Topological complexity of motion planning in projective product spaces

TL;DR

This paper investigates the topological complexity of projective product spaces, revealing that in many cases it is significantly lower than the manifold's dimension, contrasting with real projective spaces and linking to immersion problems.

Contribution

It provides new bounds and explicit calculations of topological complexity for PPS's, introduces equivariant methods, and characterizes Euclidean immersion dimensions via generalized maps.

Findings

TC of PPS's from multiple sphere factors is often lower than the manifold's dimension.

Established bounds for TC and Lusternik-Schnirelmann category of PPS's.

Connected TC and immersion dimension through generalized axial and non-singular maps.

Abstract

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.