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 $H_{2}(S^{1}\times S^{1})\cong \mathbb {Z}$.