View source for The fundamental group

{{Refactor notice|grade=A|msg=I cannot believe it's been 15 months and this still isn't complete!
* Started refactoring [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 19:55, 1 November 2016 (UTC)}}
==Definition==
Let {{Top.|X|J}} be a [[topological space]] {{M|\text{Loop}(X,b)\subseteq C(I,X)}} and consider the [[relation]] of [[path homotopic maps|path homotopic maps, {{M|\big((\cdot)\simeq(\cdot)\ (\text{rel }\{0,1\})\big)}}]] on {{M|C(I,X)}} and restricted to {{M|\text{Loop}(X,b)}}, then:
* {{M|1=\pi_1(X,b):=\frac{\text{Loop}(X,b)}{\big((\cdot)\simeq(\cdot)\ (\text{rel }\{0,1\})\big)} }} has a [[group]] structure, with the [[group operation]] being:
** {{M|:[\ell_1]\cdot[\ell_2]\mapsto[\ell_1*\ell_2]}} where {{M|\ell_1*\ell_2}} denotes the [[loop concatenation]] of {{M|\ell_1,\ell_2\in\text{Loop}(X,b)}}.
==Proof of claims==
{{Begin Inline Theorem}}
[[Proof that the fundamental group is actually a group|Outline of proof that {{M|\pi_1(X,b)}} admits a group structure with {{M|\big(:([\ell_1],[\ell_2])\mapsto[\ell_1*\ell_2]\big)}} as the operation]]
{{Begin Inline Proof}}{{:Proof that the fundamental group is actually a group/Outline}}
{{End Proof}}{{End Theorem}}
{{Begin Inline Theorem}}
[[Proof that the fundamental group is actually a group|Proof that {{M|\pi_1(X,b)}} admits a group structure with {{M|\big(:([\ell_1],[\ell_2])\mapsto[\ell_1*\ell_2]\big)}} as the operation]]
{{Begin Inline Proof}}{{:Proof that the fundamental group is actually a group/Proof}}
{{End Proof}}{{End Theorem}}

==References==
<references/>
{{Definition|Topology|Homotopy Theory}}

=OLD PAGE=
'''Requires: ''' [[Paths and loops in a topological space]] and [[Homotopic paths]]
==Definition==
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>

* <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.

{{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/>


{{Definition|Topology}}