The fundamental groups of subsets of closed surfaces inject into their first shape groups
By H. Fischer, A. Zastrow. 2005
60 citations recorded by the source corpus.
We show that for every subset X of a closed surface M 2 and every x0 ∈ X, the natural homomorphism ϕ : π1(X, x0) → y π1(X, x0), from the fundamental group to the first shape homotopy group, is injective. In particular, if X ( M 2 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. AMS Classification 55Q52, 55Q07, 57N05; 20E25, 20E26
Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/015845db835fb0b9dfc9b1937e874691d3afd1d2
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In Computational Theory and Mathematics
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Citations in the corpus
- The fundamental group as a topological group (2010, unknown): “20] but since Σ(X+) is a planar continuum, the shape group π̆1(Σ(X+)) is metrizable and φ : π1(Σ(X+))→ π̆1(Σ(X+)) is injective [15].”
- Fundamental groups of one-dimensional spaces (2013, background): “for instance [1, 3, 10, 11]) and planar (see [5, 13]) spaces were derived.”