I have come across the concept of homology groups of surfaces (so I am seeing $H_1(M, \mathbb{Z})$ and $H_1(M - X, \mathbb{Z})$ and $H_1(M, X, \mathbb{Z})$, where $M$ is a surface, and $X \subset M$ is a finite set of points, for example, section 4.5 here). I have not studied any algebraic topology yet, and just started reading Hatcher.

The surfaces I am dealing with are quite concrete. For example, torrii with a number of removed points. What do those groups look like? Are they easy to visualize like the fundamental group?

  • $\begingroup$ $H_1$ is the abelianization of $\pi_1$. You can read about that in Hatcher as well. $\endgroup$ – user144221 Oct 12 '16 at 3:29
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    $\begingroup$ I think this answers your question: math.stackexchange.com/questions/185903 $\endgroup$ – user144221 Oct 12 '16 at 3:36

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