The homotopy type of spaces of locally convex curves in the sphere

摘要

A smooth curve gamma [0; 1] -> S-2 is locally convex if its geodesic curvature is positive at every point. JA Little showed that the space of all locally convex curves gamma with gamma(0) = gamma(1) = e(1) and gamma'(0) = gamma'(1) = e(2) has three connected components L--1,L-c, L+1, L--1,L-n. The space L--1,L-c is known to be contractible. We prove that L+1 and L--1,L-n are homotopy equivalent to (Omega S-3) boolean OR S-2 boolean OR S-6 boolean OR S-10 boolean OR ... and (Omega S-3) boolean OR S-4 boolean OR S-8 boolean OR S-12 boolean OR ..., respectively. As a corollary, we deduce the homotopy type of the components of the space Free (S-1; S-2) of free curves gamma : S-1 -> S-2 (ie curves with nonzero geodesic curvature). We also determine the homotopy type of the spaces Free [0; 1], S-2/with fixed initial and final frames.