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! | ||
| + | * 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 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]] | ||
| + | {{Begin Inline Proof}} | ||
| + | {{/Proof that it is a group}} | ||
| + | {{End Proof}}{{End Theorem}} | ||
| + | ==References== | ||
| + | <references/> | ||
| + | {{Definition|Topology|Homotopy Theory}} | ||
| + | |||
| + | =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== | ||
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Contents
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]
- [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]
