Homotopy and cohomology of spaces of locally convex curves in the sphere
Abstract
We discuss the homotopy type and the cohomology of spaces of locally convex parametrized curves gamma: [0,1] -> S2, i.e., curves with positive geodesic curvature. The space of all such curves with gamma(0) = gamma(1) = e1 and gamma'(0) = gamma'(1) = e2 is known to have three connected components X-1,c, X1, X-1. We show several results concerning the homotopy type and cohomology of these spaces. In particular, X-1,c is contractible, X1 and X-1 are simply connected, pi2(X-1) contains a copy of Z and pi2(X1) contains a copy of Z2. Also, Hn(X1, R) and Hn(X-1, R) are nontrivial for all even n. More, dim H4n-2(X1, R) >= 2 and dim H4n(X-1, R) >= 2 for all positive n.
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