Note that the term torus is often used for the more specific and restricted notion of 2-torus.
The -torus is sometimes denoted , a convention we follow on this page.
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 .
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 .
Further information: homotopy of torus