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单词 FundamentalGroup
释义

fundamental group


Let (X,x0) be a pointed topological spacePlanetmathPlanetmath (that is, a topological spaceMathworldPlanetmath with a chosen basepoint x0).Denote by [(S1,1),(X,x0)]the set of homotopy classes of maps σ:S1Xsuch that σ(1)=x0.Here, 1 denotes the basepoint (1,0)S1.Define a productMathworldPlanetmathPlanetmath[(S1,1),(X,x0)]×[(S1,1),(X,x0)][(S1,1),(X,x0)]by [σ][τ]=[στ],where στ means “travel along σ and then τ”.This gives [(S1,1),(X,x0)] a group structureand we define the fundamental groupMathworldPlanetmathPlanetmath of (X,x0)to be π1(X,x0)=[(S1,1),(X,x0)].

In general, the fundamental group of a topological spacedepends upon the choice of basepoint.However, basepoints in the same path-component of the spacewill give isomorphic groupsMathworldPlanetmath.In particular, this means that the fundamental group of a (non-empty) path-connected space is well-defined, up to isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath,without the need to specify a basepoint.

Here are some examples of fundamental groups of familiar spaces:

  • π1(n){0} for each n.

  • π1(S1).

  • π1(T), where T is the torus.

It can be shown that π1 is a functorMathworldPlanetmathfrom the category of pointed topological spaces to the category of groups.In particular, the fundamental group is a topological invariant,in the sense thatif (X,x0) is homeomorphicMathworldPlanetmath to (Y,y0) via a basepoint-preserving map,then π1(X,x0) is isomorphic to π1(Y,y0).

It can also be shown that two homotopically equivalent path-connected spaceshave isomorphic fundamental groups.

Homotopy groupsMathworldPlanetmath generalize the concept of the fundamental group to higher dimensionsMathworldPlanetmath.The fundamental group is the first homotopy group,which is why the notation π1 is used.

Titlefundamental group
Canonical nameFundamentalGroup
Date of creation2013-03-22 11:58:44
Last modified on2013-03-22 11:58:44
Owneryark (2760)
Last modified byyark (2760)
Numerical id16
Authoryark (2760)
Entry typeDefinition
Classificationmsc 57M05
Classificationmsc 55Q05
Classificationmsc 20F34
Synonymfirst homotopy group
Related topicGroup
Related topicCurve
Related topicEtaleFundamentalGroup
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更新时间:2025/5/3 14:23:21