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 M2 and every x0 X, the natural homomorphism φ: π1(X,x0) π̌1(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 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