Is it possible to prove that surfaces with compact boundary are homeomorphic by glueing disks to the boundary components?

Question

Let $S_1$ and $S_2$ be two surfaces with compact boundary and the same number of boundary components. Let $M_1$ and $M_2$ be the surfaces obtained by glueing closed disks to the boundary of $S_1$ and $S_2$, respectively. Is there a direct proof that if $M_1$ and $M_2$ are homeomorphic then $S_1$ and $S_2$ are also homeomorphic? I know that this can be done in the compact case using normal forms, as in Massey's book. And if the surfaces are not compact?

Answer 1

If $M_1$ is homeomorphic to $M_2$, then it is homeomorphic via a map preserving the attached disks because we may compose the homeomorphism of $M_1$ and $M_2$ with a homeomorphism that takes the image of the disks attached $M_1$ to the disks attached to $M_2$ (such a homeomorphism exists by isotopy extension). By removing the interior of the attached disks of $M_1$, we have $S_1$ and the restriction homeomorphically maps onto $S_2$.

A similar result will be true in all higher dimensions topologically, but more care is needed in the smooth case. For example, in dimensions above 4 you can attach disks to disks along their boundary and arrive at nondiffeomorphic spheres, so the notions of diffeomorphism and diffeomorphism after capping off boundary components needs refinement (you need to specify how boundary components are capped off).