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 $\gamma'(0) = \gamma'(1) = e_2$ 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 prove that all connected components of $L_I$ are simply connected, that $H^2(L_{ +1};Z) = Z^2$ and $H^2(L_{ -1,n};Z) = Z$.
Categories: math.GT