Thom class - Knowlpedia

Let $\pi:E\to B$ be a rank- $r$ topological real vector bundle over a paracompact Hausdorff base, let $R$ be a commutative ring with identity, and write $E^\times=E\setminus\{0_b:b\in B\}$ . An $R$ -orientation is a choice of generator of the free rank-one $R$ -module $H^r(E_b,E_b\setminus\{0_b\};R)$ in each fiber, locally compatible under bundle trivializations.

Given this orientation, the Thom class is the unique relative singular cohomology class

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

whose restriction to every fiber pair is the chosen generator. For $R=\mathbb Z$ and $r>0$ , the choice is the ordinary continuous orientation of the real fibers. For $r=0$ , the canonical orientation chooses $1$ , and $u_E=1\in H^0(B;R)$ .

Thom isomorphism and naturality

Cup product with $u_E$ gives the Thom isomorphism

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

If $f:B'\to B$ is a map from another paracompact Hausdorff base, the pullback orientation on $f^*E$ is characterized by

u_{f^*E}=\widetilde f^{\,*}u_E,

where $\widetilde f:f^*E\to E$ is the canonical continuous bundle map, and $f$ is continuous.

Models and examples

After choosing a continuous positive-definite fiber metric, excision identifies the defining group with $H^r(D(E),S(E);R)$ , where $D(E)=\{v:\|v\|\le1\}$ and $S(E)=\{v:\|v\|=1\}$ are the disk and unit sphere bundles. For the trivial oriented bundle $B\times\mathbb R^r$ , the Thom class is the exterior product of $1\in H^0(B;R)$ with the preferred generator of $H^r(\mathbb R^r,\mathbb R^r\setminus\{0\};R)$ .

With integral coefficients, pulling $u_E$ back along the zero section produces the Euler class. More generally, transverse sections represent this class geometrically through their zero loci.

Coefficients and orientation

Reversing the orientation of $E$ changes $u_E$ to $-u_E$ . A nonorientable real bundle has no integral Thom class with the stated fiberwise generator property; its Thom class instead uses the orientation local system. With coefficients in $\mathbb Z/2$ , every real vector bundle has a canonical Thom class because the two signs agree.

References

John W. Milnor and James D. Stasheff, Characteristic Classes, Annals of Mathematics Studies 76, Princeton University Press, 1974. DOI record. Relevant: Chapters 9–10, oriented bundles, Thom classes, and the Thom isomorphism.
Raoul Bott and Loring W. Tu, Differential Forms in Algebraic Topology, Springer, 1982. DOI record. Relevant: Chapter 6, Thom isomorphism and characteristic classes.