Bordism and Cobordism

Monday, July 23rd, 2012 | Author:

Two connected compact manifolds N and M are said to be bordant, if there exists a manifold W with boundary consisting of two connected components isomorphic to N and M respectively. The name comes from french and means sharing a boundary. Some people say cobordant, since the manifolds don't share a boundary but "are" shared as a boundary (I don't know how to explain this better than with the definition given above). We will stick to "bordant" because we investigate precisely what "the bordism of a manifold" and "the cobordism of a manifold" are.

One can see that being bordant is an equivalence relation, so it makes sense to speak of bordism classes of manifolds. By enriching N and M with extra structure (like a tangential framing, or an orientation), we get several different notions of bordism classes.

From each of these bordism theories, we get a sequence of spaces \Omega_n such that \Omega_n is the Thom space of a universal bundle over some classifying space (I will explain that later) and \Sigma \Omega_n is homotopy equivalent to \Omega_{n+1}. Homotopy theorists like to call such a sequence then a spectrum and by standard theory one can associate to each spectrum a generalized homology theory and a generalized cohomology theory. Even better, Brown's representability theorem states that every generalized (co)homology theory comes from a spectrum, so we have a 1:1 correspondence.

The goal of this article is now to define Thom spectra and to give a geometric interpretation of the corresponding homology and cohomology theories, essentially by carrying out the Pontryagin-Thom construction relatively.

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Category: English, Mathematics | 2 Comments