Hopf Invariants for sectional category with applications to topological robotics

TL;DR

This paper develops a generalized Hopf invariant theory for sectional category, with applications to topological complexity in robotics, including new calculations and counterexamples to existing conjectures.

Contribution

It introduces a new framework for Hopf invariants in sectional category and applies it to topological complexity, providing novel insights and counterexamples.

Findings

Hopf invariants for product fibrations are shuffle joins of factors' invariants

Applied to Farber's topological complexity for 2-cell complexes

Constructed a counter-example to Ganea's conjecture analogue for topological complexity

Abstract

We develop a theory of generalized Hopf invariants in the setting of sectional category. In particular we show how Hopf invariants for a product of fibrations can be identified as shuffle joins of Hopf invariants for the factors. Our results are applied in the study of Farber's topological complexity for 2-cell complexes, as well as in the construction of a counter-example to the analogue for topological complexity of Ganea's conjecture on Lusternik-Schnirelmann category.