  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.
