Homology of compact orientable surfaces

Statement
Suppose $$g$$ is a nonnegative integer. We denote by $$\Sigma_g$$ the compact orientable surface of genus $$g$$, i.e., $$\Sigma_g$$ is the connected sum of $$g$$ copies of the 2-torus $$T = S^1 \times S^1$$(and it is the 2-sphere $$S^2$$when $$g = 0$$). In other words, $$\Sigma_0 = S^2$$, $$\Sigma_1 = T$$, $$\Sigma_2 = T \# T$$, and so on.

Unreduced version over the integers
We have:

$$H_k(\Sigma_g;\mathbb{Z}) := \lbrace\begin{array}{rl} \mathbb{Z}, & k = 0,2 \\ \mathbb{Z}^{2g}, & k = 1 \\ 0, & k > 2\end{array}$$

In other words, the zeroth and second homology groups are both free of rank one, and the first homology group is $$\mathbb{Z}^{2g}$$, i.e., the free abelian group of rank $$2g$$.

Reduced version over the integers
We have:

$$\tilde{H}_k(\Sigma_g;\mathbb{Z}) := \lbrace\begin{array}{rl} 0, & k = 0 \\\mathbb{Z}^{2g}, & k = 1 \\ \mathbb{Z}, & k = 2 \\ 0, & k > 2 \end{array}$$

Unreduced version with coefficients in $$M$$
With coefficients in a module $$M$$ over a ring $$R$$, we have:

$$H_k(\Sigma_g;M) := \lbrace\begin{array}{rl} M, & k = 0,2 \\ M^{2g}, & k = 1 \\ 0, & k > 2\end{array}$$

Reduced version with coefficients in $$M$$
$$\tilde{H}_k(\Sigma_g;M) := \lbrace\begin{array}{rl} 0, & k = 0 \\ M^{2g}, & k = 1 \\ M, & k = 2 \\  0, & k > 2 \end{array}$$

Related invariants
These are all invariants that can be computed in terms of the homology groups.

Facts used in homology computation

 * 1) uses::Homology of spheres: This tackles the $$g = 0$$ case.
 * 2) uses::Homology of torus: This tackles the $$g = 1$$ case.
 * 3) uses::Homology of connected sum: This tackles the inductive procedure of taking connected sums.