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.
Bibliographical noteJ.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