Unoriented smooth cobordism of closed manifolds

Fix an integer $n\ge 0$. In this library a manifold is Hausdorff and second-countable, and closed means compact without boundary (Smooth manifolds and their smooth charts, Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right).

Let $M_0$ and $M_1$ be closed smooth $n$-manifolds. A (unoriented) bordism from $M_0$ to $M_1$ is data $(W,{\theta }_0,{\theta }_1)$ consisting of

• a compact smooth $(n+1)$-manifold with boundary $W$ (Smooth charts, atlases, and structures with boundary) whose boundary is decomposed as a disjoint union $\mathrm{\partial }W=(\mathrm{\partial }W{)}_0\bigsqcup (\mathrm{\partial }W{)}_1$ of boundary subsets that are both open and closed in $\mathrm{\partial }W$; and
• smooth embeddings ${\theta }_0:[0,1)\times M_0\to W$ and ${\theta }_1:(-1,0]\times M_1\to W$ (Immersions and embeddings for manifolds with boundary) onto open collar neighbourhoods of $(\mathrm{\partial }W{)}_0$ and $(\mathrm{\partial }W{)}_1$ respectively, with ${\theta }_i(\{0\}\times M_i)=(\mathrm{\partial }W{)}_i$ for $i=0,1$ (Smooth collars of a manifold boundary).

Each $(\mathrm{\partial }W{)}_i$ is then a union of components of $\mathrm{\partial }W$, of which there are finitely many because $W$ is compact; and $\mathrm{\partial }W$ is a closed embedded smooth $n$-manifold (The boundary of a positive-dimensional manifold is a closed embedded smooth (n-1)-manifold). The two closed $n$-manifolds $M_0$ and $M_1$ are cobordant when such data exist; we then also say that $(W,{\theta }_0,{\theta }_1)$ is a bordism from $M_0$ to $M_1$ and that $W$ bounds $M_0\bigsqcup M_1$.

The collar embeddings are part of the data, not a choice made afterwards: every gluing argument on this page uses the supplied collars, which is why no choice principle is required for the cobordism relation. The parametrisation widths are fixed to the standard intervals $[0,1)$ and $(-1,0]$; a collar supplied with a different width is rescaled to this form before it is used, and the rescaled embedding is again a smooth embedding onto the same collar neighbourhood.

The empty manifold is allowed as $M_0$ or $M_1$ and as $W$, and $n=0$ is allowed; a bordism from the empty $n$-manifold to the empty $n$-manifold is exactly a compact smooth $(n+1)$-manifold with empty boundary. The definition depends only on the smooth structures of $M_0$, $M_1$ and $W$ and on the supplied data; it introduces no equivalence relation by itself, and the statement that cobordism is an equivalence relation is proved later.