On the Homology of the Space of Curves Immersed in The Sphere with Curvature Constrained to a Prescribed Interval

TL;DR

This paper investigates the homotopy types of spaces of sphere curves with prescribed curvature constraints, revealing they differ from the unconstrained space and identifying exotic generators influenced by endpoints and directions.

Contribution

It demonstrates that curvature-constrained subspaces are not homotopy equivalent to the full space and constructs explicit exotic generators for their homotopy and cohomology groups.

Findings

Curvature constraints alter the homotopy type of the space of immersed curves.

Explicit exotic generators for homotopy and cohomology groups are constructed.

Dimensions of generators depend on endpoints and end directions.

Abstract

While the topology of the space of all smooth immersed curves on the $2$-sphere $\mathbb{S}^2$ that start and end at given points in given directions is well known, it is an open problem to understand the homotopy type of its subspaces consisting of the curves whose geodesic curvatures are constrained to a prescribed proper open interval. In this article we prove that, under certain circumstances for endpoints and end directions, these subspaces are not homotopically equivalent to the whole space. Moreover, we give an explicit construction of exotic generators for some homotopy and cohomology groups. It turns out that the dimensions of these generators depend on endpoints and end directions. A version of the h-principle is used to prove these results.