2023/11/13 by Carlo Petronio, Petronio, Carlo
Mathematics · #37E35 #57R40 #57R42 (primary) #58D10 (secondary) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2311.07221
openalex publication_date 2023/11/13 · openalex created_date 2023/11/15 · openalex updated_date 2026/08/01
Let S be a compact surface with boundary and F be the set of the orbits of a traversing flow on S. If the flow is generic, its orbit space is a spine G of S, namely G is a graph embedded in S and S is a regular neighbourhood of G. Moreover an extra structure on G turns it into a flow-spine, from which one can reconstruct S and F. In this paper we study properly immersed curves C in S. We do this by considering generic C's and their apparent contour relative to F, namely the set of points of G corresponding to orbits that either are tangent to C, or go through a self-intersection of C, or meet the boundary of C. We translate this apparent contour into a decoration of G that allows one to reconstruct C, and then we allow C to vary up to homotopy within a fixed generic F, and next also F to vary up to homotopy, and we identify a finite set of local moves on decorated graphs that translate these homotopies.