Hopf invariants for sectional category with applications to topological robotics

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 to the study of Farber’s topological complexity for two-cell complexes, as well as to the construction of a counterexample to the analogue for topological complexity of Ganea’s conjecture on Lusternik–Schnirelmann category.