Torus

Note that the term torus is often used for the more specific and restricted notion of 2-torus.

Definition
A $$n$$-torus is defined as the product of $$n$$ copies of the defining ingredient::circle, equipped with the product topology. In other words, it is the space $$S^1 \times S^1 \times \dots \times S^1$$ with $$S^1$$ written $$n$$ times.

Cases of special interest are $$n = 1$$ (where we get the circle) and $$n = 2$$ (where we get the 2-torus).

The $$n$$-torus is sometimes denoted $$T^n$$, a convention we follow on this page.

Homology
The homology (with integer coefficients) $$H_k(T^n)$$ is a free abelian group of rank $$\binom{n}{k}$$ for $$0 \le k \le n$$, and is the zero group for $$k > n$$ (note that under one of the interpretations of binomial coefficient, we do not need to make a separate case for $$k > n$$ because $$\binom{n}{k}$$ is defined to be zero for $$k > n$$).

More generally, the homology with coefficients in a module $$M$$ over a commutative unital ring $$R$$ is $$H^k(T^n;M) \cong M^{\binom{n}{k}}$$.

Cohomology
The cohomology groups are isomorphic to the respective homology groups, both with integer coefficients and with coefficients in an arbitrary module.

The cohomology ring with coefficients in a commutative unital ring $$R$$ is the alternating algebra in $$n$$ variables over $$R$$.

Homotopy
Each torus is an aspherical space as well as a path-connected space, so its only nontrivial homotopy group is the fundamental group, which is $$\mathbb{Z}^n$$. In other words:

$$\pi_k(T^n) = \lbrace\begin{array}{rl} \mathbb{Z}^n, & k = 1 \\ 0, & k > 1 \\\end{array}$$