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 $k\geq 2$: 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 a path-connected aspherical space.

In particular, $T^{n}$ can be viewed as a classifying space for the infinite discrete group $\mathbb {Z} ^{n}$.

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