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

Hanspeter Fischer, Andreas Zastrow
2005 Algebraic and Geometric Topology  
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.
doi:10.2140/agt.2005.5.1655 fatcat:h5m7kjl62jdqjm7qmaire3usee