U(n+1)\to U(n+1)/U(n)$ is a [[Serre fibration]]. Furthermore, example identifies the [[coset]] with the [[n-sphere|(2n+1)-sphere]]
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
Since $\pi_{\leq 2n}(S^{2n+1}) = 0$, this implies that
is an isomorphism and that
is surjective. Hence now the statement follows by induction over $k-n$.
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]].
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]].
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
here the [[action]] of $U(k-n)$ is via its canonical embedding $U(k-n)\hookrightarrow U(k)$.
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]].
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]].
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]].
For all $n \leq k \in \mathbb{N}$, the canonical [[projection]] from the [[Stiefel manifold]] (def. ) to the [[Grassmannian]] is a $O(n)$-[[principal bundle]]
and the projection from the complex Stiefel manifold to the Grassmannian us a $U(n)$-[[principal bundle]]:
The real [[Grassmannians]] $Gr_n(\mathbb{R}^k)$ and the complex Grassmannians $Gr_n(\mathbb{C}^k)$ of def. admit the structure of [[CW-complexes]]. Moreover the canonical inclusions
are subcomplex incusion (hence [[relative cell complex]] inclusions).
Accordingly there is an induced CW-complex structure on the [[classifying space]] (def. ).
A proof is spelled out in (Hatcher, section 1.2 (pages 31-34)).
The [[Stiefel manifolds]] $V_n(\mathbb{R}^k)$ and $V_n(\mathbb{C}^k)$ from def. admits the structure of a [[CW-complex]].
e.g. (James 59, p. 3, James 76, p. 5 with p. 21, Blaszczyk 07)
(And I suppose with that cell structure the inclusions $V_n(\mathbb{R}^k) \hookrightarrow V_n(\mathbb{R}^{k+1})$ are subcomplex inclusions.)
The real [[Stiefel manifold]] $V_n(\mathbb{R}^k)$ (def. ) is [[n-connected topological space|(k-n-1)-connected]].
Consider the [[coset]] [[quotient]] [[projection]]
By prop. and by corollary , 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. it has the following form in degrees bounded by $n$:
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:
The complex [[Stiefel manifold]] $V_n(\mathbb{C}^k)$ (def. ) is [[n-connected topological space|2(k-n)-connected]].
Consider the [[coset]] [[quotient]] [[projection]]
By prop. and by corollary 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. it has the following form in degrees bounded by $n$:
This implies the claim.
By def. there are canonical inclusions
and
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
and
respectively.
Moreover, by def. there are canonical inclusions
and
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
and
respectively.
The inclusions are in fact compatible with the bundle structure from prop. , so that there are induced projections
and
respectively. These are the standard models for the [[universal principal bundles]] for $O$ and $U$, respectively. The corresponding [[associated bundles|associated]] [[vector bundles]]
and
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]]
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]].
There are canonical inclusions
and
given by adjoining one coordinate to the ambient space and to any subspace. Under the colimit of def. these induce maps of classifying spaces
and
There are canonical maps
and
given by sending ambient spaces and subspaces to their [[direct sum]].
Under the colimit of def. these induce maps of classifying spaces
and
The colimiting space $E O(n) = \underset{\longrightarrow}{\lim}_k V_n(\mathbb{R}^k)$ from def. is [[weakly contractible topological space|weakly contractible]].
The colimiting space $E U(n) = \underset{\longrightarrow}{\lim}_k V_n(\mathbb{C}^k)$ from def. is [[weakly contractible topological space|weakly contractible]].
By propositions , and , the Stiefel manifolds are more and more highly connected as $k$ increases. Since the inclusions are relative cell complex inclusions by prop. , the claim follows.
The [[homotopy groups]] of the classifying spaces $B O(n)$ and $B U(n)$ (def. ) are those of the [[orthogonal group]] $O(n)$ and of the [[unitary group]] $U(n)$, respectively, shifted up in degree: there are [[isomorphisms]]
and
(for homotopy groups based at the canonical basepoint).
Consider the sequence
from def. , with $O(n)$ the [[fiber]]. Since (by prop. ) the second map is a [[Serre fibration]], this is a [[fiber sequence]] and so it induces a [[long exact sequence of homotopy groups]] of the form
Since by cor. $\pi_\bullet(E O(n))= 0$, exactness of the sequence implies that
is an isomorphism.
The same kind of argument applies to the complex case.
For $n \in \mathbb{N}$ there are [[homotopy fiber sequence]] (def.)
and
exhibiting the [[n-sphere]] ($(2n+1)$-sphere) as the [[homotopy fiber]] of the canonical maps from def. .
This means (thm.), that there is a replacement of the canonical inclusion $B O(n) \hookrightarrow B O(n+1)$ (induced via def. ) by a [[Serre fibration]]
such that $S^n$ is the ordinary [[fiber]] of $B O(n)\to \tilde B O(n+1)$, and analogously for the complex case.
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]]
By prop. both bottom vertical maps are [[Serre fibrations]] and so both vertical sequences are [[fiber sequences]]. By prop. part of the induced morphisms of [[long exact sequences of homotopy groups]] looks like this
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. , which gives the [[fiber sequence]]
The claim then follows with the identification
The argument for the complex case is directly analogous, concluding instead with the identification
Given a [[smooth manifold]] $X$ of [[dimension]] $n$ and equipped with an [[embedding]]
for some $k \in \mathbb{N}$, then the classifying map of its normal bundle is the function
which sends $x \in X$ to the normal of the [[tangent space]]
regarded as a point in $G_{k-n}(\mathbb{R}^k)$.
The [[normal bundle]] of $i$ itself is the subbundle of the [[tangent bundle]]
consisting of those vectors which are [[orthogonal]] to the [[tangent vectors]] of $X$:
A $(B,f)$-structure is
for each $n\in \mathbb{N}$ a [[pointed topological space|pointed]] [[CW-complex]] $B_n \in Top_{CW}^{\ast/}$
equipped with a pointed [[Serre fibration]]
to the [[classifying space]] $B O(n)$ (def.);
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 [[commuting square|squares commute]]
where the bottom map is the canonical one from def. .
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.)
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:
and
Similarly, an $S^2$-$(B,f)$-structure is a compatible system
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
for all $n \in \mathbb{N}$, hence for all $k n \in k \mathbb{N}$.
Examples of $(B,f)$-structures (def. ) include the following:
$B_n = B O(n)$ and $f_n = id$ is orthogonal structure (or “no structure”);
$B_n = E O(n)$ and $f_n$ the [[universal principal bundle]]-projection is [[framing]]-structure;
$B_n = B SO(n) = E O(n)/SO(n)$ the classifying space of the [[special orthogonal group]] and $f_n$ the canonical projection is [[orientation]] structure;
$B_n = B Spin(n) = E O(n)/Spin(n)$ the classifying space of the [[spin group]] and $f_n$ the canonical projection is [[spin structure]].
Examples of $S^2$-$(B,f)$-structures (def. ) include
$B_{2n} = B U(n) = E O(2n)/U(n)$ the classifying space of the [[unitary group]], and $f_{2n}$ the canonical projection is [[almost complex structure]] (or rather: [[almost Hermitian structure]]).
$B_{2n} = B Sp(2n) = E O(2n)/Sp(2n)$ 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. ) include
Given a [[smooth manifold]] $X$ of [[dimension]] $n$, and given a $(B,f)$-structure as in def. , then a $(B,f)$-structure on the stable normal bundle of the manifold is an [[equivalence class]] of the following structure:
an [[embedding]] $i_X \; \colon \; X \hookrightarrow \mathbb{R}^k$ for some $k \in \mathbb{N}$;
a [[homotopy class]] of a [[lift]] $\hat g$ of the classifying map $g$ of the normal bundle (def. )
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
$k_2 \geq k_1$
the second inclusion factors through the first as
the lift of the classifying map factors accordingly (as homotopy classes)
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)
Let $X$ be a [[topological space]] and let $V \to X$ be a [[vector bundle]] over $X$ of [[rank]] $n$, which is [[associated bundle|associated]] to an [[orthogonal group|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. ) 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:
the unit disk bundle $D(V) \to X$ is the subbundle of elements of [[norm]] $\leq 1$;
the unit sphere bundle $S(V)\to X$ is the subbundle of elements of norm $= 1$;
$S(V) \overset{i_V}{\hookrightarrow} D(V) \hookrightarrow V$;
the [[Thom space]] $Th(V)$ is the [[cofiber]] (formed in [[Top]] (prop.)) of $i_V$
canonically regarded as a [[pointed topological space]].
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.
If the [[rank]] of $V$ is positive, then $S(V)$ is non-empty and then the Thom space (def. ) is the [[quotient topological space]]
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
which exhibits $Th(V)$ as the [[coproduct]] of $X$ with the point, hence as $X$ with a basepoint freely adjoined.
Let $V \to X$ be a [[vector bundle]] over a [[CW-complex]] $X$. Then the Thom space $Th(V)$ (def. ) is equivalently the [[homotopy cofiber]] (def.) of the inclusion $S(V) \longrightarrow D(V)$ of the sphere bundle into the disk bundle.
The Thom space is defined as the ordinary [[cofiber]] of $S(V)\to D(V)$. Under the given assumption, this inclusion is a [[relative cell complex]] inclusion, hence a cofibration in the [[classical model structure on topological spaces]] (thm.). Therefore in this case the ordinary cofiber represents the homotopy cofiber (def.).
The equivalence to the following alternative model for this homotopy cofiber is relevant when discussing [[Thom isomorphisms]] and [[orientation in generalized cohomology]]:
Let $V \to X$ 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. ) is [[homotopy equivalence|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]]).
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:
Let $V_1,V_2 \to X$ be two real [[vector bundles]]. Then the Thom space (def. ) 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
Notice that
$D(V_1 \oplus V_2) \simeq D(V_2|_{Int D(V_1)}) \cup S(V_1)$;
$S(V_1 \oplus V_2) \simeq S(V_2|_{Int D(V_1)}) \cup Int D(V_2|_{S(V_1)})$.
(Since a point at radius $r$ 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 $V_1$.)
For $V$ a [[vector bundle]] then the Thom space (def. ) of $\mathbb{R}^n \oplus V$, the [[direct sum of vector bundles]] with the trivial rank $n$ vector bundle, is [[homeomorphism|homeomorphic]] to the [[smash product]] of the Thom space of $V$ with the $n$-[[sphere]] (the $n$-fold [[reduced suspension]]).
Apply prop. with $V_1 = \mathbb{R}^n$ and $V_2 = V$. Since $V_1$ is a trivial bundle, then
(as a bundle over $X\times D^n$) and similarly
By prop. and remark the Thom space (def. ) of a trivial vector bundle of rank $n$ is the $n$-fold [[suspension]] of the base space
Therefore a general Thom space may be thought of as a “twisted suspension”, with twist encoded by a vector bundle (or rather by its underlying [[spherical fibration]]). See at Thom spectrum – For infinity-module bundles for more on this.
Correspondingly the [[Thom isomorphism]] (prop. below) for a given Thom space is a twisted version of the [[suspension isomorphism]] (above).
For $V_1 \to X_1$ 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. ) is the [[smash product]] of the individual Thom spaces:
Given a [[vector bundle]] $V \to X$ of [[rank]] $n$, then the [[reduced cohomology|reduced]] [[ordinary cohomology]] of its [[Thom space]] $Th(V)$ (def. ) vanishes in degrees $\lt n$:
Consider the [[long exact sequence]] of [[relative cohomology]] (from above)
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 $H^{\bullet \lt n}(D(V),S(V)) = 0$.
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
where the bottom morphism is the canonical one (def.).
(e.g. Kochmann 96, p. 25)
For each $k \in \mathbb{N}$, $k \geq n$ there is such a pullback of the canonical vector bundles over [[Grassmannians]]
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. . Together this implies (via this lemma) that we may test the universal property of the colimiting square at finite stages. And so this implies the claim by the above fact that at each finite stage there is a pullback diagram.
The universal real [[Thom spectrum]] $M O$ is the [[spectrum]], which is represented by the [[sequential prespectrum]] (def.) whose $n$th component space is the [[Thom space]] (def. )
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. , via the homeomorphisms of prop. :
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.:
Given a [[(B,f)-structure]] $\mathcal{B}$ (def. ), write $V^\mathcal{B}_n$ for the [[pullback]] of the [[universal vector bundle]] (def. ) to the corresponding space of the $(B,f)$-structure and with
and we write $e_{n_1,n_2}$ for the maps of total space of vector bundles over the $g_{n_1,n_2}$:
Observe that the analog of prop. still holds:
Given a [[(B,f)-structure]] $\mathcal{B}$ (def. ), then the pullback of its rank-$(n+1)$ vector bundle $V^{\mathcal{B}}_{n+1}$ (def. ) 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
Unwinding the definitions, the pullback in question is
Given a [[(B,f)-structure]] $\mathcal{B}$ (def. ), its universal Thom spectrum $M \mathcal{B}$ is, as a [[sequential prespectrum]], given by component spaces being the [[Thom spaces]] (def. ) of the $\mathcal{B}$-associated vector bundles of def.
and with structure maps given via prop. by the top maps in prop. :
Similarly for an $S^k-(B,f)$-structure indexed on every $k$th natural number (such as [[almost complex structure]], [[almost quaternionic structure]], example ), 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. ), then the Thom spectrum $M \mathcal{B}$ canonical becomes a [[ring spectrum]] (for more on this see [[Introduction to Stable homotopy theory – 1-2|Part 1-2]] the section on orthogonal Thom spectra ): the multiplication maps $B_{n_1} \times B_{n_2}\to B_{n_1 + n_2}$ are covered by maps of vector bundles
and under forming [[Thom spaces]] this yields (via prop. ) maps
which are [[associativity|associative]] by the associativity condition in a multiplicative $(B,f)$-structure. The unit is
The universal [[Thom spectrum]] (def. ) for [[framing]] structure (exmpl.) is equivalently the [[sphere spectrum]] (def.)
Because in this case $B_n \simeq \ast$ and so $E^{\mathcal{B}}_n \simeq \mathbb{R}^n$, whence $Th(E^{\mathcal{B}}_n) \simeq S^n$.
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
for some $\epsilon \in \mathbb{R}$, $\epsilon \gt 0$, small enough such that the map
from the [[normal bundle]] (def. ) given by
is a [[diffeomorphism]].
([[tubular neighbourhood theorem]])
For every [[embedding]] of [[smooth manifolds]], there exists a [[tubular neighbourhood]] according to def. .
Given an embedding $i \colon X \hookrightarrow \mathbb{R}^k$ with a tubuluar neighbourhood $\tau_i X \hookrigtharrow \mathbb{R}^k$ (def. ) then by construction:
the [[Thom space]] (def. ) of the [[normal bundle]] (def. ) is [[homeomorphism|homeomorphic]] to the [[quotient topological space]] of the [[topological closure]] of the tubular neighbourhood by its [[boundary]]:
$Th(N_i(X)) \simeq \overline{ \tau_i(X)}/\partial \overline{\tau_i(X)}$;
there exists a continous function
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.
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. ) the composite map of [[pointed topological spaces]]
where the first map identifies the [[n-sphere|k-sphere]] as the [[one-point compactification]] of $\mathbb{R}^k$; and where the second and third maps are those of remark .
The Pontrjagin-Thom construction is the further composite
with the image under the [[Thom space]] construction of the morphism of vector bundles
induced by the classifying map $g_i$ of the normal bundle (def. ).
This defines an element
in the $n$th [[stable homotopy group]] (def.) of the [[Thom spectrum]] $M O$ (def. ).
More generally, for $X$ a smooth manifold with normal [[(B,f)-structure]] $(X,i,\hat g_i)$ according to def. , then its Pontrjagin-Thom construction is the composite
with
The [[Pontrjagin-Thom construction]] (def. ) respects the equivalence classes entering the definition of manifolds with stable normal $\mathcal{B}$-structure (def. ) hence descends to a [[function]] (of [[sets]])
It is clear that the homotopies of classifying maps of $\mathcal{B}$-structures that are devided out in def. map to homotopies of representatives of stable homotopy groups. What needs to be shown is that the construction respects the enlargement of the embedding spaces.
Given a embedded manifold $X \overset{i}{\hookrightarrow}\mathbb{R}^{k_1}$ with normal $\mathcal{B}$-structure
write
for its image under the [[Pontrjagin-Thom construction]] (def. ). 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
By prop. 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
Hence the Pontrjagin-Thom construction for the enlarged embedding space is (using prop. ) the composite
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. ):
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 $\pi_{\bullet}(M \mathcal{B})$.
Idea. By the [[Pontryagin-Thom collapse]] construction above, there is an assignment
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]]
[[Thom’s theorem]] states that this homomorphism is an [[isomorphism]].
More generally, for $\mathcal{B}$ a multiplicative [[(B,f)-structure]], def. , there is such an identification
between the ring of $\mathcal{B}$-cobordism classes of manifolds with $\mathcal{B}$-structure and the [[stable homotopy groups]] of the universal $\mathcal{B}$-[[Thom spectrum]].
Literature. (Kochman 96, 1.5)
Throughout, let $\mathcal{B}$ be a multiplicative [[(B,f)-structure]] (def. ).
Write $I \coloneqq [0,1]$ for the standard interval, regarded as a [[smooth manifold]] [[manifold with boundary|with boundary]]. For $c \in \mathbb{R}_+$ Consider its embedding
as the arc
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 ) by equipping it with the canonical framing
of its [[normal bundle]].
Let now $\mathcal{B}$ be a [[(B,f)-structure]] (def. ). 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. ), define its negative or orientation reversal $-(X,i,\hat g)$ of $(X,i, \hat g)$ to be the restriction of the structured manifold
to $t = 1$.
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.
The [[relation]] of $\mathcal{B}$-[[bordism]] (def. ) is an [[equivalence relation]].
Write $\Omega^\mathcal{B}_{\bullet}$ for the $\mathbb{N}$-graded set of $\mathcal{B}$-bordism classes of $\mathcal{B}$-manifolds.
Under [[disjoint union]] of manifolds, then the set of $\mathcal{B}$-bordism equivalence classes of def. becomes an $\mathbb{Z}$-graded [[abelian group]]
(that happens to be concentrated in non-negative degrees). This is called the $\mathcal{B}$-bordism group.
Moreover, if the [[(B,f)-structure]] $\mathcal{B}$ is multiplicative (def. ), then [[Cartesian product]] of manifolds followed by the multiplicative composition operation of $\mathcal{B}$-structures makes the $\mathcal{B}$-bordism ring into a [[commutative ring]], called the $\mathcal{B}$-bordism ring.
e.g. (Kochmann 96, prop. 1.5.3)
Recall that the [[Pontrjagin-Thom construction]] (def. ) associates to an embbeded manifold $(X,i,\hat g)$ with normal $\mathcal{B}$-structure (def. ) 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.
For $\mathcal{B}$ be a multiplicative [[(B,f)-structure]] (def. ), the $\mathcal{B}$-[[Pontrjagin-Thom construction]] (def. ) is compatible with all the relations involved to yield a graded [[ring]] [[homomorphism]]
from the $\mathcal{B}$-[[bordism ring]] (def. ) to the [[stable homotopy groups]] of the universal $\mathcal{B}$-[[Thom spectrum]] equipped with the ring structure induced from the canonical [[ring spectrum]] structure (def. ).
By prop. the underlying function of sets is well-defined before dividing out the bordism relation (def. ). To descend this further to a function out of the set underlying the bordism ring, we need to see that the Pontrjagin-Thom construction respects the bordism relation. But the definition of bordism is just so as to exhibit under $\xi$ a [[left homotopy]] of representatives of homotopy groups.
Next we need to show that it is
a group homomorphism;
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 [[n-cube|k-cube]], glued at one face
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 ) 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. ) 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. there is a diagram of the form
To the Pontrjagin-Thom construction of the product manifold is by definition the top composite in the diagram
which hence is equivalently the bottom composite, which in turn manifestly represents the product of the separate PT constructions in $\pi_\bullet(M\mathcal{B})$.
Due to (Thom 54, Pontrjagin 55). See for instance (Kochmann 96, theorem 1.5.10).
Observe that given the result $\alpha \colon S^{n+(k-n)} \to Th(V_{k-n})$ of the Pontrjagin-Thom construction map, the original manifold $X \overset{i}{\hookrightarrow} \mathbb{R}^k$ may be recovered as this [[pullback]]:
To see this more explicitly, break it up into pieces:
Moreover, since the [[n-spheres]] are [[compact topological spaces]], and since the [[classifying space]] $B O(n)$, and hence its universal Thom space, is a [[sequential colimit]] over [[relative cell complex]] inclusions, the right vertical map factors through some finite stage (by this lemma), the manifold $X$ is equivalently recovered as a pullback of the form
(Recall that $V^{\mathcal{B}}_{k-n}$ is our notation for the [[universal vector bundle]] with $\mathcal{B}$-structure, while $V_{k-n}(\mathbb{R}^k)$ denotes a [[Stiefel manifold]].)
The idea of the proof now is to use this property as the blueprint of the construction of an [[inverse]] $\zeta$ to $\xi$: given an element in $\pi_{n}(M \mathcal{B})$ represented by a map as on the right of the above diagram, try to define $X$ and the structure map $g_i$ of its normal bundle as the pullback on the left.
The technical problem to be overcome is that for a general continuous function as on the right, the pullback has no reason to be a smooth manifold, and for two reasons:
the map $S^{n+(k-n)} \to Th(V_{k-n})$ may not be smooth around the image of $i$;
even if it is smooth around the image of $i$, it may not be [[transversal map|transversal]] to $i$, and the intersection of two non-transversal smooth functions is in general still not a smooth manifold.
The heart of the proof is in showing that for any $\alpha$ there are small homotopies relating it to an $\alpha'$ that is both smooth around the image of $i$ and transversal to $i$.
The first condition is guaranteed by [[Sard’s theorem]], the second by [[Thom’s transversality theorem]].
(…)
Idea. If a [[vector bundle]] $E \stackrel{p}{\longrightarrow} X$ of [[rank]] $n$ carries a cohomology class $\omega \in H^n(Th(E),R)$ that looks fiberwise like a [[volume form]] – a [[Thom class]] – then the operation of pulling back from base space and then forming the [[cup product]] with this [[Thom class]] is an [[isomorphism]] on (reduced) cohomology
This is the [[Thom isomorphism]]. It follows from the [[Serre spectral sequence]] (or else from the [[Leray-Hirsch theorem]]). A closely related statement gives the [[Thom-Gysin sequence]].
In the special case that the vector bundle is trivial of rank $n$, then its [[Thom space]] coincides with the $n$-fold [[suspension]] of the base space (example ) and the Thom isomorphism coincides with the [[suspension isomorphism]]. In this sense the Thom isomorphism may be regarded as a twisted suspension isomorphism.
We need this below to compute (co)homology of universal Thom spectra $M U$ in terms of that of the [[classifying spaces]] $B U$.
Composed with pullback along the [[Pontryagin-Thom collapse map]], the Thom isomorphism produces maps in cohomology that covariantly follow the underlying maps of spaces. These “[[Umkehr maps]]” have the interpretation of [[fiber integration]] against the Thom class.
Literature. (Kochman 96, 2.6)
The [[Thom-Gysin sequence]] is a type of [[long exact sequence in cohomology]] induced by a [[spherical fibration]] and expressing the [[cohomology groups]] of the total space in terms of those of the base plus correction. The sequence may be obtained as a corollary of the [[Serre spectral sequence]] for the given fibration. It induces, and is induced by, the [[Thom isomorphism]].
Let $R$ be a [[commutative ring]] and let
be a [[Serre fibration]] over a [[simply connected topological space|simply connected]] [[CW-complex]] with typical [[fiber]] (exmpl.) the [[n-sphere]].
Then there exists an element $c \in H^{n+1}(E; R)$ (in the [[ordinary cohomology]] of the total space with [[coefficients]] in $R$, called the Euler class of $\pi$) such that the [[cup product]] operation $c \cup (-)$ sits in a [[long exact sequence]] of [[cohomology groups]] of the form
(e.g. Switzer 75, section 15.30, Kochman 96, corollary 2.2.6)
Under the given assumptions there is the corresponding [[Serre spectral sequence]]
Since the [[ordinary cohomology]] of the [[n-sphere]] [[fiber]] is concentrated in just two degees
the only possibly non-vanishing terms on the $E_2$ page of this spectral sequence, and hence on all the further pages, are in bidegrees $(\bullet,0)$ and $(\bullet,n)$:
As a consequence, since the differentials $d_r$ on the $r$th page of the Serre spectral sequence have bidegree $(r+1,-r)$, the only possibly non-vanishing differentials are those on the $(n+1)$-page of the form
Now since the [[coefficients]] $R$ is a [[ring]], the [[Serre spectral sequence]] is [[multiplicative spectral sequence|multiplicative]] under [[cup product]] and the [[differential]] is a [[derivation]] (of total degree 1) with respect to this product. (See at multiplicative spectral sequence – Examples – AHSS for multiplicative cohomology.)
To make use of this, write
for the unit in the [[cohomology ring]] $H^\bullet(B;R)$, but regarded as an element in bidegree $(0,n)$ on the $(n+1)$-page of the spectral sequence. (In particular $\iota$ does not denote the unit in bidegree $(0,0)$, and hence $d_{n+1}(\iota)$ need not vanish; while by the [[derivation]] property, it does vanish on the actual unit $1 \in H^0(B;R) \simeq E_{n+1}^{0,0}$.)
Write
for the image of this element under the differential. We will show that this is the Euler class in question.
To that end, notice that every element in $E_{n+1}^{\bullet,n}$ is of the form $\iota \cdot b$ for $b\in E_{n+1}^{\bullet,0} \simeq H^\bullet(B;R)$.
(Because the [[multiplicative spectral sequence|multiplicative structure]] gives a group homomorphism $\iota \cdot(-) \colon H^\bullet(B;R) \simeq E_{n+1}^{0,0} \to E^{0,n}_{n+1} \simeq H^\bullet(B;R)$, which is an isomorphism because the product in the spectral sequence does come from the [[cup product]] in the [[cohomology ring]], see for instance (Kochman 96, first equation in the proof of prop. 4.2.9), and since hence $\iota$ does act like the unit that it is in $H^\bullet(B;R)$).
Now since $d_{n+1}$ is a graded [[derivation]] and vanishes on $E_{n+1}^{\bullet,0}$ (by the above degree reasoning), it follows that its action on any element is uniquely fixed to be given by the product with $c$:
This shows that $d_{n+1}$ is identified with the cup product operation in question:
In summary, the non-vanishing entries of the $E_\infty$-page of the spectral sequence sit in [[exact sequences]] like so
Finally observe (lemma ) that due to the sparseness of the $E_\infty$-page, there are also [[short exact sequences]] of the form
Concatenating these with the above exact sequences yields the desired [[long exact sequence]].
Consider a cohomology [[spectral sequence]] converging to some [[filtered object|filtered]] [[graded abelian group]] $F^\bullet C^\bullet$ such that
$F^0 C^\bullet = C^\bullet$;
$F^{s} C^{\lt s} = 0$;
$E_\infty^{s,t} = 0$ unless $t = 0$ or $t = n$,
for some $n \in \mathbb{N}$, $n \geq 1$. Then there are [[short exact sequences]] of the form
(e.g. Switzer 75, p. 356)
By definition of convergence of a spectral sequence, the $E_{\infty}^{s,t}$ sit in [[short exact sequences]] of the form
So when $E_\infty^{s,t} = 0$ then the morphism $i$ above is an [[isomorphism]].
We may use this to either shift away the filtering degree
or to shift away the offset of the filtering to the total degree:
Moreover, by the assumption that if $t \lt 0$ then $F^{s}C^{s+t} = 0$, we also get
In summary this yields the vertical isomorphisms
and hence with the top sequence here being exact, so is the bottom sequence.
Let $V \to B$ be a topological [[vector bundle]] of [[rank]] $n \gt 0$ over a [[simply connected topological space|simply connected]] [[CW-complex]] $B$. Let $R$ be a [[commutative ring]].
There exists an element $c \in H^n(Th(V);R)$ (in the [[ordinary cohomology]], with [[coefficients]] in $R$, of the [[Thom space]] of $V$, called a [[Thom class]]) such that forming the [[cup product]] with $c$ induces an [[isomorphism]]
of degree $n$ from the unreduced [[cohomology group]] of $B$ to the [[reduced cohomology]] of the [[Thom space]] of $V$.
Choose an [[orthogonal structure]] on $V$. Consider the fiberwise [[cofiber]]
of the inclusion of the unit sphere bundle into the unit disk bundle of $V$ (def. ).
Observe that this has the following properties
$E \overset{p}{\to} B$ is an [[n-sphere]] [[fiber bundle]], hence in particular a [[Serre fibration]];
the [[Thom space]] $Th(V)\simeq E/B$ is the quotient of $E$ by the base space, because of the [[pasting law]] applied to the following pasting diagram of [[pushout]] squares
hence the [[reduced cohomology]] of the Thom space is (def.) the [[relative cohomology]] of $E$ relative $B$
$E \overset{p}{\to} B$ has a global [[section]] $B \overset{s}{\to} E$ (given over any point $b \in B$ by the class of any point in the fiber of $S(V) \to B$ over $b$; or abstractly: induced via the above pushout by the commutation of the projections from $D(V)$ and from $S(V)$, respectively).
In the following we write $H^\bullet(-)\coloneqq H^\bullet(-;R)$, for short.
By the first point, there is the [[Thom-Gysin sequence]] (prop. ), an [[exact sequence]] running vertically in the following diagram
By the second point above this is [[split exact sequence|split]], as shown by the diagonal isomorphism in the top right. By the third point above there is the horizontal exact sequence, as shown, which is the exact sequence in relative cohomology $\cdots \to H^\bullet(E,B) \to H^\bullet(E) \to H^\bullet(B) \to \cdots$ induced from the section $B \hookrightarrow E$.
Hence using the splitting to decompose the term in the middle as a [[direct sum]], and then using horizontal and vertical exactness at that term yields
and hence an isomorphism
To see that this is the inverse of a morphism of the form $c \cup (-)$, inspect the proof of the Gysin sequence. This shows that $H^{\bullet-n}(B)$ here is identified with elements that on the second page of the corresponding [[Serre spectral sequence]] are cup products
with $\iota$ fiberwise the canonical class $1 \in H^n(S^n)$ and with $b \in H^\bullet(B)$ any element. Since $H^\bullet(-;R)$ is a [[multiplicative cohomology theory]] (because the [[coefficients]] form a [[ring]] $R$), cup producs are preserved as one passes to the $E_\infty$-page of the spectral sequence, and the morphism $H^\bullet(E) \to B^\bullet(B)$ above, hence also the isomorphism $\tilde H^\bullet(Th(V)) \to H^\bullet(B)$, factors through the $E_\infty$-page (see towards the end of the proof of the Gysin sequence). Hence the image of $\iota$ on the $E_\infty$-page is the Thom class in question.
Idea. From the way the [[Thom isomorphism]] via a [[Thom class]] works in [[ordinary cohomology]] (as above), one sees what the general concept of [[orientation in generalized cohomology]] and of [[fiber integration in generalized cohomology]] is to be.
Specifically we are interested in [[complex oriented cohomology]] theories $E$, characterized by an orientation class on infinity [[complex projective space]] $\mathbb{C}P^\infty$ (def. ), the [[classifying space]] for [[complex line bundles]], which restricts to a generator on $S^2 \hookrightarrow \mathbb{C}P^\infty$.
(Another important application is given by taking $E =$ [[KU]] to be [[topological K-theory]]. Then [[orientation in generalized cohomology|orientation]] is [[spin^c structure]] and fiber integration with coefficients in $E$ is [[fiber integration in K-theory]]. This is classical [[index theory]].)
Literature. (Kochman 96, section 4.3, Adams 74, part III, section 10, Lurie 10, lecture 5)
$\,$
Let $E$ be a [[multiplicative cohomology theory]] (def. ) and let $V \to X$ be a topological [[vector bundle]] of [[rank]] $n$. Then an $E$-[[orientation in generalized cohomology|orientation]] or $E$-[[Thom class]] on $V$ is an element of degree $n$
in the [[reduced cohomology|reduced]] $E$-[[cohomology ring]] of the [[Thom space]] (def. ) of $V$, such that for every point $x \in X$ its restriction $i_x^* u$ along
(for $\mathbb{R}^n \overset{fib_x}{\hookrightarrow} V$ the [[fiber]] of $V$ over $x$) is a generator, in that it is of the form
for
$\epsilon \in \tilde E^0(S^0)$ a [[unit]] in $E^\bullet$;
$\gamma_n \in \tilde E^n(S^n)$ the image of the multiplicative unit under the [[suspension isomorphism]] $\tilde E^0(S^0) \stackrel{\simeq}{\to}\tilde E^n(S^n)$.
(e.g. Kochmann 96, def. 4.3.4)
Recall that a [[(B,f)-structure]] $\mathcal{B}$ (def. ) is a system of [[Serre fibrations]] $B_n \overset{f_n}{\longrightarrow} B O(n)$ over the [[classifying spaces]] for [[orthogonal structure]] equipped with maps
covering the canonical inclusions of classifying spaces. For instance for $G_n \to O(n)$ a compatible system of [[topological group]] [[homomorphisms]], then the $(B,f)$-structure given by the [[classifying spaces]] $B G_n$ (possibly suitably resolved for the maps $B G_n \to B O(n)$ to become Serre fibrations) defines [[G-structure]].
Given a $(B,f)$-structure, then there are the [[pullbacks]] $V^{\mathcal{B}}_n \coloneqq f_n^\ast (E O(n)\underset{O(n)}{\times}\mathbb{R}^n)$ of the [[universal vector bundles]] over $B O(n)$, which are the universal vector bundles equipped with $(B,f)$-structure
Finally recall that there are canonical morphisms (prop.)
Let $E$ be a [[multiplicative cohomology theory]] and let $\mathcal{B}$ be a multiplicative [[(B,f)-structure]]. Then a universal $E$-orientation for vector bundles with $\mathcal{B}$-structure is an $E$-orientation, according to def. , for each rank-$n$ universal vector bundle with $\mathcal{B}$-structure:
such that these are compatible in that
for all $n \in \mathbb{N}$ then
where
(with the first isomorphism is the [[suspension isomorphism]] of $E$ and the second exhibiting the [[homeomorphism]] of Thom spaces $Th(\mathbb{R} \oplus V)\simeq \Sigma Th(V)$ (prop. ) and where
is pullback along the canonical $\phi_n \colon \mathbb{R}\oplus V_n \to V_{n+1}$ (prop. ).
for all $n_1, n_2 \in \mathbb{N}$ then
A universal $E$-orientation, in the sense of def. , for vector bundles with [[(B,f)-structure]] $\mathcal{B}$, is equivalently (the homotopy class of) a homomorphism of [[ring spectra]]
from the universal $\mathcal{B}$-[[Thom spectrum]] to a spectrum which via the [[Brown representability theorem]] (theorem ) represents the given [[generalized (Eilenberg-Steenrod) cohomology theory]] $E$ (and which we denote by the same symbol).
The [[Thom spectrum]] $M\mathcal{B}$ has a standard structure of a [[CW-spectrum]]. Let now $E$ denote a [[sequential spectrum|sequential]] [[Omega-spectrum]] representing the multiplicative cohomology theory of the same name. Since, in the standard [[model structure on topological sequential spectra]], [[CW-spectra]] are cofibrant (prop.) and Omega-spectra are fibrant (thm.) we may represent all morphisms in the [[stable homotopy category]] (def.) by actual morphisms
of sequential spectra (due to this lemma).
Now by definition (def.) such a homomorphism is precissely a sequence of base-point preserving [[continuous functions]]
for $n \in \mathbb{N}$, such that they are compatible with the structure maps $\sigma_n$ and equivalently with their $(S^1 \wedge(-)\dashv Maps(S^1,-)_\ast)$-[[adjuncts]] $\tilde \sigma_n$, in that these diagrams commute:
for all $n \in \mathbb{N}$.
First of all this means (via the identification given by the [[Brown representability theorem]], see prop. , that the components $\xi_n$ are equivalently representatives of elements in the [[cohomology groups]]
(which we denote by the same symbol, for brevity).
Now by the definition of universal [[Thom spectra]] (def. , def. ), the structure map $\sigma_n^{M\mathcal{B}}$ is just the map $\phi_n \colon \mathbb{R}\oplus Th(V^{\mathcal{B}}_n)\to Th(V_{n+1}^{\mathcal{B}})$ from above.
Moreover, by the [[Brown representability theorem]], the [[adjunct]] $\tilde \sigma_n^E \circ \xi_n$