Definition:Fundamental Group
(Redirected from Definition:Fundamental Homotopy Group)
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Definition
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
- Fundamental Group is Group
- Definition:Fundamental Group Functor
- Definition:Homotopy Group
- Fundamental Group is Independent of Base Point for Path-Connected Space
- 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 $1935$.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): fundamental homotopy group or groupoid
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): fundamental group
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): homotopy group
- 2000: James R. Munkres: Topology (2nd ed.): $9$: The Fundamental Group $\S 52$: The Fundamental Group
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): fundamental group
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): homotopy group
- 2011: John M. Lee: Introduction to Topological Manifolds (2nd ed.) ... (previous) ... (next): $\S 7$: Homotopy and the Fundamental Group. Homotopy