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 $x_0\in X$, the natural homomorphism $\phi :{\pi }_1\left(X,x_0\right)\to {\check{\pi }}_1\left(X,x_0\right)$, from the fundamental group to the first shape homotopy group, is injective. In particular, if $X⊊M^2$ is a proper compact subset, then ${\pi }_1\left(X,x_0\right)$ 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.