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 and , where d(B)=B×B is a fibrewise pointed space over B whose projection and section are given by the canonical projection to the second factor and the diagonal. In addition, our method in studying fibrewise L–S category is able to treat a fibrewise space with singular fibres.