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

TL;DR

This paper investigates the topological properties, including homotopy and cohomology groups, of spaces of locally convex curves on the sphere, revealing complex structures in their connected components.

Contribution

It provides new insights into the homotopy and cohomology of specific components of locally convex curves on the sphere, extending previous knowledge about their topology.

Findings

The cohomology groups have dimensions at least 1 or 2 in certain degrees.

The second homotopy group of L_{+1} contains Z^2.

Higher homotopy groups contain copies of Z.

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 $\gamma'(0) = \gamma'(1) = e_2$ 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 $Z$.