  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 study
the homotopy and cohomology of these spaces. In particular, for $L_{-1} =
L_{-1,c} \sqcup L_{-1,n}$, we show that $\dim H^{2k}(L_{(-1)^{k}}, \RR) \ge 1$,
that $\dim H^{2k}(L_{(-1)^{(k+1)}}, \RR) \ge 2$, that $\pi_2(L_{+1})$ contains
a copy of $Z^2$ and that $\pi_{2k}(L_{(-1)^{(k+1)}})$ contains a copy of $Z$.
