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Let $F$ be an orientable, compact, connected surface of genus $g$ with $n$ boundary components.

Claim. The first (singular) homology group is $H_1(F)=\bigoplus_{2g+n-1}\mathbb{Z}$.

I know that $H_1(F)$ is isomorphic to the abilianization of its fundamental group. So maybe the theorem of Seifert-van Kampen helps?

Is such an $F$ always of the form like here?

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If $n = 1$ $F$ does retract on a bouquet of circle, hence the claim follows. Now, if $n>1$, use one disk to retract everything on a wedge of circle except little "neighborhood" of every other boundary component, they will also add one more circle to the wedge, i.e add one more copy of $\mathbb Z$ to $H_1(F)$.

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  • $\begingroup$ Well, thank you! I would like to ask a few questions about this if you don't mind. 1.) (general question, nothing to do with your answer) I have linked a figure. Why does this figure have genug $g$? Isn't $g$ the number of wholes of $F$ here? 2.) I don't see what happens in your induction step, could you say a few more words please? $\endgroup$ – user444847 May 10 '17 at 20:03
  • $\begingroup$ 1) : you can see that gluing a disk or removing a disk is not changing the genus as the genus is the number of handles you add. You can see this way : start from the sphere, add $g$ handles and remove $d$ disks, and you get your surface. For be honest I know knot theorist like to draw surfaces of genus $g$ this way but I was never convinced by this picture. I prefer to see it as a $2g$-gon with face identified. 2) With this vision in mind, you can assume that the first disk you remove is in the center, and you can retract everything on the boundary. If you have only one disk you will $\endgroup$ – user171326 May 10 '17 at 20:13
  • $\begingroup$ (continued) get the wedge. But now, if you remove another disk, by continuity you move this disk such that its boundary touches the common points of the wedge. When you will retract, you will simply add a new circle (a picture might help). I hope it's clear. $\endgroup$ – user171326 May 10 '17 at 20:15
  • $\begingroup$ 1) What is an handle in my picture? For me it is the things on the bottom row, hence there are $n-1$ many. $\endgroup$ – user444847 May 10 '17 at 20:28
  • $\begingroup$ This is the point : you start with an handle, but using the fact that you deleted a disk you somehow can "convert" every handle into a pair of "strips" (which corresponds to $f_1,f_2, \dots$). But as I said, for me this is simpler to write a surface of genus $g$ like on this picture : i.stack.imgur.com/oXWIa.png $\endgroup$ – user171326 May 10 '17 at 20:34

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