Difference between revisions of "The fundamental group"
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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]] | [[/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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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.
