Sectional Category of the Ganea Fibrations and Higher Relative Category

We first compute James’ sectional category (secat) of the Ganea map gk of any map ιX in terms of the sectional category of ιX: we show that secatg k is the integer part of secat ι X /(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 (relcatk) of a map and show that relcatgk is the integer part of relcatkιX/(k + 1). Then we establish some inequalities linking secat and relcat of any order: we show that secat ιX ⩽ relcatkιX ⩽ secat ιX + k + 1 and relcatkιX ⩽ relcatk+1ιX ⩽ relcatkιX + 1. We give examples that show that these inequalities may be strict.


Introduction
The "Lusternik-Schnirelmann category" cat of a topological space  is the least integer  ⩾ 0 such that  can be covered by  + 1 open subsets   (0 ⩽  ⩽ ) such that each inclusion   →  is nullhomotopic; that is, the based path-space fibration  →  has a partial section on   .More generally, the "sectional category" secat  of a fibration  :  → , originally defined by Schwarz [1], is the least integer  ⩾ 0 such that  can be covered by  + 1 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 secat   = secat .So cat = secat( * → ).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 Δ :  →  ×  or equivalently of the (unbased) fibration  :   →  ×  :   → ((0),  (1)).
For a given space , Ganea [3] defined a sequence of fibrations   :   →  for  ⩾ 0, starting with  0 :  → .The fundamental property of the sequence is that it gives another criterion for detecting the category: cat 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  0 (  ) =   , and secat(  ) 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 cat/( + 1).More generally, the sectional category of the Ganea map   (  ) associated with any map   is the integer part of secat   /( + 1).
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 cat = 7, the successive values of secat(  ()) for 0 ⩽  ⩽ 7 are In [4], we used the same Ganea-type construction to define the "relative category" of a map (relcat for short).As a particular case, the relative category of the diagonal map Δ :  → × is the "monoidal topological complexity" of  defined in [5].It turns out that the relative category can differ 2 Chinese Journal of Mathematics from the sectional category by at most one.More precisely, we have secat   ⩽ relcat   ⩽ secat   + 1. ( 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 " (relcat  ) and show that the relative category of th Ganea map   (  ) associated with a map   is the integer part of relcat    /( + 1).When   : * → , we write relcat    = cat  .
Warning.Despite cat  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   (  ), relcat    = secat   for all  ⩽ .

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 ( ⩾ 0): where the outside square is a homotopy pullback, the inside square is a homotopy pushout, and the map  +1 = (  ,   ) :  +1 →  is the whisker map induced by this homotopy pushout.The iteration starts with  0 =   :  → .
In other words, the map  +1 is the join of   and   over ; namely,  +1 ≃   ⋈    .When we need to be precise, we denote   by   (  ) and   by   (  ).If  ≃ * , we also write   () and   (), respectively.
Notice that, as the outside square is a homotopy pullback,   and   have a common homotopy fiber, so their connectivity is equal.
For coherence, let  0 = id  .For any  ⩾ 0, there is a whisker map   = (id  ,   ) :  →   induced by the homotopy pullback.Thus,   is a homotopy section of   .Moreover, we have   ∘   ≃  +1 .Proposition 2. For any map   :  → , we have ( Proof.This is just an application of the "associativity of the join" (see [7,Theorem 4.8], for instance): Definition 3. Let   :  →  be any map.
(1) The sectional category of   is the least integer  such that the map   :   (  ) →  has a homotopy section: that is, there exists a map  :  →   (  ) such that   ∘  ≃ id  .(2) The relative category of   is the least integer  such that the map   :   (  ) →  has a homotopy section  and  ∘   ≃   .
We denote the sectional category by secat(  ) and the relative category by relcat(  ).If  ≃ * , secat(  ) = relcat(  ) and it is denoted simply by cat(); this is the "normalized" version of the Lusternik-Schnirelmann category.

Higher Relative Category
For any map   :  →  and two integers 0 ⩽  < , consider the following homotopy commutative diagram: where the outside square is a homotopy pullback and the inside square is a homotopy pushout.
Because of the associativity of the join, we also have We denote this integer by relcat    .In order to avoid the prefix "rel" when  ≃ * , we write cat   = relcat    in this case.
Following the same reasoning as in Proposition 4, we have the following.Proposition 7.For any map   :  → , we have Proposition 8.For any map   :  → , any , we have Proof.Only the second inequality needs a proof.Let  = secat   and let  be a homotopy section of   .Consider the following homotopy commutative diagram: where   = ( ∘   , id   ) is the whisker map induced by the right homotopy pullback.We have The map  + =   ∘  is a homotopy section of  +1 and  + ∘  +1 ≃  +1 +1 , so relcat +1   ⩽  + 1. Example 13.Let  ̸ ≃ * and consider the map  * :  → * .We have secat  * = 0 because  * has a (unique) section.By Proposition 8, relcat   * =  or 1+.Indeed, for any , the map

Let us compute cat
For coherence, let    = id   .
Definition 5. Let   :  →  be any map.The relative category of order  of   is the least integer  ⩾  such that the map   :   (  ) →  has a homotopy section  and  ∘   ≃    .