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 ${\binom {n}{k}}$ for $0\leq k\leq n$, 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

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:

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