Up-to-one approximations of sectional category and topological complexity

Jamesʼ sectional category and Farberʼs topological complexity are studied in a general and unified framework.We introduce ‘relative’ and ‘strong relative’ forms of the category for a map and show that both can differ from sectional category by just one. A map has sectional or relative category at most n if, and only if, it is ‘dominated’ in a (different) sense by a map with strong relative category at most n. A homotopy pushout can increase sectional category but neither homotopy pushouts, nor homotopy pullbacks, can increase (strong) relative category. This makes (strong) relative category a convenient tool to study sectional category. We completely determine the sectional and relative categories of the fibres of the Ganea fibrations.In particular, the ‘topological complexity’ of a space is the sectional category of the diagonal map, and so it can differ from the (strong) relative category of the diagonal by just one. We call the strong relative category of the diagonal ‘strong complexity’. We show that the strong complexity of a suspension is at most two.