2-torus

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Definition

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 ${\mathbb{R}}^3$

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

Algebraic topology

Homology

Further information: homology of torus

The homology groups with coefficients in $\mathbb{Z}$ are as follows: $H_0(S^1\times S^1)\cong \mathbb{Z}$, $H_1(S^1\times S^1)\cong \mathbb{Z}\oplus \mathbb{Z}$, and $H_2(S^1\times S^1)\cong \mathbb{Z}$.