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?