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?