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Thom isomorphism

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.


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Manifolds and cobordisms

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cohomology

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Contents

Idea

For HH being ordinary cohomology with coefficients in a ring, and VXV \to X a vector bundle of rank nn over a simply connected CW-complex, the Thom isomorphism is the morphism

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

from the cohomology of XX to the reduced cohomology of the Thom space Th(V)Th(V), given by pullback to the Thom space followed by cup product with a Thom class cH n(Th(V))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. Kochmann 96, section 2.6).

In the special case that the vector bundle is trivial of rank nn, then its Thom space coincides with the nn-fold suspension of the base space (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 EE a multiplicative cohomology theory, and VXV \to X a vector bundle of rank nn, which is EE-orientable, there is a generalization to a Thom-Dold isomorphism

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

(e.g. Rudyak 98, chapter V, 1.3, Kochmann 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 VBV \to B be a topological vector bundle of rank n>0n \gt 0 over a simply connected CW-complex BB. Let RR be a commutative ring.

There exists an element cH n(Th(V);R)c \in H^n(Th(V);R) (in the ordinary cohomology, with coefficients in RR, of the Thom space of VV, called a Thom class) such that forming the cup product with cc induces an isomorphism

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

of degree nn from the unreduced cohomology group of BB to the reduced cohomology of the Thom space of VV.

Proof

(of Thom isomorphism via fiberwise Thom spaces)

Choose an orthogonal structure on VV. Consider the fiberwise cofiber

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

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

S n1 D n S n S(V) D(V) E p B = B = B \array{ S^{n-1} &\hookrightarrow& D^n &\longrightarrow& S^n \\ \downarrow && \downarrow && \downarrow \\ S(V) &\hookrightarrow& D(V) &\longrightarrow& E \\ \downarrow && \downarrow && \downarrow^{\mathrlap{p}} \\ B &=& B &=& B }

Observe that this has the following properties

  1. EpBE \overset{p}{\to} B is an n-sphere fiber bundle, hence in particular a Serre fibration;

  2. the Thom space Th(V)E/BTh(V)\simeq E/B is the quotient of EE by the base space, because of the pasting law applied to the following pasting diagram of pushout squares

    S(V) D(V) (po) B D(V)/ BS(V) (po) * Th(V) \array{ S(V) &\longrightarrow& D(V) \\ \downarrow &(po)& \downarrow \\ B &\longrightarrow& D(V)/_B S(V) \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& Th(V) }
  3. hence the reduced cohomology of the Thom space is (def.) the relative cohomology of EE relative BB

    H˜ (Th(V);R)H (E,B;R). \tilde H^\bullet(Th(V);R) \simeq H^\bullet(E,B;R) \,.
  4. EpBE \overset{p}{\to} B has a global section BsEB \overset{s}{\to} E (given over any point bBb \in B by the class of any point in the fiber of S(V)BS(V) \to B over bb; or abstractly: induced via the above pushout by the commutation of the projections from D(V)D(V) and from S(V)S(V), respectively).

In the following we write H ()H (;R)H^\bullet(-)\coloneqq H^\bullet(-;R), for short.

By the first point, there is the Thom-Gysin sequence, an exact sequence running vertically in the following diagram

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

By the second point above this is split, as shown by the diagonal isomorphism in the top right. By the third point above there is the horizontal exact sequence, as shown, which is the exact sequence in relative cohomology H (E,B)H (E)H (B)\cdots \to H^\bullet(E,B) \to H^\bullet(E) \to H^\bullet(B) \to \cdots induced from the section BEB \hookrightarrow E.

Hence using the splitting to decompose the term in the middle as a direct sum, and then using horizontal and vertical exactness at that term yields

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

and hence an isomorphism

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

To see that this is the inverse of a morphism of the form c()c \cup (-), inspect the proof of the Gysin sequence. This shows that H n(B)H^{\bullet-n}(B) here is identified with elements that on the second page of the corresponding Serre spectral sequence are cup products

ιb \iota \cup b

with ι\iota fiberwise the canonical class 1H n(S n)1 \in H^n(S^n) and with bH (B)b \in H^\bullet(B) any element. Since H (;R)H^\bullet(-;R) is a multiplicative cohomology theory (because the coefficients form a ring RR), cup producs are preserved as one passes to the E E_\infty-page of the spectral sequence, and the morphism H (E)B (B)H^\bullet(E) \to B^\bullet(B) above, hence also the isomorphism H˜ (Th(V))H (B)\tilde H^\bullet(Th(V)) \to H^\bullet(B), factors through the E E_\infty-page (see towards the end of the proof of the Gysin sequence). Hence the image of ι\iota on the E E_\infty-page is the Thom class in question.

In generalized cohomology

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

To see this, let’s assume EE is connective. Coonsider the relative Atiyah-Hirzebruch spectral sequence?

H˜ p(Th(V),E q(*))H p(D(V),S(V),E q(*))E (D(V),S(V))E˜ (Th(V)) \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))(D(V), S(V)) is (n1)(n-1)-connected for a rank nn vector bundle, then E 2 p<n,q=0E_2^{p \lt n, q} = 0. Hence the edge homomorphism

H˜ k(Th(V),E 0(*))H˜ 0(Th(V)) \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.

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 include

Lecture notes include

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

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

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)

See also

  • Albrecht Dold, Relations between ordinary and extraordinary homology , Colloq. Algebraic Topology, August 1–10, 1962 , Inst. Math. Aarhus Univ. (1962) pp. 2–9

  • Yuli Rudyak, On the Thom–Dold isomorphism for nonorientable bundles Soviet Math. Dokl. , 22 (1980) pp. 842–844 Dokl. Akad. Nauk. SSSR , 255 : 6 (1980) pp. 1323–1325

  • Robert Switzer, Algebraic topology - homotopy and homology , Springer (1975)

  • myyn.org (Planetmath) Thom space, Thom class, Thom isomorphism theorem

Formalization in homotopy type theory is discussed in

Revised on June 21, 2016 03:06:55 by Urs Schreiber (131.220.184.222)