Chinese Journal of Mathematics

Volume 2016, Article ID 8320742, 5 pages

http://dx.doi.org/10.1155/2016/8320742

## Sectional Category of the Ganea Fibrations and Higher Relative Category

Département de Mathématiques, Université de Lille 1, 59655 Villeneuve d’Ascq Cedex, France

Received 7 April 2016; Accepted 29 June 2016

Academic Editor: Dan Burghelea

Copyright © 2016 Jean-Paul Doeraene. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We first compute James’ sectional category (secat) of the Ganea map of any map in terms of the sectional category of : we show that is the integer part of . Next we compute the relative category (relcat) of . In order to do this, we introduce the relative category of order () of a map and show that is the integer part of . Then we establish some inequalities linking secat and relcat of any order: we show that and . We give examples that show that these inequalities may be strict.

#### 1. Introduction

The “Lusternik-Schnirelmann category” of a topological space is the least integer such that can be covered by open subsets () such that each inclusion is nullhomotopic; that is, the based path-space fibration has a partial section on . More generally, the “sectional category” of a fibration , originally defined by Schwarz [1], is the least integer such that can be covered by open subsets with a partial section of on each of these sets. This notion extends to any continuous map by taking the standard homotopy replacement of by a fibration and setting . So . Sectional category earned its renown recently as Farber’s notion of “topological complexity” [2] of a space , which measures the difficulty of solving the motion planning problem: the topological complexity of is the sectional category of the diagonal or equivalently of the (unbased) fibration .

For a given space , Ganea [3] defined a sequence of fibrations for , starting with . The fundamental property of the sequence is that it gives another criterion for detecting the category: is the least such that has a section (at least for a sufficiently nice space: normal, well pointed). This construction can be generalized for any map ; that is, there is a sequence of maps , starting with , and is the least such that has a homotopy section; see Definition 3. We recover the Ganea construction when ; in this case we write instead of .

In this paper, we first show that the sectional category of th Ganea map of is the integer part of . More generally, the sectional category of the Ganea map associated with any map is the integer part of .

As we may “think of” the sectional category as the degree of obstruction for a map to have a homotopy section, this shows us how this degree of obstruction decreases when we consider the successive Ganea maps. For instance, for a space with , the successive values of for are

In [4], we used the same Ganea-type construction to define the “relative category” of a map ( for short). As a particular case, the relative category of the diagonal map is the “monoidal topological complexity” of defined in [5]. It turns out that the relative category can differ from the sectional category by at most one. More precisely, we have This establishes a dichotomy between maps: those for which the sectional category equals the relative category and those for which they differ by 1.

In this paper we introduce the “relative category of order ” () and show that the relative category of th Ganea map associated with a map is the integer part of . When , we write .

*Warning*. Despite is sometimes used in the literature for Fox’s -dimensional category, this is* not* the meaning of this notation in this paper.

We link all these invariants together by several inequalities:

Finally, we show that, with some hypothesis on the connectivity of and the homotopical dimension of the source of , for all .

For a given space (resp.: map ), the set of integers for which the equality (resp., ) holds is an interesting datum about this space (resp., map). The maximum number of such integers is (resp., ). For instance, for , there is just one such , which is 0: namely,

#### 2. Sectional Category of the Ganea Maps

We use the symbol both to mean that maps are homotopic and to mean that spaces are of the same homotopy type. We denote the integer part of a rational number by .

We build all our spaces and maps with “homotopy commutative diagrams,” especially “homotopy pullbacks” and “homotopy pushouts,” in the spirit of [6].

Recall the following construction.

*Definition 1. *For any map , the* Ganea construction* of is the following sequence of homotopy commutative diagrams ():