2007/03/27 by Nicolau C. Saldanha, Boris Shapiro, Saldanha, Nicolau C. +1
Computer Science · Mathematics · #52A10 #53A04 #55P15 #58B05 #Differential Geometry (math.DG) #Digital Image Processing Techniques #FOS: Mathematics #Geometric Topology (math.GT) #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #math.DG #math.GT #msc:52A10 #msc:53A04 #msc:55P15 #msc:58B05
paper · pdf · doi:10.48550/arxiv.math/0703803
22 pages, 4 figures; full rewrite of previous version
openalex publication_date 2007/03/27 · arxiv created 2012/01/05 · arxiv updated 2012/01/06 · openalex created_date 2022/10/17 · openalex updated_date 2026/07/28
In the 1920's Marston Morse developed what is now known as Morse theory trying to study the topology of the space of closed curves on S2. We propose to attack a very similar problem, which 80 years later remains open, about the topology of the space of closed curves on S2 which are locally convex (i.e., without inflection points). One of the main difficulties is the absence of the covering homotopy principle for the map sending a non-closed locally convex curve to the Frenet frame at its endpoint. In the present paper we study the spaces of locally convex curves in Sn with a given initial and final Frenet frames. Using combinatorics of B+n+1 = Bn+1 ∩ SOn+1, where Bn+1 ⊂ On+1 is the usual Coxeter-Weyl group, we show that for any n ≥ 2 these spaces fall in at most \lceil(n)/(2)\rceil+1 equivalence classes up to homeomorphism. We also study this classification in the double cover Spin(n+1). For n = 2 our results complete the classification of the corresponding spaces into two topologically distinct classes, or three classes in the spin case.