The fundamental groups of subsets of closed surfaces inject into their first shape groups

Abstract

We show that for every subset X of a closed surface M^2 and every basepoint x_0, the natural homomorphism from the fundamental group to the first shape homotopy group, is injective. In particular, if X is a proper compact subset of M^2, then pi_1(X,x_0) is isomorphic to a subgroup of the limit of an inverse sequence of finitely generated free groups; it is therefore locally free, fully residually free and residually finite.

Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-67.abs.html

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