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 M2 and every basepoint x0, 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 M2, then pi1(X,x0) 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.
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