Cobordism - Cobordism As An Extraordinary Cohomology Theory

Cobordism As An Extraordinary Cohomology Theory

Every vector bundle theory (real, complex etc.) has an extraordinary cohomology theory called K-theory. Similarly, every cobordism theory has an extraordinary cohomology theory, with homology ("bordism") groups and cohomology ("cobordism") groups for any space . The generalized homology groups are covariant in, and the generalized cohomology groups are contravariant in . The cobordism groups defined above are, from this point of view, the homology groups of a point: . Then is the group of bordism classes of pairs with a closed -dimensional manifold (with G-structure) and a map. Such pairs, are bordant if there exists a G-cobordism with a map, which restricts to on, and to on .

An -dimensional manifold has a fundamental homology class (with coefficients in in general, and in in the oriented case), defining a natural transformation

which is far from being an isomorphism in general.

The bordism and cobordism theories of a space satisfy the Eilenberg–Steenrod axioms apart from the dimension axiom. This does not mean that the groups can be effectively computed once one knows the cobordism theory of a point and the homology of the space X, though the Atiyah–Hirzebruch spectral sequence gives a starting point for calculations. The computation is only easy if the particular cobordism theory reduces to a product of ordinary homology theories, in which case the bordism groups are the ordinary homology groups

This is true for unoriented cobordism. Other cobordism theories do not reduce to ordinary homology in this way, notably framed cobordism, oriented cobordism and complex cobordism. The last-named theory in particular is much used by algebraic topologists as a computational tool (e.g., for the homotopy groups of spheres).

Cobordism theories are represented by Thom spectra : given a group G, the Thom spectrum is composed from the Thom spaces of the standard vector bundles over the classifying spaces . Note that even for similar groups, Thom spectra can be very different: and are very different, reflecting the difference between oriented and unoriented cobordism.

From the point of view of spectra, unoriented cobordism is a product of Eilenberg–MacLane spectra – – while oriented cobordism is a product of Eilenberg–MacLane spectra rationally, and at 2, but not at odd primes: the oriented cobordism spectrum is rather more complicated than .