Sectional Category of the Ganea Fibrations and Higher Relative Category
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.
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 , 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”  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  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 , 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 . 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 .
Recall the following construction.
Definition 1. For any map , the Ganea construction of is the following sequence of homotopy commutative diagrams ():
where the outside square is a homotopy pullback, the inside square is a homotopy pushout, and the map is the whisker map induced by this homotopy pushout. The iteration starts with .
In other words, the map is the join of and over ; namely, . 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 . For any , there is a whisker map induced by the homotopy pullback. Thus, is a homotopy section of . Moreover, we have .
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 .(2)The relative category of is the least integer such that the map has a homotopy section and .
We denote the sectional category by and the relative category by . If , and it is denoted simply by ; this is the “normalized” version of the Lusternik-Schnirelmann category.
Proposition 4. For any map , we have
Proof. By definition, is the least integer such that , that is, , has a homotopy section. Thus, if , will be such that and : that is, and , so .
3. Higher Relative Category
For any map and two integers , 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 . For coherence, let .
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 .
We denote this integer by . In order to avoid the prefix “rel” when , we write in this case.
Remark 6. Notice that and that, clearly, for any . Also notice that if and only if is a homotopy equivalence. In particular, for any .
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 and let be a homotopy section of . Consider the following homotopy commutative diagram:
where is the whisker map induced by the right homotopy pullback. We have and the left square is a homotopy pullback by the Prism lemma (see [7, Lemma ], for instance). The map is a homotopy section of and, moreover, . So .
Theorem 9. For any map , any , we have
Proof. The first two inequalities are our Remark 6; only the third needs a proof. Let and let be a homotopy section of such that . Consider the following homotopy commutative diagram:
The map is a homotopy section of and , so .
So increases at most by one when increases by one.
Corollary 10. For any map , any , we have
Example 12. If is a homotopy equivalence, then is a homotopy equivalence for all . So for all .
Example 13. Let and consider the map . We have because has a (unique) section. By Proposition 8, or . Indeed, for any , the map is homotopic to the null map, so , where . But we cannot have unless is a homotopy equivalence.
If we choose a space such that but (the 2-skeleton of the Poincaré homology 3 spheres, for instance), then and is a homotopy equivalence for all , so and for all . However, if we chose a simply connected CW-complex (in order that ), then for all .
Example 14. Consider any CW-complex with and the map . We have . Let us compute . Notice that . By Theorem 9, we know that . But we cannot have because is not a homotopy equivalence, so . By the way, we can say that factorizes up to homotopy through .
Example 15. More generally, if , we have for any by Corollary 10. Thus, while is not a homotopy equivalence (and if any exists such that is a homotopy equivalence, then for all ).
Suppose we are given any map with and any homotopy section of . For any , consider the following homotopy pullbacks:
where is the whisker map induced by the homotopy pullback . Notice that . By the Prism lemma, we know that the homotopy pullback of and is indeed and that . Also notice that since .
Proposition 16. For any map with and any homotopy section of , with the same definitions and notations as above, the following conditions are equivalent: (i).(ii) has a homotopy section.(iii) is a homotopy epimorphism.(iv).
Proof. We have the following sequence of implications: (i) ⇒ (ii): since , we have a whisker map induced by the homotopy pullback which is a homotopy section of . (ii) ⇒ (iii): it is obvious. (iii) ⇒ (iv): we have since . Thus, since is a homotopy epimorphism. (iv) ⇒ (i): we have .
Theorem 17. Let be a () connected map. If for some , has the homotopy type of a CW-complex with dimension strictly less than , then for all .
This is an immediate consequence of the following.
Proposition 18. Let be a () connected map with . If for some , has the homotopy type of a CW-complex with dimension strictly less than , then for any homotopy section of , so .
Proof. Recall that, for any , is the -fold join of . Thus, by [11, Theorem ], we obtain that is -connected. As and have the same homotopy fiber, which is -connected, we see that is -connected, too. By [12, Theorem IV.7.16], this means that, for every CW-complex with , induces a one-to-one correspondence . Apply this to and : since and are both homotopy sections of , we obtain , and Proposition 16 gives the desired result.
Example 19. Let be the Eilenberg-Mac Lane space . It is known that and that has the homotopy type of a wedge of circles (see [8, Example and Remark ], for instance). By Theorem 9, we know that . Because , we have for any homotopy section of and, thus, . Moreover, is never a homotopy equivalence, so for any ; thus, for .
The author declares that there is no conflict of interests regarding the publication of this paper.
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