Definition:Fundamental 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 is also known more explicitly as the fundamental homotopy group.

Also see

• Fundamental Group is Group
• Definition:Fundamental Group Functor
• Definition:Homotopy Group
• Fundamental Group is Independent of Base Point for Path-Connected Space

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
• 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
• 2011: John M. Lee: Introduction to Topological Manifolds (2nd ed.) ... (previous) ... (next): $\S 7$: Homotopy and the Fundamental Group. Homotopy