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
Get more specific information about torus | Get more computations of homotopy group
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