#### Volume 5, issue 4 (2005)

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

### Hanspeter Fischer and Andreas Zastrow

Algebraic & Geometric Topology 5 (2005) 1655–1676
 arXiv: math.GR/0512343
##### Abstract

We show that for every subset $X$ of a closed surface ${M}^{2}$ and every ${x}_{0}\in X$, the natural homomorphism $\phi :\phantom{\rule{0.3em}{0ex}}{\pi }_{1}\left(X,{x}_{0}\right)\to {\stackrel{̌}{\pi }}_{1}\left(X,{x}_{0}\right)$, 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.

##### Keywords
fundamental group, planar sets, subsets of closed surfaces, shape group, locally free, fully residually free
##### Mathematical Subject Classification 2000
Primary: 55Q52, 55Q07, 57N05
Secondary: 20E25, 20E26