fundamental group

plural fundamental groups

1. mathematicsFor a specified topological space, the group whose elements are homotopy classes of loops (images of some arbitrary closed interval whose endpoints are both mapped to a designated point) and whose group operation is concatenation.

   During the more than 100 years of its existence, the notion of the fundamental group has undergone a considerable evolution.

   2012, Jakob Stix, editor, The Arithmetic of Fundamental Groups: PIA 2010, Springer, page ix

   2 more examples

   From the definition it will be clear the group is a topological invariant of X; i.e., if two spaces are homeomorphic, their fundamental groups are isomorphic.

   1991, William S. Massey, A Basic Course in Algebraic Topology, Springer, page 35

   Therein came into focus the far-reaching vision that the perspective of non-abelian fundamental groups could lead to a fundamentally new understanding of deep arithmetic phenomena, including the arithmetic theory of moduli and Diophantine finiteness on hyperbolic curves.

   2011, John Coates, Minhyong Kim, Florian Pop, Mohamed Saïdi, Peter Schneider, editors, Non-abelian Fundamental Groups and Iwasawa Theory, Cambridge University Press, page vii