Editing Homotopy group (section)

== Methods of calculation ==
Calculation of homotopy groups is in general much more difficult than some of the other homotopy [[Invariant (mathematics)|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 known 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 2010 paper by Ellis and Mikhailov.<ref>{{cite journal|first1=Graham J.|last1=Ellis|first2=Roman|last2=Mikhailov|title=A colimit of classifying spaces|journal=[[Advances in Mathematics]]|volume=223|year=2010|issue=6|pages=2097–2113|arxiv=0804.3581|doi=10.1016/j.aim.2009.11.003|doi-access=free|mr=2601009}}</ref>

For some spaces, such as [[Torus|tori]], all higher homotopy groups (that is, second and higher homotopy groups) are [[Trivial group|trivial]]. These are the so-called [[aspherical space]]s. 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 <math>S^2</math> 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|''n''-connected]] spaces can be calculated by comparison with [[homology group]]s via the [[Hurewicz theorem]].