how to compute the homology groups of a torus with n points removed?

Question

i have really big problems in computing the homology groups and even don't really know how to write the exact sequence for different spaces. i tried to take the torus with n points removed,homotopic to a circle with n circles attached around and use my intuition to get the answer which i'm not sure if is true... can someone please give a detailed answer? i really want to learn this subject.

Answer 1

Since you know how to compute its fundamental group, the first homology group is the abelianization of the fundamental group. The zero homology groups is isomorphic to $\mathbb{Z} $ since it is connected. And top homology group is zero since it is not compact.

Top homology of an oriented, compact, connected smooth manifold with boundary

Answer 2

After removing $1$ point, and seeing the torus as a $CW$ complex (given by making the usual identifications $I^2/\sim$ of a square) then before gluing, we can enlarge the hole to the boundary of the square. After making identifications, this gives the wedge of two circles.

One can continue in this way to see that the torus with $n$ points removed gives the wedge of $n+1$ circles. In doing this, we can compute the (reduced) homology with Mayer-Vietoris (which is done in Hatcher, if I remember correctly.)

The easiest way to do this is probably induction on the number of circles, and noting that for the wedge of two circles, you can find neighborhoods so that $A \cup B=X$ and $A \cap B$ is contractible.