Symmetric Motion Planning

TL;DR

This paper investigates symmetric motion planning algorithms, analyzing their topological complexity using cohomological methods, and introduces new concepts to establish lower bounds for their complexity.

Contribution

It introduces the sectional category weight of a cohomology class and applies it to derive bounds on symmetric topological complexity for aspherical manifolds.

Findings

Symmetric motion planning complexity exceeds twice the cup-length for aspherical manifolds.

Cohomological lower bounds are established using equivariant cohomology and Schwarz genus.

The new sectional category weight generalizes previous notions of category weight.

Abstract

In this paper we study symmetric motion planning algorithms, i.e. such that the motion from one state A to another B, prescribed by the algorithm, is the time reverse of the motion from B to A. We experiment with several different notions of topological complexity of such algorithms and compare them with each other and with the usual (non-symmetric) concept of topological complexity. Using equivariant cohomology and the theory of Schwarz genus we obtain cohomological lower bounds for symmetric topological complexity. One of our main results states that in the case of aspherical manifolds the complexity of symmetric motion planning algorithms with fixed midpoint map exceeds twice the cup-length. We introduce a new concept, the sectional category weight of a cohomology class, which generalises the notion of category weight developed earlier by E. Fadell and S. Husseini. We apply this notion to study the symmetric topological complexity of aspherical manifolds.