On the resolution of kinks of curves on punctured surfaces

Geiss, Christof; Labardini-Fragoso, Daniel

doi:10.2140/agt.2025.25.3679

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On the resolution of kinks of curves on punctured surfaces

Christof Geiss and Daniel Labardini-Fragoso

Algebraic & Geometric Topology 25 (2025) 3679–3706

DOI: 10.2140/agt.2025.25.3679

Abstract

Let $(Σ, 𝕄, ℙ)$ be a surface with marked points $𝕄 \subseteq \partial Σ \neq \emptyset$ and punctures $ℙ \subseteq Σ ∖ \partial Σ$ . We show that for every curve $γ$ on $Σ ∖ ℙ$ , the curve obtained by resolving the kinks of $γ$ in any order is uniquely determined, up to homotopy in $Σ ∖ ℙ$ , by the $2$ -orbifold homotopy class of $γ$ , in which the punctures are interpreted to be orbifold points of order $2$ . Our proof relies on an application of the diamond lemma.

Keywords

surface with marked points, triangulation, fundamental groupoid, orbifold fundamental groupoid, kink, resolution of kink

Mathematical Subject Classification

Primary: 57K20

Secondary: 13F60, 18B40

References

Publication

Received: 12 September 2023

Revised: 9 April 2024

Accepted: 19 May 2024

Published: 1 October 2025

Authors

Christof Geiss
Instituto de Matemáticas Universidad Nacional Autónoma de México Mexico City Mexico
https://www.matem.unam.mx/~christof
Daniel Labardini-Fragoso
Dipartimento di Matematica “Tullio Levi-Civita” Università degli Studi di Padova Padova Italy
https://www.math.unipd.it/~labardin

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Volume 25, issue 6 (2025)