Spaces of locally convex curves in S^n and combinatorics of the group B^+_{n+1}

Nicolau C. Saldanha and Boris Shapiro

Journal of Singularities
volume 4 (2012), 1-22

Received: 9 September 2009. Received in revised form: 14 July 2011.

DOI: 10.5427/jsing.2012.4a

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Abstract:

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. 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 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 S^n with a given initial and final Frenet frames. Using combinatorics of B^+_{n+1} = B_{n+1} \cap SO_{n+1}, where B_{n+1} \subset O_{n+1} is the usual Coxeter-Weyl group, we show that for any n\ge 2 these spaces fall in at most 1+ceiling(n/2) 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.

Keywords:

Locally convex curves, Weyl group, homotopy equivalence

Mathematical Subject Classification:

Primary 58B05, 53A04, Secondary 52A10, 55P15

Author(s) information:

N. C. Saldanha	B. Shapiro
Departamento de Matemática, PUC-Rio	Stockholm University
R. Marquês de S. Vicente 225	S-10691
Rio de Janeiro, RJ 22453-900, Brazil	Stockholm, Sweden
email: saldanha@puc-rio.br	email: shapiro@math.su.se
web: http://www.mat.puc-rio.br/~nicolau	web: http://www.math.su.se/~shapiro