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

Abstract

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 ${\gamma }^{\prime }(0)={\gamma }^{\prime }(1)=e_2$ 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 $L_{-1}=L_{-1,c}\bigsqcup L_{-1,n}$, 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 $Z$.

arXiv comment18 pages, 12 figures

Cited by in corpus (1)

• On the components of spaces of curves on the 2-sphere with geodesic curvature in a prescribed interval