ON THE TOPOLOGY OF THE SPACES OF PLANE CURVES

Abstract. It is well known that any planar curves with the same endpoints are homotopic. An analogous claim for curves with the same endpoints and bounded curvature still remains open. In this note we find necessary and suffi-cient conditions for any planar curves with bounded curvature to be deformed one into another by a continuous one-parameter family of curves with bounded curvature answering a question stated in 1961 by L. Dubins. As a result of our analysis we prove the existence of a connected component whose elements cannot be deformed to curves having self-intersections. 1. Preliminaries It is well known that any planar curves with the same endpoints are homotopic. Surprisingly enough an analogous claim for curves with the same endpoints and bounded curvature still remains open. In this note we find necessary and sufficient conditions for any planar curves with bounded curvature to be deformed one into another by a continuous one-parameter family of curves with bounded curvature. As a result of our study we prove the existence of a connected component whos