Homotopy of torus

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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:

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