Fundamental group

The fundamental group of a pointed topological space is the group of based loops modulo homotopy that fixes the base point, with concatenation as its operation.

Based loop: A continuous path in a space that begins and ends at the same specified point. These are the objects whose homotopy classes form the group.
Topological space: A set equipped with a collection of open subsets that specifies its notion of continuity. The loops and their deformations live in this underlying space.
Seifert–van Kampen theorem: A theorem computing a space's fundamental group from groups of overlapping open subspaces. It builds complicated examples by combining groups of simpler pieces.
First homology group: The first homology group records one-dimensional cycles modulo boundaries, with coefficients specified separately. It is the abelianization of the fundamental group for path-connected spaces.
Homotopy of loops: A continuous deformation between loops that keeps their shared base point fixed. It determines when two loops represent the same group element.
Loop space: The space of continuous maps from a circle into a topological space, often with a chosen base point. Its path components correspond to based loop classes under suitable conventions.
Free group: A group generated by a set with no relations beyond those required by the group axioms. Wedges of circles have free fundamental groups, making graph examples explicit.
Higher homotopy group: A homotopy invariant generalizing the fundamental group to maps of higher-dimensional spheres. Higher homotopy groups detect sphere-shaped phenomena that loops cannot capture.
Path concatenation: The operation of traversing one path and then another, when their endpoints match. Concatenating based loops supplies the group operation.
Path-connected space: A space in which every pair of points can be joined by a continuous path. Path-connectedness makes fundamental groups at different base points isomorphic.

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