Homotopy of torus

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 $$\R^n$$, and in fact $$T^n \cong \R^n/\mathbb{Z}^n$$ 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