Cobordism - Cobordism of Manifolds With Additional Structure

Cobordism of Manifolds With Additional Structure

Cobordism can also be defined for manifolds that have additional structure, notably an orientation. This is made formal in a general way using the notion of X-structure (or G-structure). Very briefly, the normal bundle ν of an immersion of M into a sufficiently high-dimensional Euclidean space gives rise to a map from M to the Grassmannian, which in turn is a subspace of the classifying space of the orthogonal group. Given a collection of spaces and maps with maps (compatible with the inclusions, an X-structure is a lift of ν to a map . Considering only manifolds and cobordisms with X-structure gives rise to a more general notion of cobordism. In particular, may be given by, where is some group homomorphism. This is referred to as a G-structure. Examples include G = O, the orthogonal group, giving back the unoriented cobordism, but also the subgroup SO(k), giving rise to oriented cobordism, the spin group, the unitary group U(k), and the trivial group, giving rise to framed cobordism.

The resulting cobordism groups are then defined analogously to the unoriented case. They are denote by .