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Suppose we have a cobordism $W$ of manifolds $M_0$ and $M_1$ and suppose the inclusion of $M_0$ into $W$ is a homotopy equivalence. Is the same true for the inclusion of $M_1$ (ie. is $W$ already an h-cobordism)?

Using Poincare Lefschetz duality one can show that this map induces isomorphisms on homology. Hence it suffices to show that the inclusion $M_1\rightarrow W$ induces an isomorphism on $\pi_1$.

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3 Answers 3

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For a counterexample take a non-simply connected homology sphere bounding a contractible manifold and remove the interior of a small ball from the contractible manifold. Such homology spheres exist in abundance.

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I think the answer should be no, since people study so-called semi-s-cobordisms, which (if they exist) give counter-examples.

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However, the answer is "yes" after stabilizing three times: The product $W \times J^3$ (where $J^3$ is a $3$-cube) is an $h$-cobordism from $M_0 \times J^3$ to the closure of the remaining part of the boundary. There are details in Remark 1.1.3 of my book project "Spaces of PL manifolds and categories of simple maps" with Jahren and Waldhausen.

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