Homotopy and cohomology of spaces of locally convex curves in the sphere

Details Homotopy and cohomology of spaces of locally convex curves in the sphere Homotopy and cohomology of spaces of locally convex curves in the sphere

2004-07-23

arxiv.org/abs/math/0407410v1

Mathematics

Geometric Topology

25 pages, 19 figures

Scientific paper

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.

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