# Homotopy of torus

From Topospaces

This article describes the value (and the process used to compute it) of some homotopy invariant(s) for a topological space or family of topological spaces. The invariant is homotopy group and the topological space/family is torus

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This article gives the key facts about the computation of homotopy groups for the -torus , defined as the product of copies of the circle.

## Statement

is given as follows:

- Case : The set of path components is the one-point set, and we can think of it as the trivial group.
- Case : The fundamental group is the group , i.e., the product of copies of the infinite cyclic group. In other words, it is the free abelian group of rank .
- Case : Any higher homotopy group is the trivial group.

In particular, this means that any torus is an aspherical space.

## Relation with universal covering space

The universal covering space of the torus is Euclidean space , and in fact where the latter is the lattice of points with integer coordinates. The universal cover is a contractible space, and this is equivalent to the observation that is a path-connected aspherical space.

In particular, can be viewed as a classifying space for the infinite discrete group .

## Facts used in computation

- Homotopy of spheres (in particular, homotopy groups of the circle)
- Homotopy group of product is product of homotopy groups