Hopf invariants for sectional category with applications to topological robotics

• Published in: Quarterly journal of mathematics Vol. 70; no. 4; pp. 1209 - 1252
• Main Authors: González, Jesús, Grant, Mark, Vandembroucq, Lucile
• Format: Journal Article
• Language: English
• Published: Oxford University Press 01-12-2019
• Subjects: Ciências Naturais; Ganea conjecture; Generalized Hopf invariants; Join and shuffle maps; Matemáticas; Science & Technology; Sectional category; Topological complexity
• ISSN: 0033-5606; 1464-3847
• DOI: 10.1093/qmath/haz019

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.
The first author was partially supported by Conacyt Research Grant 221221. The third author was partially supported by the Research Centre of Mathematics of the University of Minho with the Portuguese Funds from the “Fundação para a Ciência e a Tecnologia”, through the Project UID/MAT/0013/2013.