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

Hanspeter Fischer·Andreas Zastrow
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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.

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Hanspeter Fischer, Andreas Zastrow. 2005-12-14. The fundamental groups of subsets of closed surfaces inject into their first shape groups. https://doi.org/10.2140/agt.2005.5.1655

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