Difference between revisions of "The fundamental group"

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{{Refactor notice|grade=A|msg=I cannot believe it's been 15 months and this still isn't complete!
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* Started refactoring [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 19:55, 1 November 2016 (UTC)}}
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==Definition==
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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:
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* {{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:
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** {{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)}}.
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==Proof of claims==
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{{Begin Inline Theorem}}
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[[/Proof that it is 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]]
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{{Begin Inline Proof}}
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{{/Proof that it is a group}}
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{{End Proof}}{{End Theorem}}
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==References==
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<references/>
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{{Definition|Topology|Homotopy Theory}}
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=OLD PAGE=
 
'''Requires: ''' [[Paths and loops in a topological space]] and [[Homotopic paths]]
 
'''Requires: ''' [[Paths and loops in a topological space]] and [[Homotopic paths]]
 
==Definition==
 
==Definition==

Revision as of 19:55, 1 November 2016

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I cannot believe it's been 15 months and this still isn't complete!
  • Started refactoring Alec (talk) 19:55, 1 November 2016 (UTC)

Definition

Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space [ilmath]\text{Loop}(X,b)\subseteq C(I,X)[/ilmath] and consider the relation of path homotopic maps, [ilmath]\big((\cdot)\simeq(\cdot)\ (\text{rel }\{0,1\})\big)[/ilmath] on [ilmath]C(I,X)[/ilmath] and restricted to [ilmath]\text{Loop}(X,b)[/ilmath], then:

  • [ilmath]\pi_1(X,b):=\frac{\text{Loop}(X,b)}{\big((\cdot)\simeq(\cdot)\ (\text{rel }\{0,1\})\big)}[/ilmath] has a group structure, with the group operation being:
    • [ilmath]:[\ell_1]\cdot[\ell_2]\mapsto[\ell_1*\ell_2][/ilmath] where [ilmath]\ell_1*\ell_2[/ilmath] denotes the loop concatenation of [ilmath]\ell_1,\ell_2\in\text{Loop}(X,b)[/ilmath].

Proof of claims

References


OLD PAGE

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]

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]


  • Identity element
  • Inverses
  • Association

See Homotopy class for these properties


TODO: Mond p30



See also

References

  1. Introduction to Topology - Second Edition - Theodore W. Gamelin and Rober Everist Greene
  2. Introduction to topology - lecture notes nov 2013 - David Mond