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

TL;DR

This paper determines the homotopy types of spaces of locally convex curves on the sphere, revealing their complex topological structures and relating them to loop spaces and spheres.

Contribution

It proves the homotopy equivalences of specific components of locally convex curve spaces, extending understanding of their topological structure and homotopy types.

Findings

Homotopy type of _{+1} is a wedge of spheres and loop spaces.

Homotopy type of _{-1,n} is a different wedge of spheres and loop spaces.

Identifies homotopy types of spaces of free curves with fixed endpoints and frames.

Abstract

A smooth curve $\gamma: [0,1] \to \Ss^2$ is locally convex if its geodesic curvature is positive at every point. J. A. 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 $L_{-1,c}$, $L_{+1}$, $L_{-1,n}$. The space $\cL_{-1,c}$ is known to be contractible. We prove that $\cL_{+1}$ and $\cL_{-1,n}$ are homotopy equivalent to $(\Omega\Ss^3) \vee \Ss^2 \vee \Ss^6 \vee \Ss^{10} \vee \cdots$ and $(\Omega\Ss^3) \vee \Ss^4 \vee \Ss^8 \vee \Ss^{12} \vee \cdots$, respectively. As a corollary, we deduce the homotopy type of the components of the space $\Free(\Ss^1,\Ss^2)$ of free curves $\gamma: \Ss^1 \to \Ss^2$ (i.e., curves with nonzero geodesic curvature). We also determine the homotopy type of the spaces $\Free([0,1], \Ss^2)$ with fixed initial and final frames.