Topological complexity is a fibrewise L-S category

Topological complexity TC(B) of a space B is introduced by M. Farber to measure how much complex the space is, which is first considered on a configuration space of a motion planning of a robot arm. We also consider a stronger version TCM(B) of topological complexity with an additional condition: in a robot motion planning, a motion must be stasis if the initial and the terminal states are the same. Our main goal is to show the equalities TC(B) = cat(B)*(d(B)) + 1 and TCM(B) = cat(B)(B)(d(B)) + 1, where d(B) = B x B is a fibrewise pointed space over B whose projection and section are given by p(d(B)) = pr(2) : B x B --> B the canonical projection to the second factor and S-d(B) = Delta(B) : B --> B x B the diagonal. In addition, our method in studying fibrewise L-S category is able to treat a fibrewise space with singular fibres. (C) 2009 Elsevier B.V. All rights reserved.