2023/01/10 by W. Patrick Hooper, Hooper, W. Patrick, Ferrán Valdez +3
Mathematics · #51M05 #57M50 (Primary) 57K20 #57R30 (Secondary) #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #FOS: Mathematics #Geometric Topology (math.GT) #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2301.03727
openalex publication_date 2023/01/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce the notion of a zebra structure on a surface, which is a more general geometric structure than a translation structure or a dilation structure that still gives a directional foliation of every slope. We are concerned with the question of when a free homotopy class of loops (or a homotopy class of arcs relative to endpoints) has a canonical representative or family of representatives, either as closed leaves or chains of leaves joining singularities. We prove that such representations exist if the surface has a triangulation with edges joining singularities (in the zebra structure sense). In the special case when the surface is closed, we describe several geometric conditions that are equivalent to the existence of canonical representations in every homotopy class of closed curves.