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

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

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 $X$ and a point $x_0\in X$ the fundamental group is

• $$\pi_1(X,x_0)$$ denotes the set of homotopy classes of loops based at $x_0$

forms a group under the operation of multiplication of the homotopy classes.

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Theorem: $\pi_1(X,x_0)$ with the binary operation $*$ 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