The fundamental groups of subsets of closed surfaces inject into their first shape groups
Group Theory
2014-10-01 v1 Algebraic Topology
Abstract
We show that for every subset X of a closed surface M^2 and every basepoint x_0, the natural homomorphism from the fundamental group to the first shape homotopy group, is injective. In particular, if X is a proper compact subset of M^2, then pi_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
Cite
@article{arxiv.math/0512343,
title = {The fundamental groups of subsets of closed surfaces inject into their first shape groups},
author = {Hanspeter Fischer and Andreas Zastrow},
journal= {arXiv preprint arXiv:math/0512343},
year = {2014}
}
Comments
Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-67.abs.html