Edit: I already received a good answer to my second question. I'd be interested in a hint about the first one, as well. Thanks in advance!
I'm interested in compact Riemann surfaces and their homology. In this question, Kundor proposes a nice drawing of the connected sum of tori, saying that it is clearer than the traditional drawing of regular $4g$-gon whose sides are to be identified.
At first I was convinced that this was a smarter way to draw the $4g$-gon (just a continuous deformation), but then I realized that this is not the case (it works for genus 2, though).
So I was wondering if there exists a way to make the transition from the $4g$-gon to the "connected sum of squares", that makes the construction of the genus $g$ surface a lot easier to visualize.
Also, I had in mind a theorem, saying that the homology groups of the connected sum of two spaces are the direct sum of the homology groups of the single spaces (well, apart from $H_0$ and $H_n$).
Only, I wasn't able to find a reference: I was convinced I saw the result in Nakahara, Geometry, Topology and Physics, but I couldn't find it anymore. After some googling, I found these two sources (link1, link2), but nothing conclusive on any book I consulted.
Does this theorem have a name? Do you know a book where it is stated/proved?
Best Answer
Computing the homology of a connected sum is a matter of applying Mayer--Vietoris.
We have (let's say closed, connected, orientable) manifolds $M_1$ and $M_2$. In each of them we choose an $n$-ball $B_i$, and a slightly smaller $n$-ball $B_i'$ contained in $B_i$ (here $i = 1,2$).
For a moment, let's omit the subscripts and just let $M$ be a closed $n$-manifold. Let $B$ be an $n$-ball in $M$, let $B'$ be a slightly smaller ball contained in $B$, so that $B \setminus B'$ is a spherical shell, while $M = M\setminus B' \cup B.$ Then by Mayer--Vietotris, one computes that $H_n(M\setminus B') = 0$, while $H_i(M \setminus B') \cong H_i(M)$ for $i < n$.
Now reintroduce the subscripts: the connected sum $M_1 \# M_2$ of $M_1$ and $M_2$ is obtained by gluing $M_1 \setminus B'_1$ and $M_2 \setminus B_2'$ along their (homeomorphic) spherical shells $B_1 \setminus B_1'$ and $B_2 \setminus B_2'$.
Applying Mayer--Vietoris again, we find that $H_n(M_1 \# M_2) = H_0(M_1 \# M_2) = \mathbb Z$, and that $H_i(M_1 \# M_2) = H_i(M_1) \oplus H_i(M_2)$ if $0 < i < n$.