Let $M^2,N^2$ be connected closed surfaces. Suppose there exists region $D$ in the interior of $M \times [-2,2]$ such that (a) $D$ is homeomorphic to $N \times [0,1]$; (b) $D$ contains $M \times [-1,1]$.


Can we prove the following statements?


1. $M$ and $N$ are homotopic.
2. $M$ and $N$ are homeomorphic.


If true, are there any generalizations for higher dimensional case?



Compatibility of two cylindrical regions
Yes.


For simplicity, we set $M\_1=M \times \{-2\}, M\_2=M \times \{0\},N\_1=N \times \{0\}$ and $N\_2=N \times \{1\}$ such that $N\_1$ is contained in $M \times (-2,0)$. Moreover, we denote the region bounded by $M\_1$ and $N\_2$ by $\Omega$.


There is a natural projection from $\Omega \to M\_1$ and the induced map $\pi\_1(\Omega) \to \pi\_1(M\_1)$ is surjective. We claim that this map is also injective. Indeed, this follows from the fact that $\Omega$ has a deformation retraction to the region bounded by $M\_1$ and $N\_1$ and the latter is contained in $M \times [-2,0]$, which is contained in $\Omega$.


Therefore, the inclusion $M\_1 \to \Omega$ induces an isomorphism between fundamental groups. By the same reason, it also induces isomorphism between higher homotopy groups. In other words, $M\_1$ is homotopic to $\Omega$.


So far, all arguments apply to general dimensions. If the dimension is 2, one can prove $\Omega$ is homeomorphic to $M\_1 \times [0,1]$ from Hempel's book Theorem 10.2. In particular, $N$ is homeomorphic to $M$.


