I have a specific example in mind. Consider $S_1,S_2$ two surfaces. Remove two discs to obtain surfaces with boundary $S_1',S_2'.$ If $S_1 \cong S_2,$ does it necessarily follow that $S_1' \cong S_2'$? It seems intuitively true (having the classification of surfaces in mind), but the homeomorphism does not descend to an obvious homeomorphism to the $S_i'$ (since one of the removed discs is not necessarily sent to the other).
More generally, does this hold for $n$-manifolds with $n$-balls? What if we remove more complicated (e.g. non simply connected) submanifolds?
Thanks!
It holds for all manifolds in all dimensions. This is a consequence of the Annulus Theorem. The immediate corollary of the annulus theorem that you need is the following. Consider two balls $B,B'$ in a connected manifold $M$. You'll need to assume these are "coordinate balls" meaning that they are closed unit balls in some coordinate chart; I will guess that you surely intended this to be the case. Then there exists a homeomorphism of $M$ taking $B$ to $B'$.
For surfaces, it does indeed follow from the classification of surfaces, because removal of a ball does not affect genus, it does not affect orientability, and it increases the number of boundary components by $1$, and those quantities classify the surface up to homeomorphism.