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General expansion
'''Homotopy groups''' are tools used in [[algebraic topology]] to classify [[topological space]]s. The different ways to map an '''n-[[sphere]]''' continuously into a given topological space are divided into [[equivalence class]]es, called '''homotopy classes'''.
'''Homotopy groups''' are tools used in [[algebraic topology]] to classify [[topological space]]s. The different ways to map an '''n-[[sphere]]''' continuously into a given topological space are divided into [[equivalence class]]es, called '''homotopy classes'''. The set of homotopy classes of maps of the n-sphere into a space may be endowed with a group structure by a means analogous to the concatenation operation used to construct the [[fundamental group]]; this group is usually denoted <math>\pi_n</math>. However, as long as <math>n \geq 2</math>, the homotopy groups <math>\pi_n(X)</math> are [[Abelian group|abelian groups]].
Homotopy groups are notoriously difficult to compute, in contrast with homology and cohomology groups, where are generally computable: even the higher homotopy groups of spheres are not fully understood. For example, the group <math>\pi_3(S^2)</math> is isomorphic to the group of integers, generated by the Hopf fibration. Spectral sequences are an important tool in the computation of higher homotopy groups.
[[category: Topology]]
[[category: Topology]]
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