Spaces of Locally Convex Curves in S^n and Combinatorics of the Group B_(n+1)^+
Nicolau Saladanha, Boris Zalmanovich Shapiro · Journal of Singularities · 2012
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 S 2 ([7], [5]).We propose to attack a very similar problem, which 80 years later remains open, about the topology of the space of closed curves on S 2 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 nonclosed locally convex curve to the Frenet frame at its endpoint.In the present paper we study the spaces of locally convex curves in S n with a given initial and final Frenet frames.Using combinatorics of B + n+1 = B n+1 ∩ SO n+1 , where B n+1 ⊂ O n+1 is the usual Coxeter-Weyl group, we show that for any n ≥ 2 these spaces fall in at most ⌈ n 2 ⌉ + 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.