The homotopy and cohomology of spaces of locally convex curves in the sphere -- I

A smooth curve $\gamma :[0,1]\to S^2$ is locally convex if its geodesic curvature is positive at every point. J. A. Little showed that the space of all locally positive 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 $L_{-1,c}$ is known to be contractible but the topology of the other two connected components is not well understood. We study the homotopy and cohomology of these spaces. In particular, for $L_{-1} = L_{-1,c} \sqcup L_{-1,n}$,weshowthat$\dim H^{2k}(L_{(-1)^{k}}, \RR) \ge 1$, that $\dim H^{2k}(L_{(-1{)}^{(k+1)}},\text{RR})\ge 2$, that ${\pi }_2(L_{+1})$ contains a copy of $Z^2$ and that ${\pi }_{2k}(L_{(-1{)}^{(k+1)}})$ contains a copy of $Z$.