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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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.


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: email:
web: web: