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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. 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Saldanha","submitted_at":"2009-05-13T14:05:00Z","abstract_excerpt":"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. 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