# Sectional Category of the Ganea Fibrations and Higher Relative Category

Authors: Jean-Paul Doeraene
Journal: Chinese Journal of Mathematics
 (2016)

---

## Abstract

We first compute James’ sectional category (secat) of the Ganea mapg
k of any map ι
X in terms of the sectional category of ι
X: we show that s
e
c
a
t
g
k is the integer part of s
e
c
a
t
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 (r
e
l
c
a
t
k) of a map and show that relcat
g
k is the integer part of relcat
k
ι
X
/
(
k
+
1
). Then we establish some inequalities linking secat and relcat of any order: we show that s
e
c
a
t
ι
X
⩽
relcat
k
ι
X
⩽
s
e
c
a
t
ι
X
+
k
+
1 and relcat
k
ι
X
⩽
relcat
k
+
1
ι
X
⩽
relcat
k
ι
X
+
1. We give examples that show that these inequalities may be strict.


---

## Body

## 1. Introduction

The “Lusternik-Schnirelmann category”cat
X of a topological space X is the least integer n
⩾
0 such that X can be covered by n
+
1 open subsets U
i (0
⩽
i
⩽
n) such that each inclusion U
i
↪
X is nullhomotopic; that is, the based path-space fibration P
X
→
X has a partial section on U
i. More generally, the “sectional category” s
e
c
a
t
p of a fibration p
:
E
→
X, originally defined by Schwarz [1], is the least integer n
⩾
0 such that X can be covered by n
+
1 open subsets with a partial section of p on each of these sets. This notion extends to any continuous map ι
X
:
A
→
X by taking the standard homotopy replacement of ι
X by a fibration p
:
E
→
X and setting secat
⁡
ι
X
=
secat
⁡
p. So cat
X
=
secat
⁡
(
∗
→
X
). Sectional category earned its renown recently as Farber’s notion of “topological complexity” [2] of a space A, which measures the difficulty of solving the motion planning problem: the topological complexity of A is the sectional category of the diagonal Δ
:
A
→
A
×
A or equivalently of the (unbased) fibration π
:
A
I
→
A
×
A
:
α
↦
(
α
(
0
)
,
α
(
1
)
).For a given spaceX, Ganea [3] defined a sequence of fibrations p
k
:
E
k
→
X for k
⩾
0, starting with p
0
:
P
X
→
X. The fundamental property of the sequence is that it gives another criterion for detecting the category: cat
X is the least n such that p
n has a section (at least for a sufficiently nice space: normal, well pointed). This construction can be generalized for any map ι
X
:
A
→
X; that is, there is a sequence of maps g
k
(
ι
X
)
:
G
k
(
ι
X
)
→
X, starting with g
0
(
ι
X
)
=
ι
X, and secat
⁡
(
ι
X
) is the least n such that g
n
(
ι
X
) has a homotopy section; see Definition 3. We recover the Ganea construction when A
=
∗; in this case we write g
k
(
X
) instead of g
k
(
ι
X
).In this paper, we first show that the sectional category ofkth Ganea map g
k
(
X
) of X is the integer part of cat
X
/
(
k
+
1
). More generally, the sectional category of the Ganea map g
k
(
ι
X
) associated with any map ι
X is the integer part of secat
⁡
ι
X
/
(
k
+
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 spaceX with cat
X
=
7, the successive values of secat
⁡
g
k
X for 0
⩽
k
⩽
7 are (1)
7
3
2
1
1
1
1
0
.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
:
X
→
X
×
X is the “monoidal topological complexity” of X defined in [5]. It turns out that the relative category can differ from the sectional category by at most one. More precisely, we have (2)
secat
⁡
ι
X
⩽
relcat
ι
X
⩽
secat
⁡
ι
X
+
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 orderk” (relcat
k) and show that the relative category of kth Ganea map g
k
(
ι
X
) associated with a map ι
X is the integer part of relcat
k
ι
X
/
(
k
+
1
). When ι
X
:
∗
→
X, we write relcat
k
ι
X
=
cat
k
X.Warning. Despite cat
k is sometimes used in the literature for Fox’s k-dimensional category, this isnot the meaning of this notation in this paper.We link all these invariants together by several inequalities:(3)
secat
⁡
ι
X
⩽
relcat
k
ι
X
⩽
secat
⁡
ι
X
+
k
+
1
,
relcat
k
ι
X
⩽
relcat
k
+
1
ι
X
⩽
relcat
k
ι
X
+
1
.Finally, we show that, with some hypothesis on the connectivity ofι
X and the homotopical dimension of the source of g
k
(
ι
X
), relcat
j
ι
X
=
secat
⁡
ι
X for all j
⩽
k.For a given spaceX (resp.: map ι
X), the set of integers k for which the equality cat
k
+
1
X
=
cat
k
X (resp., relcat
k
+
1
ι
X
=
relcat
k
ι
X) holds is an interesting datum about this space (resp., map). The maximum number of such integers is cat
X (resp., relcat
ι
X). For instance, for X
=
K
(
Q
,
1
), there is just one such k, which is 0: namely, (4)
cat
0
X
=
cat
1
X
=
2
,
cat
k
X
=
k
+
1
for
k
>
1
.


## 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 q by q.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 mapι
X
:
A
→
X, theGanea construction of ι
X is the following sequence of homotopy commutative diagrams (i
⩾
0):
where the outside square is a homotopy pullback, the inside square is a homotopy pushout, and the mapg
i
+
1
=
(
g
i
,
ι
X
)
:
G
i
+
1
→
X is the whisker map induced by this homotopy pushout. The iteration starts with g
0
=
ι
X
:
A
→
X.In other words, the mapg
i
+
1 is the join of g
i and ι
X over X; namely, g
i
+
1
≃
g
i
⋈
X
ι
X. When we need to be precise, we denote G
i by G
i
(
ι
X
) and g
i by g
i
(
ι
X
). If A
≃
∗, we also write G
i
(
X
) and g
i
(
X
), respectively.Notice that, as the outside square is a homotopy pullback,g
i and η
i have a common homotopy fiber, so their connectivity is equal.For coherence, letα
0
=
id
A. For any i
⩾
0, there is a whisker map θ
i
=
(
id
A
,
α
i
)
:
A
→
F
i induced by the homotopy pullback. Thus, θ
i is a homotopy section of η
i. Moreover, we have γ
i
∘
α
i
≃
α
i
+
1.Proposition 2.
For any mapι
X
:
A
→
X, we have (5)
g
j
g
i
ι
X
≃
g
i
j
+
i
+
j
ι
X
.Proof.
This is just an application of the “associativity of the join” (see [7, Theorem  4.8], for instance):(6)
g
j
g
i
ι
X
≃
g
i
ι
X
⋈
X
⋯
⋈
X
g
i
ι
X
j
+
1
times
≃
ι
X
⋈
X
⋯
⋈
X
ι
X
⋯
ι
X
⋈
X
⋯
⋈
X
ι
X
≃
ι
X
⋈
X
⋯
⋈
X
ι
X
j
+
1
i
+
1
times
≃
g
j
+
1
i
+
1
-
1
ι
X
.Definition 3.
Letι
X
:
A
→
X be any map.(1)
Thesectional category of ι
X is the least integer n such that the map g
n
:
G
n
(
ι
X
)
→
X has a homotopy section: that is, there exists a map σ
:
X
→
G
n
(
ι
X
) such that g
n
∘
σ
≃
id
X.
(2)
Therelative category of ι
X is the least integer n such that the map g
n
:
G
n
(
ι
X
)
→
X has a homotopy section σ and σ
∘
ι
X
≃
α
n.We denote the sectional category bysecat
⁡
(
ι
X
) and the relative category by relcat
(
ι
X
). If A
≃
∗, secat
⁡
(
ι
X
)
=
relcat
(
ι
X
) and it is denoted simply by cat
(
X
); this is the “normalized” version of the Lusternik-Schnirelmann category.A lot about the integers cat and secat is collected in [8]. The integer relcat is introduced in [4] and further studied in [9, 10].Proposition 4.
For any mapι
X
:
A
→
X, we have (7)
secat
⁡
g
k
ι
X
=
secat
⁡
ι
X
k
+
1
.Proof.
By definition,secat
⁡
g
k
(
ι
X
) is the least integer n such that g
n
(
g
k
(
ι
X
)
), that is, g
k
n
+
k
+
n
(
ι
X
), has a homotopy section. Thus, if secat
⁡
ι
X
=
m, n will be such that k
n
+
k
+
n
⩾
m and k
(
n
-
1
)
+
k
+
(
n
-
1
)
<
m: that is, n
⩾
m
/
(
k
+
1
)
-
k
/
(
k
+
1
) and n
<
m
/
(
k
+
1
)
+
1
/
(
k
+
1
), so n
=
m
/
(
k
+
1
).


## 3. Higher Relative Category

For any mapι
X
:
A
→
X and two integers 0
⩽
k
<
r, 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 haveγ
k
r
≃
γ
r
-
1
∘
γ
r
-
2
∘
⋯
∘
γ
k
+
1
∘
γ
k. For coherence, let γ
k
k
=
id
G
k.Definition 5.
Letι
X
:
A
→
X be any map. Therelative category of order
k of ι
X is the least integer n
⩾
k such that the map g
n
:
G
n
(
ι
X
)
→
X has a homotopy section σ and σ
∘
g
k
≃
γ
k
n.We denote this integer byrelcat
k
ι
X. In order to avoid the prefix “rel” when A
≃
∗, we write cat
k
X
=
relcat
k
ι
X in this case.Remark 6.
Notice thatrelcat
0
ι
X
=
relcat
ι
X and that, clearly, k
⩽
relcat
k
ι
X
⩽
relcat
k
+
1
ι
X for any k. Also notice that relcat
k
ι
X
=
k if and only if g
k
(
ι
X
) is a homotopy equivalence. In particular, cat
k
∗
=
k for any k.Following the same reasoning as in Proposition4, we have the following.Proposition 7.
For any mapι
X
:
A
→
X, we have (8)
relcat
⁡
g
k
ι
X
=
relcat
k
⁡
ι
X
k
+
1
.Proposition 8.
For any mapι
X
:
A
→
X, any k, we have (9)
secat
⁡
ι
X
⩽
relcat
k
⁡
ι
X
⩽
secat
⁡
ι
X
+
k
+
1
.Proof.
Only the second inequality needs a proof. Letn
=
secat
⁡
ι
X and let σ be a homotopy section of g
n. Consider the following homotopy commutative diagram:
whereσ
′
=
(
σ
∘
g
k
,
id
G
k
) is the whisker map induced by the right homotopy pullback. We have g
′
∘
σ
′
≃
id
G
k and the left square is a homotopy pullback by the Prism lemma (see [7, Lemma 1.3], for instance). The map σ
+
=
γ
n
n
+
k
+
1
∘
σ is a homotopy section of g
n
+
k
+
1 and, moreover, σ
+
∘
g
k
≃
γ
k
n
+
k
+
1
∘
g
′
∘
σ
′
≃
γ
k
n
+
k
+
1. So relcat
k
ι
X
⩽
n
+
k
+
1.Theorem 9.
For any mapι
X
:
A
→
X, any k, we have (10)
k
⩽
relcat
k
⁡
ι
X
⩽
relcat
k
+
1
⁡
ι
X
⩽
relcat
k
⁡
ι
X
+
1
.Proof.
The first two inequalities are our Remark6; only the third needs a proof. Let n
=
relcat
k
ι
X and let σ be a homotopy section of g
n such that σ
∘
g
k
≃
γ
k
n. Consider the following homotopy commutative diagram:
The mapσ
+
=
γ
n
∘
σ is a homotopy section of g
n
+
1 and σ
+
∘
g
k
+
1
≃
γ
k
+
1
n
+
1, so relcat
k
+
1
ι
X
⩽
n
+
1.Sorelcat
k
ι
X increases at most by one when k increases by one.Corollary 10.
For any mapι
X
:
A
→
X, any k, we have (11)
relcat
⁡
ι
X
⩽
relcat
k
⁡
ι
X
⩽
relcat
⁡
ι
X
+
k
.Remark 11.
As a consequence of Theorem9 and Corollary 10, if n
=
relcat
ι
X, there are at most n integers k for which relcat
k
+
1
ι
X
=
relcat
k
ι
X.Example 12.
Ifι
X is a homotopy equivalence, then g
k is a homotopy equivalence for all k. So relcat
k
ι
X
=
k for all k.Example 13.
LetA
≄
∗ and consider the map ι
∗
:
A
→
∗. We have secat
⁡
ι
∗
=
0 because ι
∗ has a (unique) section. By Proposition 8, relcat
k
ι
∗
=
k or 1
+
k. Indeed, for any k, the map γ
k
k
+
1
:
A
⋈
⋯
⋈
A
(
k
+
1
t
i
m
e
s
)
→
A
⋈
⋯
⋈
A
(
k
+
2
t
i
m
e
s
) is homotopic to the null map, so σ
∘
g
k
≃
γ
k
k
+
1, where σ
:
∗
→
G
k
+
1
(
ι
∗
). But we cannot have relcat
k
ι
∗
=
k unless g
k
(
ι
∗
)
:
A
⋈
⋯
⋈
A
(
k
+
1
t
i
m
e
s
)
→
∗ is a homotopy equivalence.
If we choose a spaceA such that A
≄
∗ but Σ
A
≃
∗ (the 2-skeleton of the Poincaré homology 3 spheres, for instance), then A
⋈
A
≃
Σ
A
∧
A
≃
∗ and g
k is a homotopy equivalence for all k
>
0, so relcat
0
ι
∗
=
1 and relcat
k
ι
∗
=
k for all k
>
0. However, if we chose a simply connected CW-complex A (in order that A
⋈
⋯
⋈
A
≄
∗), then relcat
k
ι
∗
=
k
+
1 for all k.Example 14.
Consider any CW-complexX with cat
X
=
1 and the map ι
X
:
∗
→
X. We have secat
⁡
ι
X
=
relcat
ι
X
=
cat
X
=
1. Let us compute cat
1
X
=
relcat
1
ι
X. Notice that G
1
(
X
)
≃
Σ
Ω
X. By Theorem 9, we know that 1
⩽
cat
1
X
⩽
2. But we cannot have cat
1
X
=
1 because g
1 is not a homotopy equivalence, so cat
1
X
=
2. By the way, we can say that γ
1
2
:
Σ
Ω
X
→
G
2
(
X
) factorizes up to homotopy through g
1
:
Σ
Ω
X
→
X.Example 15.
More generally, ifrelcat
ι
X
=
1, we have k
⩽
relcat
k
ι
X
⩽
1
+
k for any k by Corollary 10. Thus, relcat
k
ι
X
=
k
+
1 while g
k
(
ι
X
) is not a homotopy equivalence (and if any n exists such that g
n
(
ι
X
) is a homotopy equivalence, then relcat
k
ι
X
=
k for all k
⩾
n).Suppose we are given any mapι
X
:
A
→
X with secat
⁡
(
ι
X
)
⩽
n and any homotopy section σ
:
X
→
G
n of g
n
:
G
n
→
X. For any k
⩽
n, consider the following homotopy pullbacks:whereθ
k
n
=
(
γ
k
n
,
id
G
k
) is the whisker map induced by the homotopy pullback H
k
n. Notice that η
n
k
∘
θ
k
n
≃
id
G
k. By the Prism lemma, we know that the homotopy pullback of σ and η
k
n is indeed G
k and that η
n
k
∘
σ
¯
≃
id
G
k. Also notice that π
≃
π
′ since π
≃
η
n
k
∘
θ
k
n
∘
π
≃
η
n
k
∘
σ
¯
∘
π
′
≃
π
′.Proposition 16.
For any mapι
X
:
A
→
X with secat
⁡
(
ι
X
)
⩽
n and any homotopy section σ
:
X
→
G
n of g
n
:
G
n
→
X, with the same definitions and notations as above, the following conditions are equivalent: (i)
σ
∘
g
k
≃
γ
k
n.
(ii)
π has a homotopy section.
(iii)
π is a homotopy epimorphism.
(iv)
θ
k
n
≃
σ
¯.Proof.
We have the following sequence of implications:(i) ⇒ (ii): sinceσ
∘
g
k
≃
γ
k
n
≃
η
k
n
∘
θ
k
n
∘
id
G
k, we have a whisker map (
g
k
,
id
G
k
)
:
G
k
→
Q induced by the homotopy pullback Q which is a homotopy section of π.
(ii) ⇒ (iii): it is obvious.
(iii) ⇒ (iv): we haveθ
k
n
∘
π
≃
σ
¯
∘
π
′
≃
σ
¯
∘
π since π
≃
π
′. Thus, θ
k
n
≃
σ
¯ since π is a homotopy epimorphism.
(iv) ⇒ (i): we haveσ
∘
g
k
≃
η
k
n
∘
σ
¯
≃
η
k
n
∘
θ
k
n
≃
γ
k
n.Theorem 17.
Letι
X
:
A
→
X be a (q
-
1) connected map. If for some k
⩽
secat
⁡
ι
X, G
k has the homotopy type of a CW-complex with dimension strictly less than (
secat
⁡
ι
X
+
1
)
q
-
1, then relcat
j
⁡
ι
X
=
secat
⁡
ι
X for all j
⩽
k.This is an immediate consequence of the following.Proposition 18.
Letι
X
:
A
→
X be a (q
-
1) connected map with secat
⁡
ι
X
⩽
n. If for some k
⩽
n, G
k has the homotopy type of a CW-complex with dimension strictly less than (
n
+
1
)
q
-
1, then σ
∘
g
k
≃
γ
k
n for any homotopy section σ of g
n, so relcat
k
⁡
ι
X
⩽
n.Proof.
Recall that, for anyi
⩾
0, g
i is the (
i
+
1
)-fold join of ι
X. Thus, by [11, Theorem 47], we obtain that g
i
:
G
i
→
X is (
i
+
1
)
q
-
1-connected. As g
i and η
i
k have the same homotopy fiber, which is (
i
+
1
)
q
-
2-connected, we see that η
i
k
:
H
k
i
→
G
k is (
i
+
1
)
q
-
1-connected, too. By [12, Theorem IV.7.16], this means that, for every CW-complex K with dim
⁡
K
<
(
i
+
1
)
q
-
1, η
i
k induces a one-to-one correspondence [
K
,
H
k
i
]
→
[
K
,
G
k
]. Apply this to K
≃
G
k and i
=
n: since θ
k
n and σ
¯ are both homotopy sections of η
n
k, we obtain θ
k
n
≃
σ
¯, and Proposition 16 gives the desired result.Example 19.
LetX be the Eilenberg-Mac Lane space K
(
Q
,
1
). It is known that cat
(
X
)
=
2 and that G
1
(
X
)
≃
Σ
Ω
X has the homotopy type of a wedge of circles (see [8, Example 1.9 and Remark 1.62], for instance). By Theorem 9, we know that 2
⩽
cat
1
X
⩽
3. Because dim
⁡
G
1
(
X
)
=
1
<
(
cat
X
+
1
)
-
1
=
2, we have σ
∘
g
1
≃
γ
1
2 for any homotopy section σ of g
2
(
X
) and, thus, cat
1
X
=
2. Moreover, g
k is never a homotopy equivalence, so cat
k
X
>
k for any k; thus, cat
k
X
=
k
+
1 for k
⩾
1.



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*Source: 8320742-2016-08-23.xml*