Homotopy type of spaces of curves with constrained curvature on flat surfaces

TL;DR

This paper classifies the topological structure of spaces of curves on flat surfaces with bounded curvature, showing they are either contractible or homotopy equivalent to spheres, with explicit constructions.

Contribution

It determines the homotopy types of spaces of constrained-curvature curves on flat surfaces, providing explicit homotopy equivalences and a complete classification.

Findings

Connected components are either contractible or homotopy equivalent to spheres.

Every sphere dimension n≥1 is realizable as a component.

Explicit homotopy equivalences are constructed.

Abstract

Let $S$ be a complete flat surface, such as the Euclidean plane. We determine the homeomorphism class of the space of all curves on $S$ which start and end at given points in given directions and whose curvatures are constrained to lie in a given open interval, in terms of all parameters involved. Any connected component of such a space is either contractible or homotopy equivalent to an $n$-sphere, and every $n\geq 1$ is realizable. Explicit homotopy equivalences between the components and the corresponding spheres are constructed.