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cobordism ring

Contents

Idea

The cobordism ring Ω *= n0Ω n\Omega_*=\oplus_{n\geq 0}\Omega_n 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 Ω * SO\Omega^{SO}_*, the spin cobordism ring Ω * Spin\Omega^{Spin}_*, etc. In this context the bare cobordism ring is also denoted Ω * O\Omega^O_* or Ω * un\Omega^{un}_*.

A ring homomorphism out of the cobordism ring is a genus.

For TT a fixed manifold there is a relative version Ω (T)\Omega_\bullet(T) of the cobordism ring:

  • elements are classes modulo cobordism over TT of manifolds equipped with smooth functions to TT;

  • multiplication of [f:XT][f : X \to T] with [g:YT][g : Y \to T] is given by transversal intersection X TYX \cap_T Y over TT: perturb ff such (f,g)(f',g) becomes transversal maps and then form the pullback X× (f,g)YX \times_{(f',g)} Y in Diff.

This product is graded in that it satisfies the dimension formula

(dimTdimX)+(dimTdimY)=dimTdim(X TY) (dim T - dim X) + (dim T - dim Y) = dim T - dim (X \cap_T Y)

hence

dim(X TY)=(dimX+dimY)dimT. dim (X \cap_T Y ) = (dim X + dim Y) - dim T \,.

Properties

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.

Theorem

(Thom)

The degree nn component Ω n\Omega_n of the cobordism ring Ω *\Omega_* is the nnth homotopy group of the Thom spectrum MOM O

Ω n Oπ n(MO) \Omega^O_n \simeq \pi_n (M O)

The Thom spectrum MOM O is a connected spectrum hence essentially a symmetric monoidal ∞-groupoid (infinite loop space) Ω MO\Omega^\infty M O.

By one aspect of the (proof of the) cobordism hypothesis-theorem, this is the (∞,n)-category of cobordisms for nn \to \infty

Bord (,)Ω MO. Bord_{(\infty,\infty)} \simeq \Omega^\infty M O \,.

Really on the left we have the \infty-groupoidification of that ∞-category, but since Bord (,)Bord_{(\infty,\infty)} has duals for k-morphisms for all kk, it is already itself an \infty-groupoid: the Thom spectrum. See (Francis).

Hence the cobordism ring in degree nn is the decategorification of the nn-fold looping of the \infty-category of cobordisms.

References

An useful review of the central definitions and theorems about the cobordism ring is in chapter 1 of

  • Gerald Höhn, Komplexe elliptische Geschlechter und S 1S^1-äquivariante Kobordismustheorie (german) (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 n=n = \infty is in

On fibered cobordism groups?:

  • Astey, Greenberg, Micha, Pastor, Some fibered cobordisms groups are not finitely generated (pdf)

Revised on November 14, 2013 11:13:09 by Urs Schreiber (188.200.54.65)