The homotopy and cohomology of spaces of locally convex curves in the sphere -- I

Details The homotopy and cohomology of spaces of locally convex curves in the sphere -- I The homotopy and cohomology of spaces of locally convex curves in the sphere -- I

2009-05-13

arxiv.org/abs/0905.2111v2

Mathematics

Geometric Topology

18 pages, 12 figures

Scientific paper

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. In particular, for $L_{-1} = L_{-1,c} \sqcup L_{-1,n}$, we show that $\dim H^{2k}(L_{(-1)^{k}}, \RR) \ge 1$, that $\dim H^{2k}(L_{(-1)^{(k+1)}}, \RR) \ge 2$, that $\pi_2(L_{+1})$ contains a copy of $Z^2$ and that $\pi_{2k}(L_{(-1)^{(k+1)}})$ contains a copy of $Z$.

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