Let $M$ and $N$ be oriented, closed, $n-1$ manifolds and $F$ a cobordism from $M$ to $N$ and $G$ a cobordism from $N$ to $M$ such that the composite cobordism $G\circ F\cong M\times I$ and $F\circ G\cong N\times I$. Does the existence of an "inverse" cobordism imply that $M$ is diffeomorphic to $N$?
My first, naive strategy is to take the following composite: if $j: N\to F$ is the inclusion of $N$ into $F$, $\psi: G\circ F\to M\times I$ the diffeomorphism, and $\pi: M\times I\to M$ the projection on the first factor, then the composite $\pi \circ \psi \circ j$ is a map from $N\to M$. However, it is easy to foresee that this map may fail to be injective or surjective.
Is there reason to believe this statement to be true?
Best Answer
This statement is false for general $n$ (I'll write $n$ where you've written $n-1$).
For $n\geq5$ we can disprove the statement using a rather common trick. Suppose we are given two (closed) manifolds $M$ and $N$ of dimension $n\geq5$, and suppose that the manifolds are $h$-cobordant (recall that this means that their inclusions $M \hookrightarrow W$ and $N \hookrightarrow W$ into the cobordism $W$ are homotopy equivalences). The Whitehead torsion of this cobordism is an element $\tau = \tau(W,M) \in \operatorname{Wh}(\pi_1(M))$. The $s$-cobordism theorem says that $\tau$ vanishes if and only if the cobordism is a cylinder (i.e., iff $M$ and $N$ are diffeomorphic).
Onto either end of $W$ it is possible to glue an $h$-cobordism whose torsion is $-\tau$; the composite cobordism then has Whitehead torsion $0$ (these facts follow from properties of Whitehead torsion, plus a part of the $s$-cobordism theorem that says you can realize every element of the Whitehead group as the torsion of some $h$-cobordism). Our composite cobordism is hence a cylinder by the $s$-cobordism theorem.
This means that it's actually always possible to get the setup you describe in your question; you get the "inverse" of $W$ by this construction and it's not difficult to see that this will actually be a $2$-sided inverse to $W$. But if the statement you hope to be true actually holds, it would follow that all $h$-cobordant manifolds are diffeomorphic, contradicting the $s$-cobordism theorem.