Difference between revisions of "Homotopy group"
DavidB4-bot (talk | contribs) (→top: clean up & uniformity) |
|||
| (7 intermediate revisions by 6 users not shown) | |||
| Line 1: | Line 1: | ||
| − | 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 | + | '''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]]s. |
| − | [[Category: | + | |
| + | 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. | ||
| + | |||
| + | |||
| + | [[Category:Topology]] | ||
Latest revision as of 14:05, July 13, 2016
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi_n} . However, as long as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \geq 2} , the homotopy groups Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi_n(X)} 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi_3(S^2)} 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.