2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/71988We 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.25 pages, 19 figuresGeometric TopologyAlgebraic Topology58B05; 58D10text