---
tag: WEGT4EZCX2
kind: definition
name: Cobordism ring
msc: [57R75, 55N22]
defines: C02WQY819R
uses: [T8BVGPPSZP, F45VQ47GPV, 7DS6B2BFK8, CAPQBQDRA7]
sources:
  - url: https://en.wikipedia.org/wiki/Cobordism_ring
---

## Definition

Let $n \ge 0$.
Let $M$ and $N$ be closed [[smooth manifolds|F45VQ47GPV]] of dimension $n$, that is, compact and without boundary.
They are *cobordant* if there is a compact smooth [[manifold with boundary|7DS6B2BFK8]] $W$ of dimension $n + 1$ whose boundary is diffeomorphic to $M \sqcup N$.
This is an equivalence relation; let $\mathfrak{N}_n$ be the set of its classes.
The *unoriented cobordism ring* is the [[graded ring|CAPQBQDRA7]]
$$\mathfrak{N}_* = \bigoplus_{n \ge 0} \mathfrak{N}_n,$$
with addition $[M] + [N] = [M \sqcup N]$, zero the class of the empty manifold, and multiplication $[M] \cdot [N] = [M \times N]$.

## Remarks

Every element has order $2$, since $M \sqcup M$ is the boundary of $M \times [0, 1]$; so $\mathfrak{N}_*$ is an algebra over $\F_2$.
Thom proved that $\mathfrak{N}_* \cong \F_2[x_i : i \ge 2,\ i \neq 2^j - 1]$ with $\deg x_i = i$, and that $\mathfrak{N}_n$ is isomorphic to the homotopy group $\pi_n(MO)$ of the Thom spectrum $MO$.

The *oriented cobordism ring* $\Omega^{SO}_*$ uses closed oriented manifolds and requires $\partial W \cong M \sqcup (-N)$ as oriented manifolds, where $-N$ is $N$ with the opposite orientation.
It is graded commutative, and $\Omega^{SO}_* \otimes \Q$ is a polynomial ring over $\Q$ on the classes of the complex projective spaces $\C P^{2k}$, $k \ge 1$.

Example: $\mathfrak{N}_1 = 0$, since every closed $1$-manifold is a finite union of circles and each circle bounds a disc.
Also $\mathfrak{N}_2 \cong \Z/2$, generated by $\R P^2$, which does not bound because its Euler characteristic is odd.

Non-example: $\R P^2$ is a closed surface that is not cobordant to the empty manifold.
Indeed, if $\R P^2 = \partial W$ for a compact $3$-manifold $W$, then the Euler characteristic of $\R P^2$ would be even, but it is $1$.

Further variants use manifolds with a stably complex structure, a spin structure or a framing, and each is the coefficient ring of a Thom spectrum.