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

Fischer, Hanspeter; Zastrow, Andreas

doi:10.2140/agt.2005.5.1655

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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

DOI: 10.2140/agt.2005.5.1655

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 $φ : π_{1} (X, x_{0}) \to {\overset{̌}{π}}_{1} (X, x_{0})$ , from the fundamental group to the first shape homotopy group, is injective. In particular, if $X ⊊ M^{2}$ is a proper compact subset, then $π_{1} (X, x_{0})$ 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.

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

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Publication

Received: 7 November 2005

Accepted: 10 November 2005

Published: 1 December 2005

Authors

Hanspeter Fischer
Department of Mathematical Sciences Ball State University Muncie IN 47306 USA

Andreas Zastrow
Institute of Mathematics University of Gdańsk ul. Wita Stwosza 57 80-952 Gdańsk Poland

Volume 5, issue 4 (2005)