Contents

Idea

The cobordism ring ${\Omega }_{*}={\oplus }_{n\ge 0}{\Omega }_n$ is the ring whose

• degree $n$ elements are classes of $n$-dimensional manifolds modulo cobordisms;

• product operation is given by the Cartesian product of manifolds;

• addition operation is given by the disjoint union of manifolds.

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 ${\Omega }_{*}^{\mathrm{SO}}$, the spin cobordism ring ${\Omega }_{*}^{\mathrm{Spin}}$, etc. In this context the bare cobordism ring is also denoted ${\Omega }_{*}^O$ or ${\Omega }_{*}^{\mathrm{un}}$.

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

For $T$ a fixed manifold there is a relative version ${\Omega }_{•}(T)$ of the cobordism ring:

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

• multiplication of $[f:X\to T]$ with $[g:Y\to T]$ is given by transversal intersection $X{\cap }_{T}Y$ over $T$: perturb $f$ such $(f\prime ,g)$ becomes transversal maps and then form the pullback $X{\times }_{(f\prime ,g)}Y$ in Diff.

This product is graded in that it satisfies the dimension formula

hence

Higher category interpretation

The cobordism ring finds its natural interpretation in higher category theory.

The Thom spectrum $MO$ is a connected spectrum hence essentially a symmetric monoidal ∞-groupoid (infinite loop space) ${\Omega }^{\mathrm{\infty }}MO$.

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

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

Hence the cobordism ring in degree $n$ is the decategorification of the $n$-fold looping of the $\mathrm{\infty }$-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 $S^1$-äquivariante Kobordismustheorie (german) (pdf)

A discussion of its relation to the Thom spectrum and the (∞,n)-category of cobordisms for $n=\mathrm{\infty }$ is in

• John Francis, Cobordisms (notes by Owen Gwilliam) (pdf)

On fibered cobordism groups?:

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