Manifolds and cobordisms
The cobordism ring is the ring whose
Instead of bare manifolds one can consider manifolds with extra structure, such as orientation, spin structure, string structure, etc. and accordingly there is oriented cobordism ring , the spin cobordism ring , etc. In this context the bare cobordism ring is also denoted or .
A ring homomorphism out of the cobordism ring is a genus.
For a fixed manifold there is a relative version of the cobordism ring:
elements are classes modulo cobordism over of manifolds equipped with smooth functions to ;
multiplication of with is given by transversal intersection over : perturb such becomes transversal maps and then form the pullback in Diff.
This product is graded in that it satisfies the dimension formula
Relation to cobordism cohomology theory
See cobordism cohomology theory.
Relation to higher category theory
The cobordism ring finds its natural interpretation in higher category theory.
The degree component of the cobordism ring is the th homotopy group of the Thom spectrum
The Thom spectrum is a connected spectrum hence essentially a symmetric monoidal ∞-groupoid (infinite loop space) .
By one aspect of the (proof of the) cobordism hypothesis-theorem, this is the (∞,n)-category of cobordisms for
Really on the left we have the -groupoidification of that ∞-category, but since has duals for k-morphisms for all , it is already itself an -groupoid: the Thom spectrum. See (Francis).
Hence the cobordism ring in degree is the decategorification of the -fold looping of the -category of cobordisms.
The cobordism ring over the point for oriented manifolds starts out as
see e.g. (ManifoldAtlas)
For a CW-complex (for instance a manifold), then the oriented cobordism ring is expressed in terms of the ordinary homology of with coefficients in the cobordism ring over the point, prop. 1, as
e.g. Connor-Floyd 62, theorem 14.2
An useful review of the central definitions and theorems about the cobordism ring is in chapter 1 of
- Gerald Höhn, Komplexe elliptische Geschlechter und -äquivariante Kobordismustheorie (german) (pdf)
Discussion of oriented cobordism includes
Manifold Atlas, Oriented bordism
P. E. Conner, E. E. Floyd, Differentiable periodic maps, Bull. Amer. Math. Soc. Volume 68, Number 2 (1962), 76-86. (Euclid, pdf)
A historical review in the context of complex cobordism cohomology theory/Brown-Peterson theory is in
A discussion of its relation to the Thom spectrum and the (∞,n)-category of cobordisms for is in
On fibered cobordism groups?:
- Astey, Greenberg, Micha, Pastor, Some fibered cobordisms groups are not finitely generated (pdf)