Thom class

An element in the (generalized) cohomology group of a Thom space, generating it as a module over the cohomology ring of the base space. For a multiplicative generalized cohomology theory (cf. Generalized cohomology theories) , let be the image of under the -fold suspension isomorphism . Let be an -dimensional vector bundle over a path-connected finite cell complex , and let be the corresponding inclusion into the Thom space. An element is called a Thom class (or orientation) of the bundle if , with invertible in . A bundle need not have a Thom class. A bundle with a Thom class (in ) is called -orientable, and a bundle with a fixed Thom class is -oriented. The number of Thom classes of an -orientable bundle over is equal to the number of elements of the group . Multiplication by a Thom class gives a Thom isomorphism.

Comments

For a (topological) manifold with or without boundary , a Thom class is a Thom class for its tangent (micro) bundle. Given a Thom class , there are isomorphisms (Alexander duality), , (Lefschetz duality) and , (Poincaré duality), where is a compact triangulable manifold and are compact subpolyhedra, cf. [a1], Chapt. 14, for more details.

An element is called a fundamental class if for every one has that () is a generator of as a module over . (Here is the inclusion .) For the case of ordinary homology, cf. Fundamental class. The relation between a fundamental class and a Thom class is given by the result that if is a compact triangulable -manifold with Thom class , then there is a unique fundamental class such that takes to , cf. [a1], Prop. 14.17. Using this the Lefschetz and Poincaré duality isomorphisms defined by the Thom class (which essentially are defined by a slant product with ) are given by a cap product with .

References

[a1]	R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975) pp. Chapt. 2