  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 $H^2(L_{-1,n};Z) = Z$.
