connected sum of surface

Question

When I read Massey's Algebraic Topology:An Introduction,page 9,he points out that the topological type of $S_1$#$S_2$(here $S_i$ is surface,# is connected sum,i.e.,cutting an open disc $D_i$ in each surface,and then gluing the boundary circle through a homeomorphism h) does not depend on the choice of discs $D_i$ or the choice of the homeomorphism h.

The independence of h is quite evident,but I don't know why the choice of discs is irrelevant.Is there any refference?

Answer 1

Wikipedia has an extremely short "Disc Theorem" entry which references Palais' "Extending diffeomorphisms" paper. It says

Disc Theorem. For a smooth, connected, $n$-dimensional manifold $M$, if $f, f'\colon D^n \to M$ are two equi-oriented embeddings then they are ambiently isotopic.

This is one of the fundamental results in Differential Topology, and in particular it implies that connected sums are well-defined wrt choice of embeddings. There might be a simpler proof in $2$D, but this is the standard result which is typically cited for all dimensions.

(Here "ambiently isotopic" means there is an isotopy $H: M\times I \to M$ which begins at the identity map and induces an isotopy between $f$ and $f'$; "equi-oriented" means that $f$ preserves orientation iff $f'$ does. In fact the proof of Palais' theorem shows a bit more: the ambient isotopy can be chosen to be fixed outside of a compact, contractible subspace containing the images of $f$ and $f'$.)

It's been a while since I looked at it, but a proof sketch sort of goes like this: First choose a small open tube around the unit interval $I\subset U\subset \mathbb{R}^2$ and an embedding $\gamma\colon \bar U \to M$ where $\gamma(0) = f(0)$, $\gamma(1) = f'(0)$. Now pick a small disc $D\subset U$ centred at $0$, and construct an ambient isotopy in the tube $F\colon U\times I \to U$ which transports $D$ to a disc centred at $1$. Then you have to construct ambient isotopies $H_1, H_2$ on $M$ which shrink $f(D^n)$ and $f'(D^n)$ down to $\gamma(D)$ and $\gamma(F_1(D))$ respectively (I guess this step uses a linearization trick). Then these three isotopies are pieced together to give the result.

Answer 2

If you consider two different locations/ sizes of the disc on one surface, you can define a homotopy on that surface that transforms one disc to the other. This works because all disks are homotopically trivial. Hence the topological type of the connected sum does not depend on the particular choice of discs.