Thom isomorphism theorem

Statement

Let $R$ be a commutative ring with identity and let $\pi:E\to B$ be an $R$ -oriented real topological vector bundle of rank $r$ over a paracompact Hausdorff base, and let $E^\times=E\setminus 0_E(B)$ . Here $0_E(b)=0_b$ , and an $R$ -orientation means locally compatible choices of generators in $H^r(E_b,E_b\setminus\{0_b\};R)$ . If

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

is its Thom class, then for every integer $q\ge0$ 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 continuous positive-definite fiber metric, excision identifies

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

where $D(E)=\{v:\|v\|\le1\}$ and $S(E)=\{v:\|v\|=1\}$ . Collapsing $S(E)$ gives the reduced cohomology of the Thom space. These models express the same isomorphism with different support conventions.

Consequences

For integral coefficients, pulling $u_E$ back along the continuous 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 topological 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.