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'''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 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.

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.  Even small homotopy groups surprising turn out to be nontrivial: the group <math>\pi_3(S^2)</math> is isomorphic to the group of integers, generated by the [[Hopf fibration]].

 

A famous [[conjecture]] stated in terms of homotopy groups is the recently-proven Poincare conjecture, which states that any manifold homotopy equivalent to a [[sphere]] actually is a sphere.  The precise formulation depends on whether one works in the category of smooth, piecewise-linear, or topological manifolds.