arXiv subjects

Jean-Paul Doeraene

Publications and source records attributed to Jean-Paul Doeraene.

5 recordsLinked to original sources

arXiv2026

Axiomatic Sectional category, Topological complexity, and Homotopic distance

Many of the properties of sectional category, topological complexity and homotopic distance are in fact derived from a small number of basic properties, which, once established, lead to all the others without further recourse to topology. On the other hand, there are several variants of these notions: with open covers or else Whitehead-Ganea constructions, also with spaces and maps that are unpointed or else pointed, fibrewise, equivariant, etc. or even with algebraic models of spaces and maps. These are two reasons why we build an axiomatic approach to all these notions, based on just three simple axioms. We also introduce the notion of `lifting category' which unifies the notions of sectional category, topological complexity, and homotopic distance, all of which are special cases of lifting category.

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arXiv2016

About sectional category of the Ganea maps

We first compute the James' sectional category (secat) of the Ganea map g_k of any map f in terms of the sectional category of f: We show that secat(g_k) is the integer part of secat(f)/(k+1). Next we compute the relative category (relcat) of g_k. In order to do this, we introduce the relative category of order k (relcat_k) of a map and show that relcat(g_k) is the integer part of relcat_k(f)/(k+1). Then we establish some inequalities linking secat and relcat of any order: We show that secat(f) <= relcat_k(f) <= secat(f) + k + 1 and relcat_k(f) <= relcat_(k+1)(f) <= relcat_k(f) + 1. We give examples that show that these inequalities may be strict.

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arXiv2015

Yet another Hopf Invariant

The classical Hopf invariant is defined for a map f: S^r -> X. Here we define `hcat' which is some kind of Hopf invariant built with a construction in Ganea's style, valid for maps not only on spheres but more generally on a `relative suspension' f: Sigma_A W -> X. We study the relation between this invariant and the sectional category and the relative category of a map. In particular, for f being the `restriction' of f on A, we have relcat(i) <= hcat(f) <= relcat(i) + 1 and relcat(f) <= hcat(f).

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arXiv2013

Up to one approximations of sectional category and topological complexity

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. We show that both can differ from sectional category just by 1. A map has sectional or relative category less than or equal to $n$ if, and only if, it is `dominated' (in some sense) by a map with strong relative category less than or equal to $n$. 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. As a particular case, the `topological complexity' of a space is the sectional category of the diagonal map. So it can differ from the (strong) relative category of the diagonal just by 1. We call the strong relative category of the diagonal `strong complexity'. We show that the strong complexity of a suspension is at most 2.

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arXiv2010

Ganea and Whitehead definitions for the tangential Lusternik-Schnirelmann category of foliations

This work solves the problem of elaborating Ganea and Whitehead definitions for the tangential category of a foliated manifold. We develop these two notions in the category $\Tops$ of stratified spaces, that are topological spaces $X$ endowed with a partition $\cF$ and compare them to a third invariant defined by using open sets. More precisely, these definitions apply to an element $(X,\cF)$ of $\Tops$ together with a class $\cA$ of subsets of $X$; they are similar to invariants introduced by M. Clapp and D. Puppe. If $(X,\cF)\in\Tops$, we define a transverse subset as a subspace $A$ of $X$ such that the intersection $S\cap A$ is at most countable for any $S\in \cF$. Then we define the Whitehead and Ganea LS-categories of the stratified space by taking the infimum along the transverse subsets. When we have a closed manifold, endowed with a $C^1$-foliation, the three previous definitions, with $\cA$ the class of transverse subsets, coincide with the tangential category and are homotopical invariants.

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