Cobordism ring

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.