The fundamental group

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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

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The fundamental group/Proof that it is a group

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

• [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

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• Identity element
• Inverses
• Association

See Homotopy class for these properties

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TODO: Mond p30

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See also

• Homotopy class
• Homotopic paths
• Paths and loops in a topological space