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

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

Spaces of Locally Convex Curves in S and Combinatorics of the Group B

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 S2 ([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 S2 which are locally convex (i.e., without inflection points). One of the main difficulties is the absence of the co...

The homotopy and cohomology of spaces of locally convex curves in the sphere — I Nicolau C . Saldanha

A smooth curve γ : [0, 1] → S2 is locally convex if its geodesic curvature is positive at every point. J. A. Little showed that the space of all locally positive curves γ with γ(0) = γ(1) = e1 and γ ′(0) = γ′(1) = e2 has three connected components L−1,c, L+1, L−1,n. The space L−1,c is known to be contractible but the topology of the other two connected components is not well understood. We stud...

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

A smooth curve γ : [0, 1] → S2 is locally convex if its geodesic curvature is positive at every point. J. A. Little showed that the space of all locally convex curves γ with γ(0) = γ(1) = e1 and γ (0) = γ(1) = e2 has three connected components L−1,c, L+1, L−1,n. The space L−1,c is known to be contractible. In this paper we prove that L+1 and L−1,n are homotopy equivalent to ΩS3 ∨ S2 ∨ S6 ∨ S10 ...

Homotopy and cohomology of spaces of locally convex curves in the sphere

We discuss the homotopy type and the cohomology of spaces of locally convex parametrized curves γ : [0, 1] → S2, i.e., curves with positive geodesic curvature. The space of all such curves with γ(0) = γ(1) = e1 and γ′(0) = γ′(1) = e2 is known to have three connected components X−1,c, X1, X−1. We show several results concerning the homotopy type and cohomology of these spaces. In particular, X−1...