nLab
bordism homology theory

Contents

Contents

Idea

A generalized homology theory represented by a Thom spectrum. The dual concept of cobordism cohomology theory.

Examples

flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:

bordism theory\;M(B,f) (B-bordism):

relative bordism theories:

equivariant bordism theory:

global equivariant bordism theory:

algebraic:

Geometric model

See Section 2 in Atiyah Atiyah. Atiyah’s geometric model for bordism homology groups defines them as equivalence classes of maps of manifolds.

We concentrate on the oriented case, corresponding to the Thom spectrum MSO\mathrm{M} \mathrm{SO}.

Specifically, the kk-dimensional oriented bordism group MSO k(X)\mathrm{M} \mathrm{SO}_k(X) of a smooth manifold XX (more generally, a paracompact Hausdorff topological space, even more generally, an arbitrary topological space provided we use numerable open covers for trivializations) is defined as the quotient of the commutative monoid C k(X)C_k(X) by the equivalence relation \sim of bordism, defined below.

Elements of C k(X)C_k(X) are smooth (or continuous) maps F:MXF\colon M\to X, where MM is a compact kk-dimensional oriented smooth manifold (without boundary).

Two such maps FF and FF' are equivalent if there is a smooth (or continuous) map G:NXG\colon N\to X, where NN is a compact (k+1)(k+1)-dimensional oriented smooth manifold with boundary

N=MM\partial N = M\sqcup M'

such that G| M=FG|_M=F and G| M=FG|_{M'}=F'.

One can also defined twisted homology groups? in the same manner. Twists are principal bundles α\alpha over XX with structure group Z/2\mathbf{Z}/2. Elements of C k(X,α)C_k(X,\alpha) are maps F:(M,τ)(X,α)F\colon (M,\tau)\to (X,\alpha), where MM is a compact kk-dimensional smooth manifold equipped with a principal Z/2\mathbf{Z}/2-bundle τ\tau, and the map FF is a morphism of such principal bundles. Likewise, FFF\sim F' if there is a map G:(N,σ)(X,α)G\colon (N,\sigma)\to(X,\alpha) with the same properties as above.

References

The original article introducing bordism as a Whitehead-generalized cohomology theory:

Surveys:

  • Peter Landweber, A survey of bordism and cobordism, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 100, Issue 2 September 1986 , pp. 207-223 (doi:10.1017/S0305004100066032)

  • Max Hopkins, The Extraordinary Bordism Homology, 2016 (pdf, pdf)

For more, see the references at cobordism cohomology theory.

Last revised on January 31, 2021 at 07:33:01. See the history of this page for a list of all contributions to it.