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

Fischer, Hanspeter¹ ; Zastrow, Andreas²

¹Department of Mathematical Sciences, Ball State University, Muncie IN 47306, USA
²Institute of Mathematics, University of Gdańsk, ul. Wita Stwosza 57, 80-952 Gdańsk, Poland

Algebraic and Geometric Topology, Tome 5 (2005) no. 4, pp. 1655-1676

Résumé

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.

Détail

DOI : 10.2140/agt.2005.5.1655

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

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