Null-cobordant closed manifolds

A closed smooth $n$-manifold $M$ is null-cobordant if it is cobordant to the empty $n$-manifold, that is, if there is a bordism $(W,{\theta }_0,{\theta }_1)$ from $M$ to $\mathrm{\varnothing }$ (Unoriented smooth cobordism of closed manifolds).

Unwinding the bordism data, this is equivalent to the following statement: there is a compact smooth $(n+1)$-manifold $W$ whose boundary is carried by a collar ${\theta }_0:[0,1)\times M\to W$ onto all of $\mathrm{\partial }W$, so that ${\theta }_0(\{0\}\times M)=\mathrm{\partial }W$ and ${\theta }_0$ restricts to a diffeomorphism $\{0\}\times M\to \mathrm{\partial }W$ (Smooth collars of a manifold boundary, Diffeomorphisms and local diffeomorphisms of manifolds). Indeed, for a bordism to the empty manifold the outgoing boundary part $(\mathrm{\partial }W{)}_1$ is empty, hence $\mathrm{\partial }W=(\mathrm{\partial }W{)}_0={\theta }_0(\{0\}\times M)$; conversely such a collar gives a bordism from $M$ to $\mathrm{\varnothing }$ by taking $(\mathrm{\partial }W{)}_0=\mathrm{\partial }W$, $(\mathrm{\partial }W{)}_1=\mathrm{\varnothing }$ and ${\theta }_1:(-1,0]\times \mathrm{\varnothing }\to W$ the empty map. Thus, informally, $M$ is null-cobordant exactly when $M$ is (diffeomorphic to) the whole boundary of a compact smooth $(n+1)$-manifold.

In the oriented theory, a closed oriented $n$-manifold $(M,o)$ is null-cobordant if it is oriented cobordant to the empty oriented manifold (Oriented smooth cobordism). Because the incoming face contributes the negative orientation, a null-cobordism $W$ satisfies: the induced boundary orientation of the whole boundary $\mathrm{\partial }W=(\mathrm{\partial }W{)}_0$ is $-o$. Thus $(M,o)$ is null-cobordant exactly when $M$ occurs as the orientation opposite to the induced boundary orientation of some compact oriented $(n+1)$-manifold; reversing the orientation of $W$ presents $(M,o)$ as the induced boundary $\mathrm{\partial }(-W)$ with its outward-normal-first orientation. The empty manifold carries its unique orientation and is null-cobordant in both theories, being cobordant to itself by the cylinder.

Null-cobordism depends only on the cobordism class: if $M^{\mathrm{\prime }}$ is cobordant to $M$ and $M$ is null-cobordant, then $M^{\mathrm{\prime }}$ is cobordant to $M$ and $M$ to $\mathrm{\varnothing }$, and transitivity of the cobordism relation (Smooth cobordism is an equivalence relation) gives that $M^{\mathrm{\prime }}$ is null-cobordant; the same holds in the oriented theory. The definition uses no choice principle.