Difference between revisions of "Homotopy group"
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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. | + | 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]]. |
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| + | 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. | ||
Revision as of 20:34, June 9, 2009
Homotopy groups are tools used in algebraic topology to classify topological spaces. The different ways to map an n-sphere continuously into a given topological space are divided into equivalence classes, 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 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.