The fundamental group

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