Hopf Invariants for sectional category with applications to topological robotics

Jesús González, Mark Grant* (Corresponding Author), Lucile Vandembroucq

*Corresponding author for this work

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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 2-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.
Original languageEnglish
Pages (from-to)1209-1252
Number of pages43
JournalQuarterly Journal of Mathematics
Issue number4
Early online date15 Jul 2019
Publication statusPublished - Dec 2019

Bibliographical note

J.C. Partially supported by Conacyt Research Grant 221221.
L.V. Partially supported by Portuguese Funds through FCT (Funda¸c˜ao para a Ciˆencia e a Tecnologia) within the Project UID/MAT/00013/2013.


  • Sectional category
  • topological complexity
  • generalized Hopf invariants
  • two-cell complexes
  • Ganea conjecture
  • join and shuffle maps


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