# Compatibility of two cylindrical regions

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?