# Bulk Boundary Correspondence For Manifolds With Boundary

Let $$X$$ be an $$n+1$$ dimensional manifold. Suppose that $$M_1$$ and $$M_2$$ are compact manifolds with boundary such that $$M_1$$ and $$M_2$$ both have interior homeomorphic to $$X$$. Let $$Y_1$$ be the boundary of $$M_1$$ and $$Y_2$$ the boundary of $$M_2$$. Is it the case that $$Y_1$$ must be homeomorphic to $$Y_2$$?

As a sort of converse, suppose that $$M_1,M_2$$ are compact manifolds with boundary and moreover their boundaries are homeomorphic. Consider the interior $$X_1$$ of $$M_1$$ and the interior $$X_2$$ of $$M_2$$. Do $$X_1$$ and $$X_2$$ just differ by taking connected sum with closed $$n+1$$ manifolds? By that I mean something like: Is it the case that there exist closed $$n+1$$ manifolds $$Z_1$$ and $$Z_2$$ such that $$X_1 \# Z_1 \cong X_2 \# Z_2$$

For the first question, my idea for existence is that you can take any tubular neighborhood of the $$n+1$$ manifold $$X$$, then embed that into some $$\mathbb{R}^k$$. For any such embedding the boundary of the image of $$X$$ under the embedding should give you the same $$n$$ manifold $$Y$$ (also $$X$$ should be noncompact otherwise $$Y$$ is just the empty set viewed as an $$n$$ manifold).

For the second question certainly not, I can do all sorts of surgery on the interior to change the topological type (for instance to change the fundamental group in a more complicated way than just taking a free product). For instance, take any two manifolds $$M_1$$ and $$M_2$$ and delete a ball, so that they have sphere boundary; $$M_1$$ and $$M_2$$ will rarely just differ by a connected sum.
• For the second question, though, what about something like $X_1 \# Z_1 \cong X_2 \# Z_2$ for closed manifolds $Z_1$ and $Z_2$ as suggested in the question? In the case that the $M_i$ are closed manifolds with a ball removed, you could take the $Z_i$ to be those closed manifolds (in the reverse order). Dec 29, 2019 at 16:57