2-torus

As a product space
The 2-torus, sometimes simply called the torus, is defined as the product (equipped with the product topology) of two circles, i.e., it is defined as $$S^1 \times S^1$$. The 2-torus is also denoted $$T^2$$.

The term torus more generally refers to a product of finitely many copies of the circle, equipped with the product topology. The $$n$$-torus is sometimes denoted as $$T^n$$.

As a subspace of $$\R^3$$
A 2-torus in $$\R^3$$ is obtained as the surface of revolution achieved by revolving a circle about a line in its plane that does not intersect it.

Homology groups
The homology groups with coefficients in $$\mathbb{Z}$$ are as follows:

$$H_p(S^1 \times S^1;\mathbb{Z}) = \lbrace\begin{array}{rl} \mathbb{Z} & \qquad p = 0,2 \\ \mathbb{Z} \oplus \mathbb{Z}, & \qquad p = 1\\0, & \qquad p \ge 3\\\end{array}$$

The homology groups with coefficients in a module $$M$$ are as follows:

$$H_p(S^1 \times S^1;M) = \lbrace\begin{array}{rl} M & \qquad p = 0,2 \\ M \oplus M, & \qquad p = 1\\0, & \qquad p \ge 3\\\end{array}$$

Cohomology groups
The cohomology group with coefficients in $$\mathbb{Z}$$ are as follows:

$$H^p(S^1 \times S^1;\mathbb{Z}) = \lbrace\begin{array}{rl} \mathbb{Z} & \qquad p = 0,2 \\ \mathbb{Z} \oplus \mathbb{Z}, & \qquad p = 1\\0, & \qquad p \ge 3\\\end{array}$$

The cohomology groups with coefficients in a module $$M$$ are as follows:

$$H^p(S^1 \times S^1;\mathbb{Z}) = \lbrace\begin{array}{rl} M & \qquad p = 0,2 \\ M \oplus M, & \qquad p = 1\\0, & \qquad p \ge 3\\\end{array}$$

Homotopy groups
The homotopy groups are given as follows:

Algebraic structure
The 2-torus can be given the structure of a topological group, arising as the external direct product of two copies of the circle $$S^1$$, which itself is a group (for instance, it has a group structure if identified with the set of complex numbers of modulus 1). With this structure, the 2-torus is an abelian group.

On account of being a topological group, the 2-torus also acquires the structure of a H-space. Some consequences are:


 * The zeroth homotopy set naturally acquires a group structure now -- in this case the trivial group.
 * The fundamental group must be an abelian group, and hence must be isomorphic to the first homology group. In this case, both groups are isomorphic to $$\mathbb{Z}^2$$, the free abelian group of rank two.

Coalgebraic structure
Is there a coalgebraic structure?