nLab bordism homology theory

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):

• MFr

• MO, MSO, MSpin, MString, …

• MU, MSU, …

  MΩΩSU(n)

  MP, MR

• MSpin^c

• MSp

relative bordism theories:

• MOFr, MUFr, MSUFr

equivariant bordism theory:

• equivariant MFr

• equivariant MO

• equivariant MU

global equivariant bordism theory:

• global equivariant mO,

• global equivariant mU

algebraic:

• algebraic cobordism

Geometric model

See Section 2 in 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 $\mathrm{M} \mathrm{SO}$.

Specifically, the $k$-dimensional oriented bordism group $\mathrm{M} \mathrm{SO}_k(X)$ of a smooth manifold $X$ (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 $C_k(X)$ by the equivalence relation $\sim$ of bordism, defined below.

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

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

$$\partial N = M\sqcup M'$$

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

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

Related concepts

• cobordism cohomology theory

• bordism

• ordinary homology

• equivariant bordism homology theory

• stable homotopy homology theory

References

• René Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28, (1954). 17-86 (doi:10.1007/BF02566923, dml:139072, digiz:GDZPPN002056259, pdf)

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

• Michael Atiyah, Bordism and Cobordism, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 57, Issue 2, April 1961, pp. 200 - 208 (doi:10.1017/S0305004100035064, pdf)

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.