Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 $\pi_1$.

share|improve this question

3 Answers 3

up vote 12 down vote accepted

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.

share|improve this answer

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

share|improve this answer

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.