Torus

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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 \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 \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}