Difference between revisions of "The fundamental group"
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| − | '''Requires: ''' [[Paths and loops in a topological space]] and [[Homotopic paths]] | + | '''Requires: ''' [[Paths and loops in a topological space]] and [[Homotopic paths]] |
==Definition== | ==Definition== | ||
| − | Given a [[Topological space|topological space]] {{M|X}} and a point {{M|x_0\in X}} | + | Given a [[Topological space|topological space]] {{M|X}} and a point {{M|x_0\in X}} the fundamental group is<ref>Introduction to Topology - Second Edition - Theodore W. Gamelin and Rober Everist Greene</ref> |
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| + | * <math>\pi_1(X,x_0)</math> denotes the set of [[Homotopy class|homotopy classes]] of [[Paths and loops in a topological space|loops]] based at {{M|x_0}} | ||
| + | : forms a [[Group|group]] under the operation of multiplication of the homotopy classes.<ref>Introduction to topology - lecture notes nov 2013 - David Mond</ref> | ||
==References== | ==References== | ||
<references/> | <references/> | ||
Revision as of 01:01, 17 April 2015
Requires: Paths and loops in a topological space and Homotopic paths
Definition
Given a topological space [ilmath]X[/ilmath] and a point [ilmath]x_0\in X[/ilmath] the fundamental group is[1]
- [math]\pi_1(X,x_0)[/math] denotes the set of homotopy classes of loops based at [ilmath]x_0[/ilmath]
