Cut out an open ball from a 2-manifold and glue the boundary

Question

I have a possibly elementary question. Let $\mathcal{M}$ be a manifold with $\text{dim} \; \mathcal{M} = 2$. Let $U \subseteq \mathcal{M}$ be homeomorphic to $\overline{\mathcal{B}(0,1)}$, and let $\partial U = U \backslash \text{int} \; U$. Construct the topological space $\mathcal{N}$ by removing $\text{int} \; U$ and then identifying all points on $\partial U$. Is $\mathcal{N}$ homeomorphic to $\mathcal{M}$? Can someone provide a proof if this is true, or provide a counterexample?

The issue in my mind is that I am facing complications with is the fact that $\partial U$ is only homeomorphic to $S^{1}$, and not necessarily a $C^1$ curve (or even piecewise $C^1$). Note that it is well known that all 2 manifolds admit smooth structures, so it makes sense to talk about differentiability.

Answer 1

The following is theorem A1 in the paper by David Epstein, "Curves in 2-manifolds and isotopies", Acta Math, 1966.

Theorem. Let $M$ is a surface equipped with a PL structure. Then every topological embedding $f: S^1\to M$ is isotopic to a PL embedding. Moreover, isotopy takes place in an arbitrarily small neighborhood of $f(S^1)$.

Now, every topological surface $M$ admits a PL structure (Rado). Thus, we see every subset $A\subset M$ homeomorphic to $S^1$ has a collar: A neighborhood $N$ (which can be chosen arbitrarily close to $A$) homeomorphic to the annulus or the Moebius band, where $A$ is the "core curve". (Taking a suitable regular neighborhood of a PL curve isotopic to $A$.)

If $A$ bounds a topological disk in $M$, then the collar cannot be a Moebius band. Hence, in your situation, if $U\subset M$ is a subset homeomorphic to the closed disk, then $\partial U$ admits an annular collar. From this, it is easy to conclude that $(M- int(U))/\partial U$ is homeomorphic to $M$.

Note that this fails in dimensions $n\ge 3$: The quotient is not always a manifold. However, if you assume that $U\subset M$ has locally flat boundary, then $\partial U$ again admits a collar. This is Brown's theorem:

Morton Brown, "Locally flat imbeddings of topological manifolds", Annals of Mathematics, Vol. 75 (1962), p. 331-341.