Homotopy Group - Methods of Calculation

Methods of Calculation

Calculation of homotopy groups is in general much more difficult than some of the other homotopy invariants learned in algebraic topology. Unlike the Seifert–van Kampen theorem for the fundamental group and the Excision theorem for singular homology and cohomology, there is no simple way to calculate the homotopy groups of a space by breaking it up into smaller spaces. However methods developed in the 1980s involving a van Kampen type theorem for higher homotopy groupoids have allowed new calculations on homotopy types and so on homotopy groups. See for a sample result the 2008 paper by Ellis and Mikhailov listed below.

For some spaces, such as tori, all higher homotopy groups (that is, second and higher homotopy groups) are trivial. These are the so-called aspherical spaces. However, despite intense research in calculating the homotopy groups of spheres, even in two dimensions a complete list is not known. To calculate even the fourth homotopy group of S2 one needs much more advanced techniques than the definitions might suggest. In particular the Serre spectral sequence was constructed for just this purpose.

Certain Homotopy groups of n-connected spaces can be calculated by comparison with homology groups via the Hurewicz theorem.