h-Cobordism

An h-cobordism is a compact smooth cobordism triad $(W;M_0,M_1)$ (Smooth cobordism triad for Morse theory) with $\dim W=n+1\ge 2$ for which both inclusions

$M_0\hookrightarrow W,M_1\hookrightarrow W$

are homotopy equivalences (Homotopy equivalences, homotopy inverses and spaces of the same homotopy type). The inclusions are smooth embeddings of the faces into $W$ (Smooth embeddings), and compactness of $W$ is the compactness carried by the triad (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right).

Because the triad has $\mathrm{\partial }W=M_0\bigsqcup M_1$ with $M_0$ and $M_1$ closed embedded smooth submanifolds of $\mathrm{\partial }W$ of dimension $n=\dim W-1$, the two faces of an h-cobordism are closed smooth $n$-manifolds. By Homotopy equivalences, homotopy inverses and spaces of the same homotopy type, the inclusion ${\iota }_i:M_i\hookrightarrow W$ is a homotopy equivalence exactly when there is a continuous map $r_i:W\to M_i$ with

$r_i\circ {\iota }_i\simeq {\mathrm{id}}_{M_i},{\iota }_i\circ r_i\simeq {\mathrm{id}}_W;$

these are the homotopy-inverse identities. They do not by themselves specify a retraction fixing the face pointwise or a homotopy relative to that face. No map $W\to M_i$ is required to be smooth.

The definition is symmetric in the two faces: interchanging $M_0$ and $M_1$ leaves both conditions unchanged, so whenever $(W;M_0,M_1)$ is an h-cobordism the reversed triad $(W;M_1,M_0)$ is one as well, with the same collars read in the reversed order. A trivial h-cobordism is an h-cobordism diffeomorphic to the product $M_0\times [0,1]$ relative to $M_0$; the faces $M_0$ and $M_1$ of an h-cobordism are called h-cobordant, whether or not that h-cobordism is trivial. Equivalently, two closed smooth manifolds are h-cobordant when there exists an h-cobordism with those faces.

No simple connectivity, orientability or coefficient hypothesis is part of the definition, no hypothesis on the fundamental groups beyond what the homotopy equivalences already impose is made, and the two faces need not be diffeomorphic.