# Torus

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

## Definition

A -**torus** is defined as the product of copies of the circle, equipped with the product topology. In other words, it is the space with written times.

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

The -torus is sometimes denoted , a convention we follow on this page.

## Algebraic topology

### Homology

`Further information: homology of torus`

The homology (with integer coefficients) is a free abelian group of rank for , and is the zero group for (note that under one of the interpretations of binomial coefficient, we do not need to make a separate case for because is defined to be zero for ).

More generally, the homology with coefficients in a module over a commutative unital ring is .

### 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 is the alternating algebra in variables over .

### 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 . In other words: