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
[hide]Definition
Let (X,J) be a topological space Loop(X,b)⊆C(I,X) and consider the relation of path homotopic maps, ((⋅)≃(⋅) (rel {0,1})) on C(I,X) and restricted to Loop(X,b), then:
- π1(X,b):=Loop(X,b)((⋅)≃(⋅) (rel {0,1})) has a group structure, with the group operation being:
- :[ℓ1]⋅[ℓ2]↦[ℓ1∗ℓ2] where ℓ1∗ℓ2 denotes the loop concatenation of ℓ1,ℓ2∈Loop(X,b).
Proof of claims
References
OLD PAGE
Requires: Paths and loops in a topological space and Homotopic paths
Definition
Given a topological space X and a point x0∈X the fundamental group is[1]
- π1(X,x0) denotes the set of homotopy classes of loops based at x0
- forms a group under the operation of multiplication of the homotopy classes.
