fundamental group

English

Noun

fundamental group (plural fundamental groups)

1. (topology) For 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.
   • 1991, William S. Massey, A Basic Course in Algebraic Topology, Springer, page 35:

     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.

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

     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.

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

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

Derived terms

• algebraic fundamental group
• étale fundamental group
• orbifold fundamental group

Related terms

• fundamental groupoid

Translations

group of equivalence classes of the homotopies of loops in a given topological space

• Catalan: grup fonamental m
• Czech: fundamentální grupa f
• Dutch: fundamentaalgroep f
• French: groupe fondamental m
• German: Fundamentalgruppe f
• Hungarian: fundamentális csoport
• Icelandic: undirstöðugrúpa f
• Italian: gruppo fondamentale m
• Polish: grupa podstawowa f
• Russian: фундаментальная группа f (fundamentalʹnaja gruppa)
• Swedish: fundamentalgrupp c
• Turkish: temel grup
• Ukrainian: фундаментальна група f (fundamentalʹna hrupa)