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 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.

Algebraic topology

Homology

Further information: homology of torus

The homology (with integer coefficients) $H_k(T^n)$ is a free abelian group of rank $(\genfrac{}{}{0ex}{}{n}{k})$ 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 $(\genfrac{}{}{0ex}{}{n}{k})$ 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 $H^k(T^n;M)\cong M^{(\genfrac{}{}{0ex}{}{n}{k})}$.

Cohomology

Further information: cohomology of torus

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

Further information: homotopy of torus

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:

${\pi }_k(T^n)=\{\begin{array}{cc}{\mathbb{Z}}^n,& k=1\\ 0,& k>1\\ \end{array}$