Thom isomorphism theorem - Imported Knowlpedia Content

Statement

Let $\pi:E\to B$ be an $R$ -oriented real vector bundle of rank $r$ over a paracompact Hausdorff base, and let $E^\times=E\setminus 0_E(B)$ . If

u_E\in H^r(E,E^\times;R)

is its Thom class, then for every $q$ the map

\Phi_E:H^q(B;R)\longrightarrow H^{q+r}(E,E^\times;R), \qquad a\longmapsto \pi^*a\smile u_E,

is an isomorphism. This is the Thom isomorphism theorem. It is natural for orientation-preserving pullbacks of vector bundles and sends $1\in H^0(B;R)$ to $u_E$ .

Equivalent models

After choosing a bundle metric, excision identifies

H^*(E,E^\times;R)\cong H^*(D(E),S(E);R),

where $D(E)$ and $S(E)$ are the disk and sphere bundles. Collapsing $S(E)$ gives the reduced cohomology of the Thom space. These models express the same isomorphism with different support conventions Milnor–Stasheff, chapter 10.

Consequences

Pulling $u_E$ back along the zero section gives the Euler class. The theorem also yields Gysin maps and the Gysin long exact sequence of the sphere bundle. For the trivial oriented bundle $B\times\mathbb R^r$ , $\Phi_E$ is the suspension-type degree shift obtained by multiplying with the preferred generator of $H^r(\mathbb R^r,\mathbb R^r\setminus\{0\};R)$ .

Coefficients and scope

Integral coefficients require an orientation. Every real vector bundle is orientable over $\mathbb Z/2$ , so the mod- $2$ theorem has no orientability hypothesis. More generally, a nonorientable bundle admits a Thom isomorphism with its orientation local system.

References

John W. Milnor and James D. Stasheff, Characteristic Classes, Princeton University Press, 1974. DOI record. Relevant: chapters 9–10, orientations, Thom classes, and the Thom isomorphism theorem.
Raoul Bott and Loring W. Tu, Differential Forms in Algebraic Topology, Springer, 1982. DOI record. Relevant: chapter 6, Thom isomorphism and Euler class.