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

机译：截面类别和拓扑复杂性的最大近似值

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James' sectional category and Farber's 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.
机译：在一个通用的统一框架中研究了James的截面类别和Farber的拓扑复杂性。我们为地图引入类别的“相对”和“强相对”形式，并表明这两者与断面类别的区别仅是一个。当且仅当“占主导地位”时，地图的断面或相对类别最多为n。相对于最多具有n个相对类别的贴图在（不同）意义上说。同位异体推出可以增加截面类别，但同位异体推出和同伦回撤都不能增加（强）相对类别。这使得（强）相对类别a研究断面类别的便捷工具，我们可以完全确定Ganea纤维的断面和相对类别，特别是空间的“拓扑复杂性”是对角线图的断面类别，因此它可以不同于对角线图的断面类别。对角线的（强）相对类别仅加一，我们称对角线的强相对类别为“强复杂度”，表明悬架的强复杂度最多为两个。

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