Cobordism

Remarks

Being cobordant is an equivalence relation: $M \times [0, 1]$ gives reflexivity, and two cobordisms can be glued along a common boundary component for transitivity. The set of classes $\Omega_n^O$ is an abelian group under disjoint union, and every element has order $2$, since $M \sqcup M$ bounds $M \times [0, 1]$. In the oriented version, $W$ is oriented and $\partial W \cong M \sqcup (-N)$, where $-N$ is $N$ with the opposite orientation; this gives the groups $\Omega_n^{SO}$.

Example: the circle $S^1$ bounds the disc $D^2$, so $S^1$ is cobordant to the empty manifold. Every sphere $S^n = \partial D^{n+1}$ is likewise null-cobordant.

Non-example: the real projective plane $\R P^2$ is not the boundary of any compact $3$-manifold. A closed manifold $M$ that bounds a compact manifold $W$ has even Euler characteristic, but $\chi(\R P^2) = 1$.

A manifold cobordant to the empty manifold is called null-cobordant.