nLab Thom isomorphism

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Contents

This entry is about the isomorphisms in cohomology induced by Thom classes. For the Pontrjagin-Thom isomorphism in cobordism theory see at Thom's theorem.

Context

Manifolds and cobordisms

Cohomology

Idea
Statement

Concretely

In ordinary cohomology
In generalized cohomology

Abstractly

Applications
Related concepts
References

Idea

For $H$ denoting ordinary cohomology with coefficients in some ring, and $V \to X$ a vector bundle of rank $n$ over a simply connected CW-complex, the Thom isomorphism is the morphism

c \cup (-) \;\colon\; H^\bullet(X) \stackrel{\simeq}{\longrightarrow} \tilde H^{\bullet + n}(Th(V)) \,.

from the cohomology of $X$ to the reduced cohomology of the Thom space $Th(V)$ , given by pullback to the Thom space followed by cup product with a Thom class $c \in H^n(Th(V))$ .

That this is indeed an isomorphism follows via the Leray-Hirsch theorem (see e.g. Ebert 12, 2.3,2.4) or from running a Serre spectral sequence (e.g. Kochman 96, section 2.6).

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 (exmpl.) and the Thom isomorphism coincides with the suspension isomorphism. In this sense the Thom isomorphism may be regarded as a twisted suspension isomorphism.

More generally, for $E$ a generalized multiplicative cohomology theory, and $V \to X$ a vector bundle of rank $n$ , which is $E$ -orientable, there is a generalization to a Thom-Dold isomorphism

c \cup (-) \;\colon\; E^\bullet(X) \stackrel{\simeq}{\longrightarrow} \tilde E^{\bullet + n}(Th(V))

(e.g. Rudyak 98, chapter V, 1.3, Kochman 96, prop. 4.3.6)

One may think of the Thom isomorphism from left to right as cupping with a generalized volume form on the fibers, and from right to left as performing fiber integration against this volume form.

Statement

Concretely

In ordinary cohomology

Proposition

Let $V \to B$ be a topological vector bundle of rank $n \gt 0$ over a 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

H^\bullet(B;R) \overset{c \cup (-)}{\longrightarrow} \tilde H^{\bullet + n}(Th(V);R)

of degree $n$ from the unreduced cohomology group of $B$ to the reduced cohomology of the Thom space of $V$ .

Proof

(of Thom isomorphism via fiberwise Thom spaces)

Choose an orthogonal structure on $V$ . Consider the fiberwise cofiber

E \coloneqq D(V)/_B S(V)

of the inclusion of the unit sphere bundle into the unit disk bundle of $V$ (def.).

\array{ S^{n-1} &\hookrightarrow& D^n &\longrightarrow& S^n \\ \downarrow && \downarrow && \downarrow \\ S(V) &\hookrightarrow& D(V) &\longrightarrow& E \\ \downarrow && \downarrow && \downarrow^{\mathrlap{p}} \\ B &=& B &=& B }

Observe that this has the following properties

$E \overset{p}{\to} B$ is an n-sphere fiber bundle (spherical fibration), 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:

$\array{ S(V) &\longrightarrow& D(V) \\ \downarrow &(po)& \downarrow \\ B &\longrightarrow& D(V)/_B S(V) \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& Th(V) }$
hence the reduced cohomology of the Thom space is (def.) the relative cohomology of $E$ relative to $B$ :

$\tilde H^\bullet(Th(V);R) \simeq H^\bullet(E,B;R) \,.$
$E \overset{p}{\to} B$ 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, an exact sequence running vertically in the following diagram

\array{ && H^\bullet(B) \\ && {}^{\mathllap{p^\ast}}\downarrow & \searrow^{\mathrlap{\simeq}} \\ \tilde H^\bullet(Th(V)) &\longrightarrow& H^\bullet(E) &\underset{s^\ast}{\longrightarrow}& H^\bullet(B) \\ && \downarrow \\ && H^{\bullet-n}(B) } \,.

By the second point above this is split, 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

\array{ && H^\bullet(B) \\ && {}^{\mathllap{(0,id)}}\downarrow & \searrow^{\mathrlap{\simeq}} \\ \tilde H^\bullet(Th(V)) &\overset{(id,0)}{\hookrightarrow}& \tilde H^\bullet(Th(V)) \oplus H^\bullet(B) &\underset{(0,id)}{\longrightarrow}& H^\bullet(B) \\ && \downarrow^{\mathrlap{(id,0)}} \\ && H^{\bullet-n}(B) }

and hence an isomorphism

\tilde H^\bullet(Th(V)) \overset{\simeq}{\longrightarrow} H^{\bullet-n}(B) \,.

To see that this is the inverse of a morphism of the form $c \cup (-)$ , 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

\iota \cup b

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.

In generalized cohomology

Let $E$ be a generalized (Eilenberg-Steenrod) cohomology theory. First observe that an E-orientation on $V \to X$ induces an $H \pi_0(E)$ -orientation, i.e. in ordinary cohomology with coefficients in the degree-0 ground ring.

To see this, let’s assume $E$ is connective. Consider the relative Atiyah-Hirzebruch spectral sequence

\tilde H^p(Th(V), E^q(\ast)) \simeq H^p(D(V),S(V), E^q(\ast)) \;\Rightarrow\; E^\bullet(D(V), S(V)) \simeq \tilde E^\bullet(Th(V))

Since $(D(V), S(V))$ is $(n-1)$ -connected for a rank $n$ vector bundle, then $E_2^{p \lt n, q} = 0$ . Hence the edge homomorphism

\tilde H^k(Th(V), E^0(\ast)) \longrightarrow \tilde H^0(Th(V))

is an isomorphism, and one checks that it sends Thom classes to Thom classes.

For a fully detailed account see (Pedrotti 16).

Abstractly

A general abstract discussion is around page 30, 31 of (ABGHR).

(…)

Applications

The Thom isomorphism is used to define fiber integration of multiplicative cohomology theories.

Related concepts

References

The original proof that the Thom isomorphism is indeed an isomorphism is due to

René Thom, Quelques propriétés globales des variétés différentiables Comm. Math. Helv. , 28 (1954) pp. 17–86

The argument via a Serre spectral sequence for a relative fibration seems to be due to

W. H. Cockcroft, On the Thom isomorphism Theorem, Mathematical Proceedings of the Cambridge Philosophical Society (1962), 58 (web, pdf)

Textbook accounts:

Yuli Rudyak, chapter V of Thom spectra, Orientability and Cobordism, Springer 1998 (pdf)

Discussion in the generality of complex oriented cohomology theory:

Pierre Conner, Edwin Floyd, Section 6 of: The Relation of Cobordism to K-Theories, Lecture Notes in Mathematics 28 Springer 1966 (doi:10.1007/BFb0071091, MR216511)
Stanley Kochman, Section 2.6 of: Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Dai Tamaki, Akira Kono, Section 3.6 in: Generalized Cohomology, Translations of Mathematical Monographs, American Mathematical Society, 2006 (ISBN: 978-0-8218-3514-2)

Lecture notes:

Akhil Mathew, The Thom isomorphism theorem, 2010
Johannes Ebert, sections 2.3, 2.4 of A lecture course on Cobordism Theory, 2012 (pdf)
Michael Hopkins (notes by Akhil Mathew), Lecture 21 in: Spectra and stable homotopy theory, 2012 (pdf, pdf)
Dan Freed, lecture 8 of Bordism: old and new, 2013 (pdf)
Riccardo Pedrotti, Complex oriented cohomology, generalized orientation and Thom isomorphism, 2016, 2018 (pdf)

A discussion in differential geometry with fiberwise compactly supported differential forms is around theorem 6.17 of

Raoul Bott, Loring Tu, Differential Forms in Algebraic Topology

A comprehensive general abstract account for multiplicative cohomology theories in terms of E-infinity ring spectra is in

Matthew Ando, Andrew Blumberg, David Gepner, Michael Hopkins, Charles Rezk, Units of ring spectra and Thom spectra (arXiv:0810.4535)

An alternative simple formulation in terms of geometric cycles as in bivariant cohomology theory is in

Martin Jakob, A note on the Thom isomorphism in geometric (co)homology (arXiv:math/0403540)