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

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 prove that all connected components of $L_I$ are simply connected, that $H^2(L_{+1};Z)=Z^2$ and $H^2(L_{-1,n};Z)=Z$.

arXiv comment26 pages, 21 figures

References in corpus (1)

• The geometry of the critical set of nonlinear periodic Sturm-Liouville operators

Cited by in corpus (5)

• On the components of spaces of curves on the 2-sphere with geodesic curvature in a prescribed interval
• Homotopies of Curves on the 2-Sphere with Geodesic Curvature in a Prescribed Interval
• Combinatorialization of spaces of nondegenerate spherical curves
• Topology of the space of locally convex curves on the 3-sphere
• Results on the homotopy type of the spaces of locally convex curves on $S^3$