Thom class - Imported Knowlpedia Content

Definition

Let $\pi:E\to B$ be an oriented real vector bundle of rank $r$ over a paracompact Hausdorff base, and let $E^\times=E\setminus 0_E(B)$ be the complement of its zero section. The Thom class of $E$ is the unique class

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

whose restriction to every fiber pair

(E_b,E_b\setminus\{0_b\}) \cong (\mathbb R^r,\mathbb R^r\setminus\{0\})

is the generator determined by the chosen fiber orientation. Thus the local orientation generators fit together into one global relative singular cohomology class.

Thom isomorphism and naturality

Cup product with $u_E$ gives the Thom isomorphism

H^q(B;\mathbb Z) \longrightarrow H^{q+r}(E,E^\times;\mathbb Z), \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 bundle map. These statements are the Thom isomorphism theorem in Milnor and Stasheff, Chapters 9–10.

Models and examples

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

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.