Topological robotics: motion planning in projective spaces

We study an elementary problem of topological robotics: rotation of a line, which is fixed by a revolving joint at a base point: one wants to bring the line from its initial position to a final position by a continuous motion in the space. The final goal is to construct an algorithm which will perform this task once the initial and final positions are given. Any such motion planning algorithm will have instabilities, which are caused by topological reasons. A general approach to study instabilities of robot motion was suggested recently by the first named author. With any path-connected topological space X one associates a number TC(X), called the topological complexity of X. This number is of fundamental importance for the motion planning problem: TC(X) determines character of instabilities which have all motion planning algorithms in X. In the present paper we study the topological complexity of real projective spaces. In particular we compute TC(RP^n) for all n<24. Our main result is that (for n distinct from 1, 3, 7) the problem of calculating of TC(RP^n) is equivalent to finding the smallest k such that RP^n can be immersed into the Euclidean space R^{k-1}.Comment: 16 page

Topological complexity of motion planning in projective product spaces

We study Farber's topological complexity (TC) of Davis' projective product spaces (PPS's). We show that, in many non-trivial instances, the TC of PPS's coming from at least two sphere factors is (much) lower than the dimension of the manifold. This is in high contrast with the known situation for (usual) real projective spaces for which, in fact, the Euclidean immersion dimension and TC are two facets of the same problem. Low TC-values have been observed for infinite families of non-simply connected spaces only for H-spaces, for finite complexes whose fundamental group has cohomological dimension not exceeding 2, and now in this work for infinite families of PPS's. We discuss general bounds for the TC (and the Lusternik-Schnirelmann category) of PPS's, and compute these invariants for specific families of such manifolds. Some of our methods involve the use of an equivariant version of TC. We also give a characterization of the Euclidean immersion dimension of PPS's through generalized concepts of axial maps and, alternatively, non-singular maps. This gives an explicit explanation of the known relationship between the generalized vector field problem and the Euclidean immersion problem for PPS's.Comment: 16 page