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Moishe Kohan
Moishe Kohan
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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, youit is easy to conclude that $(M- int(U))/\partial U$ is homeomorphic
to 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 tame boundarylocally flat boundary homeomorphic to $S^{n-1}$, 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 (Brown's theorem1962), p. 331-341.

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, 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". 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, you 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 tame boundary homeomorphic to $S^{n-1}$ then $\partial U$ again admits a collar (Brown's theorem).

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.

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Moishe Kohan
Moishe Kohan
  • 12.2k
  • 1
  • 36
  • 58

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, 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". 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, you 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 tame boundary homeomorphic to $S^{n-1}$ then $\partial U$ again admits a collar (Brown's theorem).