Difference between revisions of "The fundamental group"
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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}} | * <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> | + | : forms a [[Group|group]] under the operation of multiplication of the homotopy classes. |
| + | |||
| + | {{Begin Theorem}} | ||
| + | Theorem: {{M|\pi_1(X,x_0)}} with the binary operation {{M|*}} forms a [[Group|group]]<ref>Introduction to topology - lecture notes nov 2013 - David Mond</ref> | ||
| + | {{Begin Proof}} | ||
| + | * Identity element | ||
| + | * Inverses | ||
| + | * Association | ||
| + | See [[Homotopy class]] for these properties | ||
| + | {{Todo|Mond p30}} | ||
| + | {{End Proof}} | ||
| + | {{End Theorem}} | ||
| + | |||
| + | ==See also== | ||
| + | * [[Homotopy class]] | ||
| + | * [[Homotopic paths]] | ||
| + | * [[Paths and loops in a topological space]] | ||
| + | |||
==References== | ==References== | ||
<references/> | <references/> | ||
Revision as of 12:57, 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]
- forms a group under the operation of multiplication of the homotopy classes.
Theorem: [ilmath]\pi_1(X,x_0)[/ilmath] with the binary operation [ilmath]*[/ilmath] forms a group[2]
