  We discuss the homotopy type and the cohomology of spaces of locally convex
parametrized curves gamma: [0,1] -> S^2, i.e., curves with positive geodesic
curvature. The space of all such curves with gamma(0) = gamma(1) = e_1 and
gamma'(0) = gamma'(1) = e_2 is known to have three connected components
X_{-1,c}, X_1, X_{-1}.
  We show several results concerning the homotopy type and cohomology of these
spaces. In particular, X_{-1,c} is contractible, X_1 and X_{-1} are simply
connected, pi_2(X_{-1}) contains a copy of Z and pi_2(X_1) contains a copy of
Z^2. Also, H^n(X_{1}, R) and H^n(X_{-1}, R) are nontrivial for all even n.
More, dim H^{4n-2}(X_1, R) >= 2 and dim H^{4n}(X_{-1}, R) >= 2 for all positive
n.
