Algebraic & Geometric Topology 5 (2005) 1655–1676

DOI: 10.2140/agt.2005.5.1655

arXiv: math.GR/0512343

Abstract

We show that for every subset X of a closed surface M² and every x₀ in X, the natural homomorphism φ:π₁(X,x₀)→ˇπ₁(X,x₀), from the fundamental group to the first shape homotopy group, is injective. In particular, if X⊂M² is a proper compact subset, then π₁(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.

Keywords

fundamental group, planar sets, subsets of closed surfaces, shape group, locally free, fully residually free

Mathematical Subject Classification

Primary: 55Q07, 55Q52, 57N05

Secondary: 20E25, 20E26

References

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Publication

Received: 7 November 2005
Accepted: 10 November 2005
Published: 1 December 2005

Authors