Q-Topological Complexity

0.23 MB·

by  Lucía Fernández Suárez

Q-TOPOLOGICAL COMPLEXITY
7 LUC´IAFERNA´NDEZSUA´REZANDLUCILEVANDEMBROUCQ
1
0
2 Abstract. By analogy with the invariant Q-category defined by Scheerer,
Stanley and Tanr´e, we introduce the notions of Q-sectional category and Q-
n
topologicalcomplexity. Weestablishseveralpropertiesoftheseinvariants. We
a
alsoobtainaformulaforthebehaviourofthesectionalcategorywithrespectto
J
afibrationwhichgeneralizestheclassicalformulasforLusternik-Schnirelmann

0 category andtopological complexity.
2

]
T
A 1. Introduction
.
h The Q-category of a topological space X, denoted by QcatX, is a lower bound
t for the Lusternik-Schnirelmann category of X, catX, which has been introduced
a
m by H. Scheerer, D. Stanley, and D. Tanr´e in [17]. This invariant, defined using a
fibrewiseextensionofafunctorQk equivalenttoΩkΣk,hasbeeninparticularused
[
inthe study ofcriticalpoints (see [2, Chap. 7],[15]) andin the study ofthe Ganea

1 conjecture. More precisely, after N. Iwase [12] showed that the Ganea conjecture,
v
whichassertedthatcat(X×Sn)=catX+1foranyn≥1,wasnottrueingeneral,
5
although it was known to be true for many classes of spaces (e.g. [19], [14], [11],
4
8 [16], [22]), one could ask for a complete characterizationof the spaces X satisfying
5 the equality above. In [17], H. Scheerer, D. Stanley, and D. Tanr´e conjectured
0 that a finite CW-complex X satisfies the Ganea conjecture, that is the equality
.
1 cat(X ×Sn)= catX +1 holds for any n≥ 1, if and only if QcatX = catX. One
0 directionofthisequivalencehasbeenprovedin[23]butthecompleteanswerisstill
7
unknown.
1
In this paper we introduce the analogue of Q-category for Farber’s topological
:
v complexity [4] and establish some properties of this invariant. Since L.-S. category
i
X andtopologicalcomplexityarebothspecialcasesofthenotionofsectionalcategory,
introduced by A. Schwarz in [18], we naturally consider and study a notion of Q-
r
a sectional category. Our definition, given in Section 2.3, is based on a generalized
notion of Ganea fibrations and on a fibrewise extension of the functor Qk that
we respectively recall in Sections 2.1 and 2.2. We next, in Section 3, establish
variousformulasforQ-sectionalcategoryandQ-topologicalcomplexity. Itisworth
noting that our study of the behaviour of the Q-sectional category in a fibration
led us to establish a new formula for the sectional category (see Theorem 3.8),
which generalizes both the classical formula for the LS-category in a fibration and
the formula established by M. Farber and M. Grant for topologicalcomplexity [5].
Finally,weuseourresultstostudysomeexamples. Thispermitsusinparticularto
2010 Mathematics Subject Classification. 55M30.
Key words and phrases. sectionalcategory, topological complexity.
Thisresearch has beensupported byPortuguese Funds fromthe“Fundac¸a˜o paraaCiˆenciae
aTecnologia”, throughtheProjectUID/MAT/0013/2013.
1
2 L.FERNA´NDEZSUA´REZ,L.VANDEMBROUCQ
observe that the analogue of the Scheerer-Stanley-Tanr´econjecture for topological
complexity is not true. We also include a small observation about the original
Scheerer-Stanley-Tanr´econjecture.
Throughoutthis textwe workinthe categoryofcompactlygeneratedHausdorff
spaces having the homotopy type of a CW-complex.
2. Q-sectional category and Q-topological complexity
2.1. Sectionalcategory and Ganeafibrations. Letf :A→X beamapwhere
X is a well-pointed path-connected space with base point ∗ ∈ X. The sectional
category of f,secat(f), is the leastintegern (or ∞)for which there exists anopen
cover U ,...,U of X such that, for any 0 ≤ i ≤ n, f admits a local homotopy
0 n
section on U (that is, a continuous map s :U →A such that f ◦s is homotopic
i i i i
to the inclusion U ֒→ X). When f is a fibration, we can, equivalently, require
i
localstrictsectionsinsteadoflocalhomotopysections. Asspecialcasesofsectional
category,Lusternik-SchnirelmanncategoryandFarber’stopologicalcomplexityare
respectively given by
• cat(X)=secat(ev :PX →X)=secat(∗→X)
1
where PX ⊂ XI is the space of paths beginning at the base point ∗ and
ev is the evaluation map at the end of the path,
1
• TC(X)=secat(ev :XI →X×X)=secat(∆:X →X×X)
0,1
where ev evaluates a path at its extremities and ∆ is the diagonal map.
0,1
As is well-known, if f : A → X and g : B → Y are two maps with homotopy
∼ ∼
equivalences A → B and X → Y making the obvious diagram commutative then
secat(f) = secat(g). Also secat(f) can be characterized through the existence
of a global section for a certain join map which can be, for instance, explicitly
constructedthroughaniteratedfibrewisejoinoff whenf isafibration([18]). Here
we will assume (without loss of generality) that f : A → X is a (closed) pointed
cofibration and consider the following constructions which give us a natural and
explicitfibration,theGaneafibrationoff,equivalenttothejoinmapcharacterizing
secat(f):
• the fatwedge of f ([7], [9]):
Tn(f)={(x ,...,x )∈Xn+1 | ∃j,x ∈f(A)}
0 n j
which generalizes the classical fat-wedge
Tn(X)={(x ,...,x )∈Xn+1 | ∃j,x =∗},
0 n j
• the space Γ X = (γ ,...,γ )∈ XI n+1 | γ (0)=···=γ (0)
n n 0 n (cid:0) (cid:1) 0 n o
together with the fibration
Γ X →X, (γ ,...,γ )7→γ (0)=···=γ (0)
n 0 n 0 n
which is a homotopy equivalence,
• thenthGaneafibrationoff,g (f):G (f)→X,whichisobtainedbypull-
n n
back alongthe diagonalmap∆ :X →Xn+1 of the fibrationassociated
n+1
to the inclusion Tn(f)֒→Xn+1 and explicitly given by:
G (f)={(γ ,...,γ )∈Γ (X) | ∃j, γ (1)∈f(A)} → X
n 0 n n j
(γ ,...,γ ) 7→ γ (0)=···=γ (0).
0 n 0 n
Q-TOPOLOGICAL COMPLEXITY 3
Allthesespacesareconsideredwiththeobviousbasepoints(∗,...,∗)∈Tn(f)and
(ˆ∗,...,ˆ∗)∈G (f)⊂Γ (X)(whereˆ∗denotestheconstantpath). Byconstruction,
n n
thereexistsacommutativediagraminwhichthesquareis a(homotopy)pull-back.
G (f) //• oo ∼ Tn(f)
n l
gn(f)(cid:15)(cid:15) (cid:15)(cid:15)(cid:15)(cid:15) zz✈✈✈✈✈✈✈✈L
X //Xn+1
∆n+1
By [7], we know that secat(f) is the least integer n such that the diagonal map
∆ : X → Xn+1 lifts up to homotopy in the fatwedge of f, Tn(f). By the
n+1
homotopy pull-back diagram above, this is equivalent to say that secat(f) is the
least integer n such that the fibration g (f) admits a (homotopy) section. By [9,
n
Th. 8], the fibration g (f) is equivalent to the (fibrewise) join of n+1 copies of
n
f or of any map weakly equivalent to f. The fibre of g (f), denoted by F (f), is
n n
homotopically equivalent to the (usual) join of n+1 copies of the homotopy fibre
F of f.
When f is the inclusion ∗ → X (which is a cofibration since X is well-pointed)
we recover a possible description of the classical Ganea fibration of X and we will
use, in that case, the classical notation
g (X):G (X)={(γ ,...,γ )∈Γ (X) : ∃j, γ (1)=∗}→X.
n n 0 n n j
We have
cat(X)≤n ⇔ g (X):G (X)→X admits a (homotopy) section.
n n
When X is a CW-complex, f =∆:X →X×X is a cofibration and we have
TC(X)≤n ⇔ g (∆):G (∆)→X ×X admits a (homotopy) section.
n n
We finally note that, if we have a commutative diagram where f and g are
cofibrations
A ϕ //B
f g
(cid:15)(cid:15) (cid:15)(cid:15)
X //Y
ψ
then we have a commutative diagram
G (f)Gn(ψ,ϕ)//G (g)
n n
gn(f) gn(g)
(cid:15)(cid:15) (cid:15)(cid:15)
X //Y.
ψ
Moreover,we have
(a) if ϕ and ψ are homotopy equivalences, then so is G (ψ,ϕ);
n
(b) ifthe firstdiagramis ahomotopypull-back,thensoisthe seconddiagram.
Note that (b) follows (for instance) from the equivalence between g (f) and the
n
fibrewise join of n+1 copies of f together with the Join Theorem ([3]).
4 L.FERNA´NDEZSUA´REZ,L.VANDEMBROUCQ
2.2. FibrewiseQconstruction. ThenotionofQ-categorydefinedin[17]isbased
on the Dror-Frajoun fibrewise extension [6] of a functor Qk equivalent to ΩkΣk.
Instead of using the Dror-Farjoun construction, we will here use the (equivalent)
explicit fibrewise extension of Qk given in [23]. We describe the case k = 1, recall
morequicklythe generalcaseandreferto [17],[2,Sections4.5,4.6,4.7]and[23]for
further details.
We denoteby Σ1Z the unreducedsuspensionofaspaceZ, i.e. Σ1Z =Z×I/∼
with (z,0)∼ (z′,0) and (z,1)∼ (z′,1) for z,z′ ∈ Z. Denoting by [z,t] ∈ Σ1Z the
classof(z,t)∈Z×I,wehaveacanonicalmapα1 =α1 :{0,1}=∂I →Σ1Z given
Z
by α1(0)=[z,0] and α1(1)=[z,1] where z ∈Z. The functor Q1 is defined by
Q1(Z)={ω :I →Σ1Z|ω =α1}
|∂I Z
and is given with a co-augmentation η1 =η1 :Z →Q1Z which takes z ∈Z to the
Z
path z →[z,t]. If Z is pointed and ΣZ denotes the reduced suspension of Z, then
thereisanaturalmapQ1(Z)→ΩΣZe,whichisahomotopyequivalenceinducedby
the identification map Σ1Z →∼ ΣZeand which makes compatible the coaugmention
of Q1 with the usual coaugmenetation η˜:Z →ΩΣZ.
e
We nowdescribe afibrewiseextensionofQ1. Letp:E →B be afibration(over
a path-connectedspace) with fibre F. The fibrewise suspension of p, Σ1E →B, is
B
defined by the push-out:
E×{0,1} // //E×I
p×id
(cid:15)(cid:15) (cid:15)(cid:15)
B×{0,1} // //Σ1E
B ❉❉
❉❉❉pˆ
❉❉
❉"",,B(cid:20)(cid:20)
Theresultingmappˆ:Σ1E →B isafibrationwhosefibreoverbisthe(unreduced)
B
suspension Σ1F of the fibre of p : E → B over b. By construction, we have a
b
canonical map:
µ1 :B×{0,1}→Σ1E
B
whichisafibrewiseextensionofα1: foranyb∈B, µ1(b,−)=α1 :{0,1}→Σ1F .
Fb b
We define
Q1(E)={ω :I →Σ1E | ∃b∈B,pˆω =b and ω =µ1(b,−)}
B B |∂I
together with the map q1(p):Q1(E)→B givenby ω 7→pˆω(0). This is a fibration
B
whosefibreoverbisQ1(F )([23,Lemma8]). Wealsohaveafibrewisecoaugmenta-
b
tion η1 :E →Q1(E) which extends η1 : F →Q1(F) and we have a commutative
B B
diagram
E ηB1 //Q1(E) //Q1(E) ∼ //ΩΣE
B
e
p q1(p) Q1(p) ΩΣep
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
B B //Q1(B) ∼ //ΩΣB
η1
e
Q-TOPOLOGICAL COMPLEXITY 5
wherethemapQ1(E)→Q1(E)isinducedbytheidentificationmapΣ1E →Σ1E.
B B
In general, for any k≥1, we consider
• thek foldunreducedsuspensionofaspaceZ,ΣkZ,whichcanbedescribed
as (Z ×Ik)/∼ where the relation is given by
(z,t ,...,t )∼(z′,t′,...,t′)
1 k 1 k
if,forsomei,t =t′ ∈{0,1}andt =t′ forallj >i. Wewrite[z,t ,...,t ]
i i j j 1 k
for the class of an element.
• the kth fibrewise suspension of p : E → B, pˆk : ΣkE → B, whose fibre
B
overbis ΣkF . As before, ΣkE canbe describedas(E×Ik)/∼ wherethe
b B
relation is given by (e,t ,...,t ) ∼ (e′,t′,...,t′) if p(e) = p(e′) and, for
1 k 1 k
some i, t =t′ ∈{0,1} and t =t′ for all j >i. We write [e,t ,...,t ] for
i i j j 1 k
the class of an element.
• the canonical map αk : ∂Ik →ΣkZ given by αk(t ,...,t ) = [z,t ,...,t ]
1 k 1 k
(where z ∈ Z is any element) and its fibrewise extension µk : B ×∂Ik →
ΣkE satisfying µk(b,−)=αk :∂Ik →ΣkF for any b∈B.
B b
• the fibration qk(p) : Qk(E) → B where QkE is the (closed) subspace of
B B
(ΣkE)Ik given by
B
Qk(E)={ω :Ik →ΣkE | ∃b∈B,pˆkω =b and ω =µk(b,−)}
B B |∂Ik
and qk(p)(ω)=pˆkω(0). The fibre is
Qk(F)={ω :Ik →ΣkF|ω =αk}→∼ ΩkΣkF
|∂Ik
e
and is given with an obvious coaugmentation ηk : F → Qk(F), equivalent
to the classical augmentation η˜k : F → ΩkΣkF. We denote by ηk : E →
B
Qk(E) the fibrewise extension of ηk. Wheneit is relevant Qk(F) and QkE
B B
areconsideredwiththebasepointgivenbythemapu7→[∗,u]whereu∈Ik
and ∗ is the base point of F and E.
We have, for any k ≥1, a commutative diagram:
(1) E ηBk // Qk(E) //Qk(E) ∼ //ΩkΣkE
B
e
p qk(p) Qk(p) ΩkΣekp
B(cid:15)(cid:15) B(cid:15)(cid:15) ηk //Qk((cid:15)(cid:15)B) ∼ // ΩkΣ(cid:15)(cid:15)kB.
e
Setting Q0 =Q0 =id we also have a commutative diagram of the following form:
B
(2) E =Q0E //Q1(E) //··· // Qk(E) bkB // Qk+1(E) //···
B B B B
p q1(p) qk(p) qk+1(p)
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
B B ··· B B ···
wherethe mapbk is a fibrewiseextensionofthe mapbk :Qk(F)→Qk+1(F)given
B
bybk(ω)(t ,...,t )=[ω(t ,...,t ),t ]. Observethatthe mapbk isequivalent
1 k+1 1 k k+1
to the map Ωk(η˜ΣekF):Ωk(ΣkF)→Ωk(ΩΣΣkF)=Ωk+1Σk+1F. We can interpret
thesequence(2)asafibreweisestabilizationeeofthefibratioenpsince,foranyintegers
6 L.FERNA´NDEZSUA´REZ,L.VANDEMBROUCQ
k and i, we have π (QkF)=π (ΣkF).
i i+k
e
For any k ≥ 0, the functors Qk and Qk preserve homotopy equivalences. We also
B
notethattheQk constructionisnaturalinthesensethat,ifwehaveacommutative
−
diagram
E′ f // E
(†) p′ p
(cid:15)(cid:15) (cid:15)(cid:15)
B′ //B
g
where p and p′ are fibrations (over path-connected spaces), then, for any k, we
obtain commutative diagram
Qk (E′) //Qk(E)
B′ B
(‡) qk(p′) qk(p)
(cid:15)(cid:15) (cid:15)(cid:15)
B′ // B
g
Moreoverwe have
Proposition 2.1. With the notations above, if Diagram (†) is a homotopy pull-
back, then so is Diagram (‡).
Proof. SincethefunctorsQkandQk preservehomotopyequivalences,itissufficient
B
to establish the statement when Diagram (†) is a strict pull-back. In this case, the
whisker map ψ : E′ → E × B′ is a homeomorphism and its inverse φ satisfies
B
p′φ(e,b′)=b′ for (e,b′)∈E× B′. We can then check that the map φ¯:ΣkE×
B B B
B′ →Σk E′ induced by ((e,u),b′)7→[φ(e,b′),u] for ((e,u),b′)∈(E×Ik)× B′ is
B′ B
an inverse of the whisker map Σk E′ →ΣkE× B′. In other words, the diagram
B′ B B
Σk E′ //ΣkE
B′ B
pb′k (cid:15)(cid:15)(cid:15)(cid:15) (cid:15)(cid:15)(cid:15)(cid:15)pˆk
B′ // B
g
is a strict pull-back (and homotopy pull-back since pˆk is a fibration). We can next
check that the map Qk(E)× B′ →Qk (E′) that takes (ω,b′)∈Qk(E)× B′ to
B B B′ B B
the element Ik → Σk E′ of Qk (E′) given by u 7→ φ¯(ω(u),b′) is an inverse of the
B′ B′
whisker map Qk (E′) → Qk(E)× B′. This means that Diagram (‡) is a strict
B′ B B
pull-back and therefore a homotopy pull-back since qk(p) is a fibration. (cid:3)
Remark 2.2. Assuming that the fibres F and F′ of p and p′ are path-connected
spaces(havingthehomotopy typeofaCW-complex), wecangivethefollowing more
conceptual proof of Proposition 2.1. In this case, the spaces Qk(E) and Qk (E′)
B B′
are path-connected spaces having the homotopy type of a CW-complex, since B and
Qk(F) ≃ ΩkΣkF and Qk(F′) ≃ ΩkΣkF so are (see [20]). Since Diagram (†) is a
homotopy puell-back, the map e : F′e→ F induced by this diagram is a homotopy
equivalence and so is Qk(e). Therefore Diagram (‡) is a homotopy pull-back since
the whisker map from Qk (E′) to Qk(E)× B′ induces Qk(e) between the fibres
B′ B B
and is hence a homotopy equivalence.
Q-TOPOLOGICAL COMPLEXITY 7
Finally, we also note the following constructions which will be, as in [17], useful
in our study of products. If p : E → B and p′ : E′ → B′ are two fibrations then,
there exist, for any k ≥0, commutative diagrams of the following form
Qk(E)×E′ //Qk (E×E′) E×Qk (E′) //Qk (E×E′)
B ◗q◗k◗(◗p◗)×◗◗p◗′◗◗◗B◗×(( B′ (cid:15)(cid:15) qk(p×p′) B′ ◗p◗×◗◗q◗k◗(p◗′◗)◗◗◗◗B◗×(( B′ (cid:15)(cid:15)qk(p×p′)
B×B′ B×B′
which are induced by the obvious fibrewise extensions of the maps
ΣkZ ×Z′ → Σk(Z ×Z′)
([z,t ,...,t ],z′) 7→ [(z,z′),t ,...,t ]
1 k 1 k
and
Z×ΣkZ′ → Σk(Z ×Z′)
(z,[z′,t ,...,t ]) 7→ [(z,z′),t ,...,t ]
1 k 1 k
By considering the fibrewise extensions of the evaluation
ev :ΣkQkZ → ΣkZ
[ω,t ,...,t ] 7→ ω(t ,...,t )
1 k 1 k
and of the map
Qk◦Qk(Z) → Qk(Z)
ω :Ik →ΣkQkZ 7→ ev◦ω
we also get a commutative diagram
Qk ◦Qk(E) // Qk(E)
B B ◆◆◆◆◆◆◆◆◆◆◆◆&&B(cid:15)(cid:15)
B.
This permits us to establish the following result:
Proposition 2.3. Let p : E → B and p′ : E′ → B′ be two fibrations. For any
k ≥0, there exists a commutative diagram
Qk(E)×Qk (E′) //Qk (E×E′)
B qBk(P′pP)P×PqPkP(PpP′)PPPP(( ww♦♦♦♦♦q♦k♦(p♦×♦♦Bp♦′×)B′
B×B′.
Proof. Using the constructions above we obtain:
QkB(E)×QkB′(E′) //QkB×B′(E×QkB′(E′)) //QkB×B′◦QkB×B′(E×E′) //QkB×B′(E×E′)
qk(p)×qk(p′) qk(p×p′)
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
B×B′ B×B′ B×B′ B×B′.
(cid:3)
8 L.FERNA´NDEZSUA´REZ,L.VANDEMBROUCQ
2.3. Definition of Qsecat and QTC. Let f : A→X a cofibration where X is a
well-pointedpath-connectedspaceandletk ≥0. ByapplyingtheQk construction
X
to the Ganea fibrations g (f) we define
n
Definition 2.4. Qksecat(f) is the least integer n (or ∞) such that the fibration
qk(g (f)):Qk (G (f))→X
n X n
admits a (homotopy) section.
By Diagram (2), we have:
···≤Qk+1secat(f)≤Qksecat(f)≤···≤Q1secat(f)≤Q0secat(f)=secat(f)
As a limit invariant, we set: Qsecat(f):=limQksecat(f).
If f is the inclusion ∗ → X, we recover the notion of Qkcat(X) introduced by H.
Scheerer, D. Stanley and D. Tanr´e. If X is a CW-complex and f is the diagonal
map ∆:X →X ×X, we naturally use the notation QkTC(X) and QTC(X).
Remark 2.5. The notion of Qksecat can beextendedto anymap g by applying the
Q construction to any fibration equivalent to the join map characterizing secat(g)
or, equivalently, by setting Qksecat(g) := Qksecat(f) where f is any cofibration
weakly equivalent to g.
3. Some properties of Qsecat and QTC
In all the statements in this section, we consider cofibrations whose target is a
well-pointed path-connected space.
3.1. Basic properties.
We start with the following properties which permit us to generalize to QTC and
Qcat the well-known relationships [4] between TC and cat (see Corollary 3.2):
Proposition 3.1. Let f :A→X and g :B →Y be two cofibrations together with
a commutative diagram
A // B
ϕ
f g
(cid:15)(cid:15) (cid:15)(cid:15)
X //Y.
ψ
(a) Ifψisahomotopyequivalencethen,foranyk ≥0,Qksecat(f)≥Qksecat(g).
(b) If the diagram is a homotopy pull-back then, for any k ≥ 0, we have
Qksecat(f)≤Qksecat(g).
Proof. By considering the Ganea fibrations and applying the Qk construction, we
X
obtain the following commutative diagram
Qk (G (f)) // Qk(G (g))
X n Y n
qk(gn(f)) qk(gn(g))
(cid:15)(cid:15) (cid:15)(cid:15)
X // Y.
ψ
If ψ is a homotopy equivalence, we deduce from a sectionof the left-hand fibration
a homotopy section of the right-hand fibration, which proves the statement (a).
Q-TOPOLOGICAL COMPLEXITY 9
For (b), the hypothesis implies that the diagram is a homotopy pull-back (see
Proposition 2.1) which permits us to obtain a section of the left hand fibration
from a section of the right hand one. (cid:3)
Corollary 3.2. Let X be a path-connected CW-complex. For any k ≥ 0, we have
Qkcat(X)≤QkTC(X)≤Qkcat(X ×X).
Proof. It suffices to apply the two items of the propostion above to the following
two diagrams, respectively. The right hand diagram, where i is the inclusion on
2
the second factor, is a homotopy pull-back:
∗ //X ∗ //X
∆ ∆
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
X ×X X ×X X // X ×X.
i2
(cid:3)
3.2. Cohomological lower bound. Let f : A → X be a cofibration, and let
f∗: H∗(X;k)→H∗(A;k)be the morphisminducedbyf incohomologywithcoef-
ficients in a field k. As is well-known ([18]), if we consider the index of nilpotency,
nil, of the ideal kerf∗, that is, the least integer n such that any (n+1)-fold cup
product in kerf∗ is trivial, then we have
nilkerf∗ ≤secat(f).
Actually the proof of [8, Thm 5.2] permits us to see that
nilkerf∗ ≤Hsecat(f)≤secat(f)
where Hsecat(f) is the least integer n such that the morphism induced in coho-
mology by the nth Ganea fibration of f, H∗(X;k)→H∗(G (f);k), is injective.
n
Here we prove that
Theorem 3.3. For any k ≥0, nilkerf∗ ≤Hsecat(f)≤Qksecat(f).
Proof. Suppose that Qksecat(f)≤ n. By applying Diagram (1) to the nth Ganea
fibration of f, we obtain the following commutative diagram:
G (f) //Qk (G (f)) //Qk(G (f)) ∼ //ΩkΣkG (f)
n X n n n
e
gn(f) qk(gn(f)) Qk(gn) ΩkΣekgn(f)
X(cid:15)(cid:15) X(cid:15)(cid:15) ηk //Qk((cid:15)(cid:15)X) ∼ // ΩkΣ(cid:15)(cid:15)kX.
e
The composite η˜k : X → Qk(X) →∼ ΩkΣkX, which is the usual coaugmentation,
canbeidentifiedtotheadjointoftheidenetityofΣkX throughthek-foldadjunction.
From Qksecat(f)≤n we know that there existes a map ψ :X →ΩkΣkG (f) such
n
that ΩkΣk(g (f))ψ ≃ η˜k. Through the k fold adjunction, we getea homotopy
n
section oef Σk(g (f)) which implies that H∗(g (f)) is injective. (cid:3)
n n
e
When f =∆:X →X ×X, nilker∆∗ coincides with Farber’s zero-divisor cup-
length, that is, the nilpotency of the kernel of the cup-product ∪ : H∗(X;k)⊗
X
H∗(X;k)→H∗(X;k) and we have:
10 L.FERNA´NDEZSUA´REZ,L.VANDEMBROUCQ
Corollary 3.4. Let X be a path-connected CW-complex. For any k ≥ 0, we have
nilker∪ ≤QkTC(X).
X
3.3. Products and fibrations. WeherestudythebehaviourofQsecat andQTC
withrespectto productsandfibrationsandestablishgeneralizationsofwell-known
properties of LS-category and topological complexity. Classically these properties
are proven using the open cover definition of these invariants. In order to be able
to obtain generalizations to Qsecat, QTC and Qcat, we first need a proof of the
classical property based on the Ganea fibrations.
Theorem 3.5. Let f : A →X and g : B →Y be two cofibrations. Then, for any
k ≥0, Qksecat(f ×g)≤Qksecat(f)+Qksecat(g).
Proof. Fromthe diagramconstructedin[13,Pg. 27](fromtheproductoffibrewise
joins of two fibrations to the fibrewise join of the fibrationproduct) we candeduce
the existence of a commutative diagram of the following form:
G (f)×G (g) //G (f ×g)
n gn(mf❖)❖×❖g❖m❖❖(❖g❖)❖❖❖'' ww♣♣♣♣g♣n♣+♣m♣♣(♣fm♣×+gn)
X ×Y.
By applying the Qk construction and using the map of Proposition 2.3 we
X×Y
obtain:
Qk (G (f))×Qk(G (g)) //Qk (G (f)×G (g)) //Qk (G (f ×g))
X n Yqk(❱gm❱n❱(❱f❱))❱×❱q❱k❱(❱g❱m❱(❱g❱))❱X❱×❱❱Y❱❱** n qkm❯(❯gn❯(❯f❯)❯×❯g❯m❯(❯g❯))❯❯X❯❯×❯Y❯** m(cid:15)(cid:15)+qnk(gn+m(f×g))
X×Y X×Y.
From this diagram, we can establish that, if Qksecat(f)≤ n and Qksecat(g) ≤ m
then Qksecat(f ×g)≤n+m. (cid:3)
Corollary 3.6. Let X and Y be path-connected CW-complexes. For any k ≥ 0,
we have QkTC(X ×Y)≤QkTC(X)+QkTC(Y).
Proof. Sincethediagonalmap∆ :X×Y →X×Y×X×Y coincides,uptothe
X×Y
homeomorphismswitching the two middle factors, with the product ∆ ×∆ , we
X Y
haveQkTC(X×Y)=Qksecat(∆ ×∆ )andthe resultfollowsfromthe theorem
X Y
above. (cid:3)
We now turn to the study of fibrations. We first establish the following result.
Proposition 3.7. Suppose that there exists a commutative diagram
A //C
f g
(cid:15)(cid:15) (cid:15)(cid:15)
X ι // Y
in which f, g and ι are cofibrations. Then, for any k ≥0, we have
Qksecat(g)≤(Qksecat(ι)+1)(secat(f)+1)−1.