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 $H^{k} (T^{n}; M) ≅ M^{(\binom{n}{k})}$ .

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 $Z^{n}$ . In other words:

$π_{k} (T^{n}) = {\begin{matrix} Z^{n}, & k = 1 \\ 0, & k > 1 \end{matrix}$