Oriented robot motion planning in Riemannian manifolds

Abstract

We consider the problem of robot motion planning in an oriented Riemannian manifold as a topological motion planning problem in its oriented frame bundle. For this purpose, we study the topological complexity of oriented frame bundles, derive an upper bound for this invariant and certain lower bounds from cup length computations. In particular, we show that for large classes of oriented manifolds, e.g. for spin manifolds, the topological complexity of the oriented frame bundle is bounded from below by the dimension of the base manifold.Date: February 27, 2019.In the following we consider the spaces S(M) and S n as spaces with free Z 2 -actions given by the deck transformation actions of q M and q n , resp. Definition 5.7. a) Given a spin manifold M we let i(M) denote the biggest integer k ∈ N for which there exists a continuous Z 2 -equivariant map f : S k → S(M) with respect to the antipodal involution on S k .