Cobordism - Unoriented Cobordism

Unoriented Cobordism

The cobordism class of a closed -dimensional manifold is determined by the Stiefel–Whitney characteristic numbers, which depend on the stable isomorphism class of the tangent bundle. Thus if has a stably trivial tangent bundle then . Every closed manifold is such that, so for every . In 1954 René Thom computed

with one generator in each dimension . For even it is possible to choose, the cobordism class of the -dimensional real projective space.

The low-dimensional unoriented cobordism groups are

This shows, for example, that every 3-dimensional closed manifold is the boundary of a 4-manifold (with boundary).

The mod 2 Euler characteristic of an unoriented -dimensional manifold is an unoriented cobordism invariant. For example, for any

In particular such a product of real projective spaces is not null-cobordant. The mod 2 Euler characteristic map is onto for all, and an isomorphism for .