Foundations for the Fundamental Group
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The fundamental group was introduced by Poincaré 1892 (though anticipated to some extent by the study of curves on surfaces in Jordan 1866b). Poincaré defined the group in function-theoretic terms by considering analytic continuation of a many-valued function Φ around a closed path p in a manifold. Since the value obtained after completing p may differ from the initial value, p may be considered to define a transformation of Φ, and since any path p′ which is deformable into p defines the same transformation, the group of transformations of the “most general” function Φ is naturally isomorphic to the group of equivalence classes of closed paths, where “equivalent” means mutually deformable.
KeywordsFundamental Group Simplicial Complex Free Product Closed Path Deformation Retract
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