The cobordism ring $\Omega_*=\oplus_{n\geq 0}\Omega_n$ 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 $\Omega^{SO}_*$, the spin cobordism ring $\Omega^{Spin}_*$, etc. In this context the bare cobordism ring is also denoted $\Omega^O_*$ or $\Omega^{un}_*$.
A ring homomorphism out of the cobordism ring is a genus.
For $T$ a fixed manifold there is a relative version $\Omega_\bullet(T)$ 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 $(f',g)$ becomes transversal maps and then form the pullback $X \times_{(f',g)} Y$ in Diff.
This product is graded in that it satisfies the dimension formula
hence
See cobordism cohomology theory.
The cobordism ring finds its natural interpretation in higher category theory.
(Thom)
The degree $n$ component $\Omega_n$ of the cobordism ring $\Omega_*$ is the $n$th homotopy group of the Thom spectrum $M O$
The Thom spectrum $M O$ is a connected spectrum hence essentially a symmetric monoidal ∞-groupoid (infinite loop space) $\Omega^\infty M O$.
By one aspect of the (proof of the) cobordism hypothesis-theorem, this is the (∞,n)-category of cobordisms for $n \to \infty$
Really on the left we have the $\infty$-groupoidification of that ∞-category, but since $Bord_{(\infty,\infty)}$ has duals for k-morphisms for all $k$, it is already itself an $\infty$-groupoid: the Thom spectrum. See (Francis).
Hence the cobordism ring in degree $n$ is the decategorification of the $n$-fold looping of the $\infty$-category of cobordisms.
The cobordism ring over the point for oriented manifolds starts out as
$k$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | $\geq 9$ |
---|---|---|---|---|---|---|---|---|---|---|
$\Omega^{SO}_k$ | $\mathbb{Z}$ | 0 | 0 | 0 | $\mathbb{Z}$ | $\mathbb{Z}_2$ | 0 | 0 | $\mathbb{Z}\oplus \mathbb{Z}$ | $\neq 0$ |
see e.g. (ManifoldAtlas)
For $X$ a CW-complex (for instance a manifold), then the oriented cobordism ring is expressed in terms of the ordinary homology $H_q(X,\Omega^{SO}_{p-q})$ of $X$ 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
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 $n = \infty$ is in
On fibered cobordism groups?: