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

A smooth curve $\gamma :[0,1]\to {\text{Ss}}^2$ 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 ${\gamma }^{\mathrm{\prime }}(0)={\gamma }^{\mathrm{\prime }}(1)=e_2$ has three connected components $L_{-1,c}$, $L_{+1}$, $L_{-1,n}$. The space ${\text{cL}}_{-1,c}$ is known to be contractible. We prove that ${\text{cL}}_{+1}$ and ${\text{cL}}_{-1,n}$ are homotopy equivalent to $(\mathrm{\Omega }{\text{Ss}}^3)\vee {\text{Ss}}^2\vee {\text{Ss}}^6\vee {\text{Ss}}^{10}\vee \cdots$ and $(\mathrm{\Omega }{\text{Ss}}^3)\vee {\text{Ss}}^4\vee {\text{Ss}}^8\vee {\text{Ss}}^{12}\vee \cdots$, respectively. As a corollary, we deduce the homotopy type of the components of the space $\text{Free}({\text{Ss}}^1,{\text{Ss}}^2)$ of free curves $\gamma :{\text{Ss}}^1\to {\text{Ss}}^2$ (i.e., curves with nonzero geodesic curvature). We also determine the homotopy type of the spaces $\text{Free}([0,1],{\text{Ss}}^2)$ with fixed initial and final frames.