Homotopy of torus

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 $n$-torus $T^n$, defined as the product of $n$ copies of the circle.

Statement

${\pi }_k(T^n)$ is given as follows:

• Case $k=0$: The set of path components ${\pi }_0(T^n)$ is the one-point set, and we can think of it as the trivial group.
• Case $k=1$: The fundamental group ${\pi }_1(T^n)$ is the group ${\mathbb{Z}}^n$, i.e., the product of $n$ copies of the infinite cyclic group. In other words, it is the free abelian group of rank $n$.
• Case $k\ge 2$: Any higher homotopy group ${\pi }_k(T^n)$ 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 $T^n$ is Euclidean space ${\mathbb{R}}^n$, and in fact $T^n\cong {\mathbb{R}}^n/{\mathbb{Z}}^n$ 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 $T^n$ is an aspherical space.

Facts used in computation

1. Homotopy of spheres (in particular, homotopy groups of the circle)
2. Homotopy group of product is product of homotopy groups