Genus two surface

Definition
This topological space, denoted $$T^2 \# T^2$$ or $$\Sigma_2$$, is defined in the following equivalent ways:


 * 1) It is the connected sum of two copies of the defining ingredient::2-torus.
 * 2) It is the defining ingredient::compact orientable surface of genus $$2$$.

Homology
The homology groups over the integers are as follows:

$$H_k(\Sigma_2;\mathbb{Z}) = \lbrace\begin{array}{rl} \mathbb{Z}, & \qquad k = 0,2 \\ \mathbb{Z}^4, & \qquad k = 1 \\ 0, & \qquad k > 2 \\\end{array}$$

More generally, with coefficients in a module $$M$$, the homology groups are:

$$H_k(\Sigma_2;\mathbb{Z}) = \lbrace\begin{array}{rl} M, & \qquad k = 0,2 \\ M^4, & \qquad k = 1 \\ 0, & \qquad k > 2 \\\end{array}$$

The reduced homology looks the same except that the zeroth homology groups/modules are now zero.

Cohomology
The cohomology groups over the integers are as follows:

$$H^k(\Sigma_2;\mathbb{Z}) = \lbrace\begin{array}{rl} \mathbb{Z}, & \qquad k = 0,2 \\ \mathbb{Z}^4, & \qquad k = 1 \\ 0, & \qquad k > 2 \\\end{array}$$

More generally, with coefficients in a module $$M$$, the cohomology groups are:

$$H^k(\Sigma_2;\mathbb{Z}) = \lbrace\begin{array}{rl} M, & \qquad k = 0,2 \\ M^4, & \qquad k = 1 \\ 0, & \qquad k > 2 \\\end{array}$$