Definition:Fundamental Group

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Let $\struct {X, x_0}$ be a pointed topological space with base point $x_0$.

The fundamental group $\map {\pi_1} {X, x_0}$ of $X$ at the base point $x_0$ is the set of homotopy classes of loops with base point $x_0$ with multiplication of homotopy classes of paths.

Also known as

The fundamental group of a topological space $T$ is also known more explicitly as the fundamental homotopy group of $T$.

The fundamental group of $T$ is also known as the Poincaré group of $T$, for Henri Poincaré.

Also see

  • Results about fundamental groups can be found here.

Historical Note

The concept of the fundamental group was defined by Henri Poincaré in $1895$.

The definition was extended to $n > 1$ by Eduard Čech in $1932$ and Witold Hurewicz in $1935$.