- Published 2007

In 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, see [7] and [5]. 80 years later a very similar problem about the topology of the space of closed and locally convex (i.e. without inflection points) curves on S2 is still widely open. The main difficulty 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 the Weyl group Dn+1 ⊂ SO(n+1) we show that for any n ≥ 2 these spaces fall in at most ⌈n2 ⌉+ 1 equivalence classes under weak homotopy equivalence. We also study this classification in the double cover D̃n+1 ⊂ S̃O(n+1) = Spin(n+1). We show that the obtained classes are topologically distinct for n = 2.

@inproceedings{Saldanha2007ClassificationOS,
title={Classification of spaces of locally convex curves in S n and combinatorics of the Weyl group D n + 1 Nicolau C . Saldanha and Boris Shapiro March 22 , 2007},
author={Nicolau C. Saldanha and Boris Z. Shapiro},
year={2007}
}