FIBREWISE CONSTRUCTION APPLIED TO
LUSTERNIK-SCHNIRELMANN CATEGORY
HANS SCHEERER, DONALD STANLEY, AND DANIEL TANRE
Abstract.In this paper a variant of Lusternik-Schnirelmann category is p*
*resented
which is denoted by Qcat(X). It is obtained by applying a base-point fre*
*e version
of Q = 1 1 fibrewise to the Ganea fibrations. We prove cat(X) Qcat(X)
oecat(X) where oecat(X) denotes Y. Rudyak's strict category weight. How*
*ever,
Qcat(X) approximates cat(X) better, because e.g. in the case of a ration*
*al space
Qcat(X) = cat(X) and oecat(X) equals the Toomer invariant.
We show that Qcat(X xY ) Qcat(X)+Qcat(Y ). The invariant Qcat is desi*
*gned
to measure the failure of the formula cat(X x Sr) = cat(X) + 1. In fact *
*for 2-cell
complexes Qcat(X) < cat(X) , cat(X x Sr) cat(X) for some r 1.
We note that the paper is written in the more general context of a fun*
*ctor
from the category of spaces to itself satisfying certain conditions; = *
*Q, nn,
Sp1 or Lf are just particular cases.
Contents
0. Introduction 2
1. Fibrewise application of functors *
* 5
2. Specific constructions: nn, 1 1 , Sp1 7
3. Product formulae 8
4. Hopf invariants 9
Appendix A. Dror Farjoun's construction 15
Appendix B. Unpointed version of nn 16
References 17
___________
1991 Mathematics Subject Classification. Primary 55M30. Secondary 55Q10, 55Q2*
*5.
1
E
2 HANS SCHEERER, DONALD STANLEY, AND DANIEL TANR
0.Introduction
Let S (resp. S*) be the category of simplicial sets (resp. pointed simplici*
*al sets);
we will also denote convenient categories of spaces by these symbols. The base *
*point
of X 2 S* is always denoted by * 2 X.
0.1. Fibrewise application of functors and Lusternik-Schnirelmann category.
Let : S ! S (or S* ! S*) be a functor together with a natural transf*
*ormation
: id! as coaugmentation. If : S ! S is a coaugmented functor and X 2 S* then
(X) is canonically pointed by * ! X ! (X) thus defines a functor 0:S* ! S*.
Throughout this work we suppose that:
- the map * ! (*) coming from the coaugmentation is a weak equivalence;
- preserves weak equivalences.
Such a is called a regular coaugmented functor.
For any f 2 S there exists a functorial decomposition f = pf O jf such that j*
*f is
a cofibration and a weak equivalence and pf a fibration. We fix such a construc*
*tion
and by definition call pf the fibration associated to f. For any point x in the*
* target
of f the fibre of pf over x is called the homotopy fibre of f over x. If f 2*
* S* the
homotopy fibre of f indicates the homotopy fibre over *.
By [Far96]_ a regular coaugmented functor : S ! S admits an extension*
* to a
functor from the category of spaces over a space to itself such that there ar*
*e natural
transformations
_(E) __ r_(E)
E __________//(E)__________//(E)
| _
p || (p)|| (p)||
fflffl| |fflffl fflffl|
B _____________B___________//_(B)
over_ idB and (B) respectively. Moreover for p: E ! B the homotopy fi*
*bre of
(E) ! B over a point x is naturally equivalent to (F ) where F is the homotopy
fibre of p over x. We remark that the previous_consideration about pointed vers*
*ions
for maps in the image of works also with .
Applying this construction to the Ganea fibrations we obtain variants of Lust*
*ernik-
Schnirelmann category. First recall the Ganea construction for a map ss :E ! B*
* in
S*.
Definition 1. Let q0: G0(E; ss) ! B be the fibration associated to ss and sup*
*pose
the fibrations qi:Gi(E; ss) ! B have been constructed for i k - 1. Then we de*
*fine
q0k:Gk-1(E; ss)[C(Fk-1) ! B by q0k|Gk-1(E;ss):= qk-1 and q0k|C(Fk-1)= * where C*
*(Fk-1)
is the cone on the fibre Fk-1 of qk-1. Let qk: Gk(E; ss) ! B be the fibration a*
*ssociated
to q0k. In the particular case ss = (* ! B) we write qk: Gk(B) ! B.
We apply now the fibrewise construction to these fibrations (the dashed arrows
correspond to homotopy sections or liftings that are described below):
(Fn(E;ss))
Fn(E; ss)_________//_(Fn(E; ss))________//_Fn (E; ss)
in|| || ||
fflffl|_(Gn(E;ss))_ fflffl|r_(Gn(E;ss)) fflffl|
Gn(E; ss)__________//(Gn(E;Zss))________//(Gn(E;Zss))
__ YY___ ____55__________ZZ___
qn||_o____ _(qn)||oe_____s_______(qn)||________ae____
fflffl|___ fflffl|_______________fflffl|_______
B ___________________B_________________//_(B)
(B)
FIBREWISE CONSTRUCTION APPLIED TO LUSTERNIK-SCHNIRELMANN CATEGORY 3
where Fn (E; ss) is the homotopy fibre of (qn). In such a diagram we_may consid*
*er the
existence of a homotopy section o to qn, a homotopy section oe of (qn), a homo*
*topy
lifting s of (B) through (qn) or a homotopy section ae of (qn). The existence*
* of
o is the Ganea definition of the LS-category of ss, catG (ss), being less than *
*or equal
to n. For the others we set:
Definition 2. Let : S ! S be a regular coaugmented functor and ss :E ! B in S*.
Then:
__- the Ganea -category of ss, catG (ss), is the least integer n (or 1) such th*
*at
(qn) admits a section oe up to pointed homotopy;
- the Ganea [-category of ss, [catG (ss), is the least integer n (or 1) such *
*that
there exists s: B ! (Gn(E; ss)) satisfying (qn) O s ' (B);
- the Toomer -invariant of ss, e (ss), is the least integer n (or 1) such tha*
*t (qn)
admits a section ae up to pointed homotopy.
In the particular case ss = (* ! B) we write cat(B) := catG (* ! B), [cat(B) *
*:=
[catG (* ! B) and e (B) = e (* ! B).
As we will see below, this presentation unifies the following approximations *
*of the
Lusternik-Schnirelmann category:
- Mcat of a rational space [HL88 ] is a special case of catG [SS99],
- the strict category weight [Rud99 ], [Str00], [Van99 ] fits into the settin*
*g of [catG ,
- the Toomer invariant introduced in [Too74 ] is equal to eM for M the abel*
*ian
group completion of = Sp1 , cf Example 2.
__
If o is a section of qn we get a section_of (qn) by oe := _(Gn(B)) O o. In t*
*he same
way if oe is a homotopy section of (qn) the composite s := r_(Gn(B)) O oe is a*
* lifting
up to homotopy of (B) through (qn). That is:
catG (ss) catG (ss) [catG (ss) :
With an extra hypothesis the existence of a lifting up to homotopy s implies the
existence of a homotopical section to (qn):
Proposition 1. Suppose that is a regular coaugmented functor equipped with a n*
*atural
transformation 2 = O ! whose composition with ( ) is equal to the identity
! 2 ! . Let s: B ! (Gn(E; ss)) such that (qn) O s ' (B). Then there exists a
homotopical section ae: (B) ! (Gn(E; ss)) of (qn) and we have [catG (ss) = e (s*
*s).
This applies in particular if together with the coaugmentation and the trans*
*for-
mation 2 ! constitutes a triple.
In Definition 2 the subscript G is chosen to make a distinction from another *
*notion
of category of a map due to Fox [Fox41], Berstein and Ganea [BG62 ] (see also [*
*Jam78 ,
& 7]) which admits also variants with a fibrewise construction.
__
0.2. Unpointed version of pointed functors. In the fibrewise construction as-
sociated to a functor : S ! S the basepoint free situation is essential and we *
*first
meet the problem that the examples of functors that we have in mind, such as the
infinite symmetric product, need a basepoint. Therefore for any regular coaugme*
*nted
functor : S* ! S* we define a canonical functor + :S ! S called basepoint free
functor associated to :
For Y 2 S we denote by Y + 2 S* the space Y with an extra point ad*
*ded and
considered as the basepoint. Let * ! *+ ! (*+) be the map obtained from the
canonical inclusion and the coaugmentation. Denote by * ! (*+) the fi*
*bration
associated to the composition * ! (*+). The functor Y 7! + (Y ) is defined by *
*the
E
4 HANS SCHEERER, DONALD STANLEY, AND DANIEL TANR
following pullback:
+ (Y )_____//(Y +)
| |
| |
fflffl| fflffl|
* ______//_(*+)
By naturality the composite Y ! (Y ) ! (Y +) ! (*+) factorizes as Y ! * ! (*+)
and we get a coaugmentation Y ! + (Y ) from the universal property of pullbacks.
Note that + (Y ) is naturally equivalent to the homotopy fibre of (Y +) ! (*+) *
*over
* 2 (*+).
We will say that a coaugmented functor : S* ! S* has a basepoint free version*
* if
there exists a coaugmented functor : S ! S and a natural transformation between
0 and compatible with the coaugmentations and which is a weak equivalence for
any X 2 S*. Sometimes, as in Proposition 3, + is a basepoint free version of .
Let (resp. ) be the reduced suspension (resp. the loop space) in S*. In t*
*his
paper we are mainly concerned with the functors M = Sp1 , nn, Q = lim! nn
and their basepoint free functors M+ , P n= (nn)+ , Q+ . We also consider the
localization functor Lf [Far96]. The functors M and Q are particular cases of*
* a
more general construction, the infinite delooping associated to any S_-algebra *
*[Ada78 ],
[EKMM97 ].
We will see that M+ (resp. Q+ ) is a basepoint free version of M (resp. Q*
*).
For nn the situation is more complicated: we construct a basepoint free version
Qn: S ! S which is not homotopically equivalent to P n= (nn)+ . We have a
general comparison theorem between all these invariants:
Theorem 1. Let ss :E ! B in S* and n m. Then we have the following series of
inequalities:
catG (ss) Qm catG (ss) QncatG (ss) P ncatG (ss) QcatG (ss) McatG (ss) e*
*M (ss) :
For rational spaces all the invariants of Theorem 1, except the Toomer invari*
*ant,
coincide. In the last section we will give examples of spaces which show that *
*all
the inequalities can be strict except possibly QncatG P ncatG . The inequalit*
*ies in
Theorem 1 result from the existence of natural transformations between the rela*
*ted
functors.
The functions P ncatG and QcatG can be compared with stabilized var*
*iants of
Lusternik-Schnirelmann category studied in [Rud99 ], [Str00], [Van98 ], [Van99 *
*].
Definition 3. Given ss :E ! B in S*. Let oeicatG (ss) be the least integer n (*
*or 1)
such that iGn(E; ss) ! iB admits a right homotopy inverse. For simplicity we sh*
*all
write oecatG for oe1 catG .
From the adjunction formula between i and i it follows that oeicatG (ss)=Qi[c*
*atG (ss)
and therefore we can state:
Corollary 1. Let (ss :E ! B) 2 S*. Then one has QcatG (ss) oecatG (ss) *
*and
QicatG (ss) oeicatG (ss).
0.3. The invariants and cartesian products. It was a question of Ganea [Gan71 ]
called the Ganea conjecture whether the equality cat(Y x Sr) = cat(Y ) + 1 hold*
*s for
Y connected and r 1. By a result of N. Iwase [Iwa98] this is not always true*
*. It
is true however that oecat(Y x Sr) = oecat(Y ) + 1 by [Rud99 ], [Van99 ]. For *
*rational
simply connected spaces Y , Z of finite type over the rationals the general fo*
*rmula
cat(Y x Z) = cat(Y ) + cat(Z) holds [FHL98 ] (cf [Jes90] and [Hes91] for Z = Sr*
*).
About our invariants, in particular about Qcat, we can state the following:
FIBREWISE CONSTRUCTION APPLIED TO LUSTERNIK-SCHNIRELMANN CATEGORY 5
Theorem 2. Let Y; Z 2 S*. Then for = Q, P n, Qn, M or for a localization
functor Lf we have
cat(Y x Z) cat(Y ) + cat(Z) :
Moreover the corresponding inequality holds also for [cat.
Remark . The equality Qcat(X xSr) = Qcat(X)+1 is true if Qcat(X) = oecat(X). F*
*or
then Qcat(X x Sr) Qcat(X) + 1 = oecat(X) + 1 = oecat(X x Sr) and Qcat(X x Sr) =
oecat(X x Sr) = Qcat(X) + 1.
In view of the Ganea conjecture we have in particular:
Problem 1. Does Qcat satisfy the analogue of the Ganea conjecture i.e*
*. for X
connected and r 1 does Qcat(X x Sr) = Qcat(X) + 1 hold?
0.4. Hopf invariants. Finally we introduce the notion of Hopf invariants adapte*
*d to
our situation and prove that they determine if cat grows when attaching a cell.*
* We
also apply them to find examples where the invariants cat, Qncat, Qcat, Mcat, o*
*ecat
are different.
Recall that the counter-example of Iwase is a 2-cell complex. In our applica*
*tion
of Hopf invariants to complexes with 2 cells we show that Qcat(X) = cat(X) iff *
*the
Ganea conjecture is true for X. We would like to conjecture that this is a gen*
*eral
pattern. We state it as:
Problem 2. Let X be a finite CW-complex. Are the two following conditions equi*
*valent?
(1) Qcat(X) < cat(X),
(2) there exists r 1 such that cat(X x Sr) = cat(X).
We construct an infinite CW-complex X such that Qcat(X) < cat(X) and cat(XxSr) =
cat(X) + 1 for any r 1. However the implication (2) ) (1) is also open for inf*
*inite
complexes.
We mention finally that, under some restrictions on dimension and connectivit*
*y, a
mapping version of Problem 2 is proved for rational spaces in [Sta98].
The paper is organized as follows. In Section 1 we recall some more propertie*
*s of
Dror Frajoun's fibrewise application of regular functors. We also study the bas*
*epoint
free functor associated to a coaugmented functor S* ! S* and prove Proposition *
*1.
In Section 2 we discuss the case of the functors Q, the abelian group completio*
*n M
of Sp1 and nn. In fact we defer the topological construction of a base point f*
*ree
version of nn to Appendix B. In Section 3 we prove Theorem 2 and in Section 4
we present the theory of Hopf invariants for cat and [cat.
1.Fibrewise application of functors
1.1. Consequences of Dror Farjoun's construction. Let : S ! S be_a regular
coaugmented functor. Let ss :E ! B be a fibration and let E ! (E) over idB be
the construction of Dror Farjoun referred to in the introduction. For the conve*
*nience
of the reader we will describe it in Appendix A. Directly from it we deduce:
Property 1. Let ss :E ! B be a fibration with B connected. Let 1; 2: S ! S
be two regular coaugmented functors and L: 1 ! 2 be a natural transfor*
*mation
compatible_with_the_coaugmentations. Then L induces a natural transformation o*
*ver
B, L : 1 ! 2. As a consequence we have 1catG (ss) 2catG (ss). __
Moreover if L(Y ) is a weak equivalence for any Y 2 S then L(E) *
*is a weak
equivalence and 1catG (ss) = 2catG (ss).
E
6 HANS SCHEERER, DONALD STANLEY, AND DANIEL TANR
1.2. Basepoint free version of : S* ! S*. We now study the relation between
and the base point free functor + :S ! S defined in the introduction. The follo*
*wing
two properties are immediate:
Property 2. Let 1; 2: S* ! S* be two regular coaugmented functors. Let L: 1 ! 2
be a natural transformation compatible with the coaugmentations. Then L induce*
*s a
natural transformation L+ :+1! +2. Moreover if L(X) is a weak equivalence for a*
*ny
X 2 S* then L+ (Y ) is a weak equivalence for any Y 2 S.
In the particular case of the functor + :S ! S Proposition 6 of Ap*
*pendix A
implies:
Property 3. Let : S* ! S* be a regular coaugmented functor and + :S ! S the
associated basepoint free_functor. Let ss :E ! B in S* be a fibration with fibr*
*e F . Then
the homotopy fibre of + (E) ! B is equivalent to the homotopy fibre of (F +) ! *
*(*+)
over *.
If we start from a basepoint free coaugmented functor : S ! S, we may compare
with the associated free construction of the associated functor 0:S* ! S*:
Proposition 2. Let : S ! S be a regular coaugmented functor. Then there exists a
natural transformation ! (0)+ compatible with the coaugmentations.
We now state a sufficient condition for and (+ )0 to be equivalent:
Let : S* ! S* be a regular coaugmented functor with values in the category G
of grouplike spaces (we assume that the base point of a grouplike space is its *
*unit
element). For X 2 S* let * ! (*) ! *(X) ! (X) be the canonical decomposition
of * ! (*) ! (X) into a cofibration-weak equivalence followed by a fib*
*ration
*(X) ! (X).
Denote by X+ ! X the canonical map of S* and by ^Fthe pullback of *(X) ! (X)
and (X+) ! (X). The universal property of pullbacks gives a factorizat*
*ion of
(*+) ! (X+):
(*+_)_________________________________________________*
*______________________________________________________________________@
___________________________________________________*
*______________________________________________________________________@
____oe1___________________________________________*
*______________________________________________________________________@
________$$________________________________________*
*______________________##_____________________oe2
________________________________^//_(X+ )
_______________________F
________________________________||
________________________________________||
_##_________________________________________*
*__fflffl|fflffl|
*(X) _____//_(X)
Since the homotopy fibre F of (X+) ! (X) over * is equivalent to F^, we can view
oe1 as a map into F .
Proposition 3. Using the notation above suppose that oe1: (*+) ! F is a*
* weak
equivalence and ss0((X+)) ! ss0((X)) is surjective. Then the composite + (X) !
(X+) ! (X) is a weak equivalence.
Proof of Proposition 2.In the following square * ! (*) is a weak equivalence and
a cofibration and * ! (*+) a fibration. Therefore there exists the dashed arr*
*ow
making the diagram commutative:
*________//*::___
| ____|_____
| ______|_
fflffl|___fflffl|_
(*) ____//_(*+)
The result follows from the definition of (0)+ as a pullback and the existence*
*_of a
factorization of the composite (X) ! (X+) ! (*+) as (X) ! (*) ! (*+). |__|
FIBREWISE CONSTRUCTION APPLIED TO LUSTERNIK-SCHNIRELMANN CATEGORY 7
Proof of Proposition 3.First we look at the different base points. The*
* universal
property of pullbacks gives a factorization of some canonical maps:
*___________________________________________________*
*______________________________________________________________________@
____________________________________________________*
*______________________________________________________________________@
___________________________________________________*
*_______________________________
______""_____________________________##____________*
*________________
_____________________j/+/_
______________________(X(X)+)
________________________
_________________________||
____________________||
__##___________________fflffl|fflffl|*
______//_(*+)
Therefore + (X) 2 S* and j(*) = *. Note also that the canonical map (X+ ! X) 2 *
*S*
induces ((X+) ! (X)) 2 G, with neutral element + (resp. *) in (X+) (resp.
(X)).
The map ((X+) ! (*+)) 2 G admits a section up to homotopy oe = oe2O oe1 which
gives a homotopy equivalence ': (*+)x+ (X) ! (X+), (ff; fi) 7! oe2(oe1(ff)):*-1*
*:j(fi).
The result follows now from the five lemma applied to the following morphism of
homotopical fibrations:
oe2
FO__________//(X+)O_________//(X)OOOO
oe1|| |'| ||
| | |
(*+) ____//_(*+) x + (X)____//_+ (X)
*
*__
with (*+) ! (*+) x + (X), ff 7! (ff; *). |*
*__|
We end this section with the
Proof of Proposition 1.This follows directly from the following diagram
2(Gn(E;8ss)))____//(Gn(E;8ss))
(s) qqqq
qqqq 2(qn)|| |(qn)|
qqq fflffl| fflffl|
(B) _(_____//_2(B)__________//_(B) :
(B))
The left triangle homotopy commutes because (qn) O s ' (B) and , preserving *
* __
weak equivalences, preserves the homotopy relation. *
* |__|
2.Specific constructions: nn, 1 1 , Sp1
Example 1. The functor Q = 1 1 satisfies the assumptions of Proposition 3. *
*In
fact let X 2 S*. The homotopy groups of Q(X) constitute a reduced ho*
*mology
theory. From the cofibration sequence (*+) ! (X+) ! X (which admits a retraction
(X+) ! (*+)) we deduce that the homotopy sequence of Q(*+) ! Q(X+) ! Q(X)
decomposes into split short exact sequences. Therefore Q+ (X) ! Q(X) is a homot*
*opy
equivalence. We also note that this statement is a particular case of [BE74 , C*
*orollary
7.4].
Example 2. Let R be a commutative ring with unit 1. For X 2 S denote by R X
the free R-module generated by X. If X 2 S* we define MR (X) := R X=R *. For
R = Z we obtain in particular M(X). Proposition 3 applies to MR andPM+ (X) is
the simplicialPset with n-simplices the finite linear combinations rioei of *
*n-simplices
of X with ri = 1. This base point free version of MR (X) occurs for example*
* in
[BK72 ]. If X 2 S* is connected M(X) coincides with the infinite symmetric pro*
*duct
Sp1 (X).
E
8 HANS SCHEERER, DONALD STANLEY, AND DANIEL TANR
Example 3. For the construction of a basepoint free version Qn of nn: S* ! S**
* we
refer to Appendix B. We remark that the basepoint free construction P n= (nn)+
is not homotopically equivalent to Qn.
Proof of Theorem 1.Property 1 is the key point for the comparison betw*
*een two
invariants: we need only to exhibit natural transformations compatible *
* with the
coaugmentations.
o There is a natural transformation Q ! MZ compatible with the coaugmentations
(see e.g. [CM95 , 7.3]). An easy way to construct it is to use the combinatoria*
*l model
of Barrat and Eccles [BE74 ]. Thus we have QcatG (ss) MZcatG (ss).
o If m n the natural transformation Qm ! Qn in T op gives Qm catG*
* (ss)
QncatG (ss), cf Appendix B.
o From Proposition 2 we get a natural transformation Qn ! ((Qn)0)+ . By Ap-
pendix B there is a natural transformation (Qn)0 ! nn thus composing *
*Qn !
((Qn)0)+ ! (nn)+ provides a natural transformation Qn ! P n.
o The inequality P ncatG (ss) QcatG (ss) comes from the natural tran*
*sformation
nn ! 1 1 .
o The existence of a homotopical section to M(qn) could be chosen as a defini*
*tion
for Toomer's invariant. Therefore Mcat(X) eM (X) is a direct conseque*
*nce of
Proposition 1. *
* __
o The remaining inequality catG (ss) QncatG (ss) is obvious. *
* |__|
3.Product formulae
The proof of Theorem 2 will be based on the following result of *
*[SS99]: Let
: S ! S be a regular coaugmented functor. Suppose there is a natural transforma*
*tion
(Y )x(Z) ! (Y xZ) which is compatible with the coaugmentations then the product
formula cat(Y x Z) cat(Y ) + cat(Z) holds. The corresponding formula for [cat
follows even more easily.
Proposition 4. Suppose that : S ! S is coaugmented by : id ! . Assume that
there are natural transformations (Y ) x Z ! (Y x Z) and m: 2 ! which are
compatible with the coaugmentations.
Then there is a natural transformation (Y ) x (Z) ! (Y x Z).
Proof.The transformation consists of the composition (Y ) x (Z) ! (Y x (Z))_!_
2(Y x Z) ! (Y x Z). |__|
Remark . Usually m will have the property that m O (Y ) = id(Y ) for Y 2 S.
Corollary 2. Let : S* ! S* be coaugmented such that there exist natural transfo*
*rma-
tions (X) x X0! (X x X0) and 2(X) ! (X) compatible with the coaugmentations.
Then there is a natural transformation + (Y ) x + (Z) ! + (Y x Z) of functors
S x S ! S.
Proof.Let __(Y ) = (Y +) for Y 2 S. Then it suffices to show that __ satisfies*
* the
assumptions on of the Proposition.
(a) __(Y ) x Z = (Y +) x Z ! (Y +) x (Z+) ! ((Y +) x (Z+)) ! ((Y x Z)+) =
__(Y x Z). The last arrow is induced by the canonical map (Y +) x (Z+) ! (Y x Z*
*)+.
(b) __2(Y ) = (((Y +))+) ! ((Y +)) ! (Y +) = __(Y ). The first arrow is induc*
*ed
by the map ((Y +))+ ! (Y +) which is the identity on (Y +) and maps + to the __
base point + 2 (Y +). |__|
Proof of Theorem 2.We need only to observe from Proposition 7 of Appendix B that
the basepoint free versions of Q and nn satisfy the assumptions of Proposition *
*4.
The combinatorial models for 1 1 of [BE74 ] and n for nn of [Smi89 ] are
convenient too. It has been shown directly in [BE74 ] that in particular satis*
*fies the
FIBREWISE CONSTRUCTION APPLIED TO LUSTERNIK-SCHNIRELMANN CATEGORY 9
assumptions of Corollary 2. A close look at the combinatorial details shows tha*
*t this
is also true for n . Thus the functor P nsatisfies the conditions of Propositi*
*on 4.
For = nn we can also argue topologically. The second transformation needed in
Corollary 2 exists for but -may be- not the first one. However we show that __*
*admits
a natural transformation __(Y ) x Z ! __(Y x Z) compatible with the coaugmentat*
*ions.
It follows that + (hence P n) is a functor as in Proposition 4.
To give the required formula we write n(Y +) = Sn ^ (Y +) = Sn o Y wh*
*ere
o: S* x S ! S* is the halfsmash. Then we have n((Y x Z)+) = Sn o (Y x Z) =
(Sn oY )oZ. We define : nn(Y +)xZ ! nn((Y xZ)+) by (w; z)(t) = [w(t); z]
where w :Sn ! n(Y +), t 2 Sn, and [w(t); z] denotes the classe of (w(t*
*); z) in
(Sn o Y ) o Z.
For the localization functor, Lf, we observe [Far96, Page 21-23] the existenc*
*e of a
natural transformation Lf(Y ) x Z ! Lf(Y x Z) which gives a natural transformat*
*ion
Lf(Y ) x Lf(Z) ! LfLf(Y x Z). The coaugmentation induces a weak equivalen*
*ce_
Lf ! LfLf and we deduce Lfcat(Y xZ) = LfLfcat(Y xZ) Lfcat(Y )+Lfcat(Z). |__|
4. Hopf invariants
Let X 2 S* and ff: Sr ! X be a map with cofibre Y = X [ffer+1.
We will characterize the relationship between the different LS-type invarian*
*ts of
X and Y in terms of a homotopy class associated to ff and called a Hopf invari*
*ant.
We use a presentation as in [Iwa98], [Sta99]. For = id this coincides with t*
*he
Berstein-Hilton definition [BH60 ] (see [Van98 , Proposition 3.2.7] for a detai*
*led proof).
In this section we will make no distinction between maps and (pointed) homotopy
classes of maps.
4.1. Definition and Properties. Consider first the adjoint ff]:Sr-1 ! X *
*of ff
whose supension gives a homotopy class ff]:Sr ! X into the first Ganea space
associated to X. By composition with the maps Xn:X ! Gn(X) coming from
the construction of the Ganea fibrations we have maps XnO ff]:Sr ! Gn(X). We
work with the absolute case and the situation described in the introduction bec*
*omes:
Fn(X) __________//_(Fn(X)____________//Fn (X)
in|| || ||
XnOff] |fflffl_(Gn(X))_ fflffl|r_(Gn(X)) fflffl|
Sr __________//Gn(X)__________//(Gn(X)) __________//(Gn(X))
|| ZZ___ YY___ ___55________
|| qX| o__ _(qX )|_oe_ s_________|(qX_)
|| n| ___ n | ______________ | n
|| |fflffl__ fflffl|____________fflffl|_
Sr ______ff____//X_________________X________________//(X)
(X)
Recall that (Gn(X)) = r_(Gn(X)) O _(Gn(X)).
__ __
Definition 4. (1) Suppose that (qXn): (Gn(X)) ! X admits a homotopical section
oe. Then the Hopf-invariant associated to (oe, , ff) is:
i j __
H0oe;(ff) := _(Gn(X)) O XnO ff] - (oe O ff)2 ssr( (Gn(X))) :
(2) Suppose there exists s: X ! (Gn(X)) such that (qXn) O s ' (X). Then the
Hopf-invariant associated to (s, , ff) is:
i j
Hs;(ff) := (Gn(X)) O XnO ff] - (s O ff)2 ssr((Gn(X))) :
Remark . Consider fi :St ! Sr a coH-map (for instance a suspension) and ff: Sr*
* ! X.
Directly from Definition 4 we have Hoe;(ffOfi) = Hoe;(ff)Ofi and Hoe;(ffOfi) = *
*Hoe;(ff)Ofi.
E
10 HANS SCHEERER, DONALD STANLEY, AND DANIEL TANR
__
The element H0oe;(ff) 2 ssr( (Gn(X))) lifts in the fibre as an eleme*
*nt denoted
by Hoe;(ff)_2 ssr((Fn(X))) and there is no indeterminacy in this lifti*
*ng because
(Fn(X)) ! (Gn(X)) induces an injection between homotopy groups. Notice that we
are distinguishing between Hoe; and H0oe;. We do this because though H0oe;alwa*
*ys
determine Hoe;, O H0oe;does not determine O Hoe;. This turns out to be one
source of examples where the invariants we study differ, cf Corollary 3.
We consider the classical Hopf invariant of Berstein-Hilton [BH60 ] as a part*
*icular
case of Hoe;for = id and use, in this case, the notation Hoe(or Hoe). If there*
* is a
unique homotopy class of section we shorten the notation in H (or H).
The LS-category of the skeleton of a CW-complex is always less than or equal *
*to
the LS-category of the total space [Sta99]. This property can be extended to t*
*he
setting of cat as follows:
Theorem 3. Let : S ! S be a regular coaugmented functor preserving k-equivale*
*nces
for any k > 0. Let X be a (k - 1)-connected CW-complex and X(r)be its r-skeleto*
*n.
We suppose k 2 or (k = 1_and n _2)._
For any_section oe of (qXn): (Gn(X)) ! X, n 1, there exists a compatible s*
*ection
oer of (Gn(X(r))) ! X(r). In other words the following diagram commutes:
__ __
(Gn(X(r)))____//_(Gn(X))YY
_____ YY_______________
|_oe___________|_______________________________*
*_____________
|___r__________|__________oe___________________*
*____
fflffl|________fflffl|__________________________
X(r)___________//_X
As a consequence if X is simply connected we have cat(X(r)) cat(X), for any r.
We show now that the Hopf invariant characterizes in a certain way the growth
of the LS-category when a cell is attached to a CW-complex. The following theor*
*em
generalizes results of [BH60 ], [Iwa98], [Sta99] and [Van98 ]:
Theorem 4. Let : S ! S be a regular coaugmented functor and X be a connected
space of associated Ganea fibration qXn:Gn(X) ! X. Consider ff: Sr ! X. Denote
by Y = X [ffer+1 the space X with a cell attached along ff and by ae: X ! Y t*
*he
canonical inclusion. __
1) If there is some homotopy section oe of (qXn) such that Hoe;(*
*ff) = 0 then
cat(Y ) n.
2) We suppose n > 1 or X simply connected. If preserves (r + 1)-equivalenc*
*es,
r > 1 and dim X r then: __
cat(Y ) n iff there exists a homotopy section oe of (qXn) such that Hoe;(ff*
*) = 0:
3) Suppose that is a regular coaugmented functor equipped with a natural tra*
*nsfor-
mation 2 = O ! whose composition with ( ) is equal to the identity ! 2 ! .
If there exists s: X ! (Gn(X)) such that (qXn) O s ' (X) and Hs;(ff) = 0 then
[cat(Y ) n.
The hypothesis required on in the statements 2) and 3) are satisfied by the
functors Qn, P n, Q, M = Sp1 .
Suppose there exists a natural transformation L: 1 ! 2 compatible with*
* the
coaugmentations_between two regular coaugmented functors._ If oe1 is a homotopi*
*cal
section of 1(qXn) we define a homotopical section of 2(qXn) by oe2 := L(Gn(X)*
*) O oe1
and we have H0oe2;2= L(Gn(X)) O H0oe1;1. We may also define a lifting *
* s1 from
oe1_ and the Hopf invariant Hs1;1 is obtained from H0oe1;1by compositio*
*n with
1(Gn(X)) ! 1(Gn(X)). These considerations and Theorem 4 give us directly a
relationship between the different Hopf invariants associated to our functors:
FIBREWISE CONSTRUCTION APPLIED TO LUSTERNIK-SCHNIRELMANN CATEGORY 11
Corollary 3. Let X be a simply connected space of LS-category n with *
* a section
o :X ! Gn(X) to the Ganea fibration qXn. Let ff: Sr ! X and Y = X [ffer+1. Deno*
*te
by Ho(ff) 2 ssr(Fn(X)) and H0o(ff) 2 ssr(Gn(X)) the Hopf invariants associated *
*to (o; ff)
and by Hur the Hurewicz homomorphism. Then we have:
o iHo(ff) = 0 ) Qicat(Y ) n;
o iH0o(ff) = 0 ) oeicat(Y ) n;
o Hur Ho(ff) = 0 ) Mcat(Y ) n;
o Hur H0o(ff) = 0 ) e(Y ) n.
Coming back to the general situation we will prove that Theorem 4 implies:
Corollary 4. Let be a regular coaugmented functor and ff: Sr ! X. Then
cat(X [ffer+1) cat(X) + 1 :
The argument used in the proof of Corollary 4 does not work for *
*[cat. The
relation between e(X [ffer+1) and e(X) remains an open problem. It is not known
for example if attaching a cell can increase e by two.
We present now some particular results used in the proofs:
__ __
Lemma_1. Consider the situation of Theorem 4 and let (Gn(ae)): (Gn(*
*X)) !
(Gn(Y )) and (Gn(ae)): (Gn(X)) ! (Gn(Y )) be the maps induced by ae: X ! Y .
__
(i) If (qXn) admits a homotopy section oe we have:
__ __ 0
(Gn(ae)) O oe O ff ' (Gn(ae)) O Hoe;(ff) :
(ii) If s exists we have:
(Gn(ae)) O s O ff ' (Gn(ae)) O Hs;(ff) :
Lemma 2. Let B be a (k - 1)-connected CW-complex of dimension r. Consider
the cofibration _JSr ! B ! C = B [J er+1. Let k 2 or (k = 1 and n 2). Then
the map B ! C induces an (r + 1)-equivalence Fn(B) ! Fn(C) between the fibres of
Ganea fibrations.
ae
Lemma 3. Let Sr _ff_//_B___//_C = B [ffer+1be a cofibration and p: Y ! C be a
map such that ssr+1(p) is surjective. Let ': B ! Y be a map such that ' O ff '*
* * and
p O ' ' ae. Then there exists oe :C ! Y such that oe O ae ' ' and p O oe ' idC*
* .
The end of this section is devoted to proofs beginning with proofs of lemmas.
Proof of Lemma 1. By definition we have:
__ __ h 0 X ]i
(Gn(ae)) O oe O ff = (Gn(ae)) O Hoe;(ff) + _(Gn(X)) O n O ff :
The required equality follows from
__ X ] Y ]
(Gn(ae)) O _(Gn(X)) O n O ff ' _(Gn(Y )) O n O ae O ff ' * :
*
* __
The verification of (ii) is similar. *
* |__|
Proof of Lemma 2. Observe that the fibre Fn(B) (resp. Fn(C)) having the homotopy
type of *n+1B (resp. *n+1C) implies that it is ((n + 1)k - 2)-connected. With
the assumptions on k and n Fn(B) and Fn(C) are simply connected. An homology
argument shows that the induced map Fn(B) ! Fn(C) is an (nk + r - 1)-equivalenc*
*e__
and thus an (r + 1)-equivalence. *
*|__|
E
12 HANS SCHEERER, DONALD STANLEY, AND DANIEL TANR
Proof of Lemma 3. The map p induces a morphism between the two following long
exact sequences coming from the cofibration Sr ! B ! C:
_____//[Sr+1; Y_]_//_[C;_Y_]//_[B; Y_]_//_
| | |
| | |
fflffl| fflffl| fflffl|
_____//[Sr+1; C]__//_[C;_C]_//_[B; C]__//_
From ' O ff ' * we deduce the existence of :C ! Y such that O ae ' '. The
elements p O and idC of [C; C] satisfy p O O ae ' id O ae. By a theorem of *
*D. Puppe
[Hil67, Theorem 15.4] there exists 02 [Sr+1; C] such that (pO )0 ' idC where (p*
*O )0
denotes the cooperation of 0 on p O induced by the cofibration.
By hypothesis there exists 2 [Sr+1; Y ] such that 0' p O . Set oe = . Then*
* we __
have p O oe = p O ( ) ' (p O )pO ' idC . *
* |__|
Proof of Theorem 3.Denote by ir: X(r)! X and i0r:X(r-1)! X(r) the canonical
inclusions_and by qXn;r:Gn(X(r)) ! X(r)the Ganea fibration. Let_oe be any_secti*
*on_of
(qXn). The map ir induces a morphism of fibrations between (qXn;r) and (qXn)*
* which
is an r-equivalence between the bases and an (r + 1)-equivalence between the fi*
*bres
(by Lemma 2 and the hypothesis on ). Also the Ganea fibrations_split after loop*
*ing.
So with the homotopy long exact sequences we deduce that (ir) is an r-equivale*
*nce.
Therefore there exists __oesuch that in the following diagram
__ _(ir) __
(Gn(X(r)))__________//(Gn(X))YYYY
_(qX | ___________________|____________________*
*_____________________________
n;r)|oe____________(qXn)|_______________oe__*
*_________________________
fflffl|______________fflffl|______________*
*________
X(r-1) _____i0r_____//_X(r)_____ir_______//_X
__ __ __ __
(ir) O oe' oe O ir and (qXn;r) O oeO i0r' i0r. We need a refinement of Lemma*
* 3 applied
to_the cofibration _Sr-1 ! X(r-1)! X(r): the element 0 2 [_Sr; X(r)] such that
( (qXn;r) O __oe)0 ' id can be choosen in the kernel of [_Sr; X(r)] ! [_Sr; X] *
*(cf the
construction of m in the proof of Theorem 15.4 of [Hil67]). Now in the followi*
*ng
commutative diagram of abelian groups the vertical maps admit sections*
* and the
horizontal maps are surjective:
__ ssr(_(ir)) __
ssr( (Gn(X(r))))_________//ssr( (Gn(X)))
ssr(_(qXn;r))|| |ssr(_(qXn))|
fflffl| fflffl|
ssr(X(r))_____ssr(ir)____//ssr(X)
Therefore_with_0 in the kernel_of_[_Sr; X(r)] ! [_Sr;_X]_we can choose in the *
*kernel
of [_Sr; (Gn(X(r)))] ! [_Sr; (Gn(X))] such that 0= (qXn;r) O . As_in the pro*
*of of
Lemma 3 we check that the map oer := __oeis a homotopy section of (qXn;r). Mor*
*eover
__ __ __ _ __ __ *
* __
we have: (ir) O oer ' ( (ir) O oe) (ir)O' (ir) O oe' oe O ir. *
* |__|
FIBREWISE CONSTRUCTION APPLIED TO LUSTERNIK-SCHNIRELMANN CATEGORY 13
__
Proof of Theorem 4.Suppose that (qXn) admits a section oe. By applica*
*tion of
Lemma 1 (i) we get a commutative diagram:
__
6(Gn(X))M6_
H0oe;mmmmmm MMM(Gn(ae))M
mmmmm MMM
mmmmm MM&&_
Sr _ff_//_X_______(G________//_(Gn(Y ))
n(ae))Ooe _
ae|| ||(qYn)
fflffl| fflffl|
Y _______________________Y
__
__1) If H0oe;' * we apply Lemma_3 to construct a map oe0:Y ! (Gn(Y )) such that
(Gn(ae)) O oe ' oe0ae and (qYn) O oe0' idY . By definition we have cat(Y ) n.
__ __
2) Let oe0:Y ! _(Gn(Y_)) be a section of (qYn)._ By Theorem 3 there exists a
section oe of (qXn)__such that oe0O ae ' (Gn(ae)) O oe. From the diag*
*ram above we
deduce immediatly that (Gn(ae)) O H0oe;'_*. This implies that (Fn(ae)) O Hoe;'*
* * by
injectivity of ssr((Fn(Y ))) ! ssr( (Gn(Y ))) and that Hoe;' * by Lemma 2 and t*
*he
hypothesis on .
3) Set "ff:= (X) O ff: Sr ! (X) and __ff:= s O ff: Sr ! (Gn(X)). Note *
*that
(pXn) O __ff= "ff. From naturality of we have (ae) O "ff' * and we deduce fr*
*om
Lemma 1-(ii) that (Gn(ae)) O __ff' *. The universal property of pushout*
*s gives a
homotopy commutative diagram (without the dashed arrow):
___ff__//_____________// r+1____________
Sr X X [ffe ____________Y
||
|| (X) | | | (Y )
|| | | |
|| "ff fflffl| fflffl| fflffl|
Sr ______//_(X)________//_(X) ["ffer+1______//_(Y )
|| OO OO___ OO
|| (qX )| "q| "s___ |(qY)
|| n | n|__ n | n
|| | | _ |
Sr __ff//_(Gn(X))____//_(Gn(X)) [__ffer+1_//_(Gn(Y ))
where (Gn(X)) ! (Gn(X)) [__ffer+1 ! (Gn(Y )) is equal to (Gn(ae)) and (X) !
(X) ["ffer+1 ! (Y ) is equal to (ae).
From the hypothesis on and Proposition 1 one has a homotopical section__snto
(qXn); a look to its construction gives _snO __ff' "ff. Denote by _snand "qnth*
*e maps
induced by _snand (qXn) between the cofibres. The_map ' induced by (qXn) O _sn'*
* id
is a homotopy equivalence [Qui67]. By composing _snwith '-1 we get a homotopical
section "snof "qn. The required homotopy lifting of Y ! (Y ) through (qYn) is *
*the
following composite:
X [ffer+1_____//(X) ["ffer+1"sn//_(Gn(X)) [__ffer+1_//_(Gn(Y )):
__
|__|
Proof of Corollary 4.The triviality of the induced map Fn(Y ) ! Fn+1(Y ) implie*
*s the
triviality of (Fn(Y )) ! (Fn+1(Y )) and the image of the Hopf invariant Hoe;(ff*
*) in
ss*((Fn+1(Y ))) is zero. As in the beginning of the proof of Theorem 4 we const*
*ruct
a dashed arrow making commutative
__
X ____//_(Gn+1(Y9))9_
____
| ______|(qY_ )
| ________| n+1
fflffl|_____fflffl|_
Y __________Y
E
14 HANS SCHEERER, DONALD STANLEY, AND DANIEL TANR
__
In other words cat(Y ) n + 1. |_*
*_|
4.2. Examples. We come back to the chain of inequalities of Theorem 1 and exhib*
*it
examples of spaces for which a strict inequality occurs (except for P n and Qn)*
*. For
this we will apply Corollary 3.
Example 4. We use the notation and results of [Tod62 , Proposition 13.9 page *
*179].
The composite fi := ff1(3) O ff1(2p): S4p-3 ! S2p ! S3 is a generator of ss4p-3*
*(S3) = Zp
such that fi 6' * and 2fi ' *. Denote by w :S4 ! S3 _ S2 the Whitehead
bracket of the classes S3 and S2 and by fl := w O fi :S4p-2 ! S4 ! S3 _ S2. Set
X = S3 _ S2 [fle4p-1. Then we claim Q1cat(X) = 1 and cat(X) = 2 (cf also [Sta9*
*8]
for a different proof of cat(X) = 2).
The Hopf invariant of fl satisfies H(fl) = H(wOfi) = H(w)Ofi. Therefore H(fl)*
* ' *
and Q1cat(X) = 1 by Corollary 3. We are now reduced to proving that H(fl) is not
trivial. Denote by f] the adjoint of a map f and observe that H(fl)] = H(w)]O*
* fi.
The non-triviality of H(fl) is a consequence of the following lemma. It is cer*
*tainly
well known but we cannot find it in the litterature.
Lemma 4. Let wi;j:Si+j-1 ! Si _ Sj be the Whitehead bracket of the ca*
*nonical
inclusions ji:Si ,! Si_ Sj, jj: Sj ,! Si_ Sj. Denote by Fi;jthe homotopy fibre*
* of
the first Ganea fibration associated to Si_ Sj. The Hopf invariant associated *
*to wi;j
has for adjoint a map H(wi;j)]:Si+j-2! Fi;j.
Then there exists a map _p:Fi;j! Si+j-1 such that the adjoint of _pO H(wi;j)]*
* is
a map of degree 1: Si+j-1! Si+j-1.
Proof.First we use the construction of the Hopf invariant as in [BH60 ] which c*
*orre-
sponds to ours as mentioned before. Denote by Pi;j:Si_ Sj ! (Si1_ Sj1) _ (Si2_*
* Sj2)
the pinch map where Sik(resp. Sjk) is a copy of Si (resp. Sj) for k = 1; 2. We *
*have
Pi;j(ji) = ji1+ ji2and Pi;j(jj) = jj1+ jj2and therefore Pi;j([ji; jj]) = [ji1+ *
*ji2; jj1+ jj2].
By definition the Hopf invariant consists in cutting off the part coming from t*
*he
product (Si1_ Sj1) x (Si2_ Sj2) which corresponds here to [ji1; jj1] + [ji2; jj*
*2]. That is we
have: H(wi;j)]= [ji1; jj2] + [jj1; ji2].
Now recall that the Hilton-Milnor theorem [Whi78Q, Page 515] gives a homotopy
equivalence between (Si-11_Sj-11_Si-11_Sj-11) and k !k(Si-11; Si-12; Sj-11; S*
*j-12)
where !k(Si-11; Si-12; Sj-11; Sj-12) ranges over all admissible words. In part*
*icular we
have the admissible word [Si-11; Sj-12] corresponding to a sphere Si-11^ Sj-12~*
*=Si+j-2.
Denote by _p:Fi;j! Si+j-1 the associated projection. Then the composite _pOH(wi*
*;j)_
has for adjoint a map of degree 1: Si+j-1! Si+j-1. |_*
*_|
Example 5. Let fi :So ! S3 such that 2nfi 6' * and 2n+1fi ' * [Gra84 , Theorem
12] or [Sta99, Corollary 9.2]. Denote by w :S4 ! S3 _ S2 the Whitehead bracket*
* of
the classes S3 and S2 and let fl := w O fi. Set X = (S3 _ S2) [fleo+2. The me*
*thod
used in Example 4 gives Q2ncat(X) = 1 and Q2n-1cat(X) = 2.
The existence of fi :So ! S4 such that 2n-1fi 6' * and 2nfi ' * allows with
the same process the construction of a space X = (S3 _ S3) [fleo+2 suc*
*h that
Q2n-1cat(X) = 1 and Q2n-2cat(X) = 2.
We remark that the example of Iwase [Iwa98] is a space X satisfying 2 = cat(X*
*) =
Q1cat(X) > Q2cat(X) = 1.
Example 6. For any n 1 we denote by X(n) a CW-complex which satisfies, as
in Example 5, Qn-1cat(X(n)) > Qncat(X(n)) (by convention: Q0cat = cat). *
*Set
Y = _n1 X(n) and observe that Y (resp. Y x Sr) dominates X(n) (resp. X(n) x Sr*
*).
We deduce from Corollary 3 and from [Iwa97] that Y is an infinite CW-complex s*
*uch
FIBREWISE CONSTRUCTION APPLIED TO LUSTERNIK-SCHNIRELMANN CATEGORY 15
that Qcat(Y ) < Qn-1cat(Y ) for any n 1 and cat(Y x Sr) = cat(Y ) + 1 for any *
*r 1.
This justifies the restriction to a finite complex in Problem 2.
Example 7. Denote by ff1(3) 2 ss2p(S3) a generator of the p-component *
* and by
w :S4 ! S2 _ S3 the Whitehead bracket. We deduce from Lemma 4 that QH(w O
ff1(3)) 6' * and HurH(w O ff1(3)) ' *. Therefore the space X = (S2 _ S3) [wOff1*
*(3)
e2p+2 satisfies Qcat(X) = 2 and Mcat(X) = 1.
We address now the relation between oecat and Qcat.
Example 8. (The Lemaire-Sigrist example revisited.) Denote by w :S5 ! CP2 the
attaching map of the top cell of CP3 and by fl :S6 ! CP2 _ S2 the Whi*
*tehead
bracket of w and S2. Set Z = (CP2 _ S2) [fle7. We claim that Qcat(Z) = 3 and
oecat(Z) = oe1cat(Z) = e(Z) = 2.
Observe that the rationalized space Z0 satisfies cat(Z0) = Qcat(Z0) = *
*3 and
oecat(Z0) = e(Z0) = 2, [LS81 ]. We deduce that 3 cat(Z) Qcat(Z) Qcat(Z0) = 3.
Consider the first Ganea space G1(X) associated to X := CP2 _ S2. Fr*
*om the
decomposition (CP2) = S1 x (S5) and standard properties of and we see that
G1(X) is a wedge of spheres. Among them we have S2(1)corresponding to a generat*
*or
of ss2(CP2) = Z, S5 corresponding to a generator of ss5(CP2) = Z and S2. So we *
*have
a homotopy equivalence G1(X) ' S2(1)_ S5 _ S2 _ _iSni.
For any section o of qX2 we have (o O fl) ' *. We look now at fl]:S6 ! G1(X)
and remark that fl] = iiSni + where is decomposable and i 2 Z. The
fact that (S2(1)_ S5 _ S2) ! (X) has a homotopy section implies that Sni+ xi, xi
decomposable in S2(1)_S5_S2, is in the kernel of ss*(qX2). Therefore in the con*
*struction
of the second Ganea space each sphere Sni is killed or identified to some decom*
*posable
elements. In any case it disappears after one suspension. This implies (X2Ofl])*
* ' *.
We have proved that H0o;Q(fl) = ((X2O fl]) - (o O fl)) ' * and so oe1cat(Z) 2 *
*by
Corollary 3. Since 2 = oe1cat(Z0) oe1cat(Z) we get that oe1cat(Z) = 2.
Remark . We note that the notion of n-LS-fibration [ST97 ] does not allow an e*
*fficient
use of Hopf invariants. For instance the fact that idS3: S3 ! S3 is a 1-LS-fib*
*ration
implies that a 1-LS fibration cannot bring a characterization of the category o*
*f S3[ffek.
Proposition 5. For any space with two cells Problem 2 has a positive answer.
Proof.It is a direct consequence of [Iwa97]: __
if X = Sn [' ep then cat(X x Sr) cat(X) iff rH(') = 0. |_*
*_|
Observe that the proof given by Iwase [Iwa97] should certainly be adaptable t*
*o the
invariant Qcat. This would give a positive answer to Problem 1 for 2-cell cones.
Appendix A. Dror Farjoun's construction
In this paragraph we recall a construction from [Far96, Chapter 1.F.2]. Let :*
* S ! S
be a regular coaugmented functor and ss :E ! B in S a fibration. We consider *
*the
simplex category B defined by:
- its objects are pairs ([n]; oe), oe 2 Bn;
- a morphism ff: ([n]; oe) ! ([m]; o) is a simplicial map ff: [n] ! [m] such
that foOff= foewhere foe: [n] ! B is the characteristic map of oe.
E
16 HANS SCHEERER, DONALD STANLEY, AND DANIEL TANR
Denote by B":B ! S the forgetful functor determined by ([n]; oe) 7! [n] and
let E":B ! S be the functor defined by the following pullback
E"([n]; oe)___//_E
| ||
| ss|
fflffl| fflffl|
[n] __f____//B
oe
The projection "E([n]; oe) ! [n] defines a natural transformation "E! B"*
*. The
homotopy colimits (in S) of the functors B", O "B, E" and O "Egive a commutat*
*ive
diagram
hocolim O "Eoo___hocolim "E____//_E
| | |
| | |ss
|fflffl fflffl| fflffl|
hocolim O "Boo___hocolim "B____//_B
__
The functor is constructed _with_ a homotopy pullback-pushout operation*
*: P is
the homotopy pullback (hpb) and (E) the homotopy pushout (hpo) defined in the
following diagram
hocolimO"E_____//_nE
nnnnnn O hpo |
nn |
vvnnn fflfflO __fflffl|
hocolim O "Eoo______P ________//_(E)
| | |
| hpb | |
fflffl| fflffl| fflffl|
hocolim O "Boo___hocolim "B_____//_B
__
This induces a factorization E ! (E) ! B of ss. All diagrams
_~__//_
(E"([m]; o)) (E"([n]; oe))
| |
| |
fflffl| fflffl|
([m]; o) ___~____//([n]; oe)
are homotopy pullbacks. Hence by [Pup74 ] this implies:
Proposition_6._[Far96, Chapter 1, Theorem_F.3] For b 2 B let F be the fibre of*
*_ss
over b and F the homotopy fibre of (E) ! B over b. Then the induced map F ! F
is naturally equivalent to the coaugmentation F ! (F ).
Appendix B. Unpointed version of nn
We now construct an unpointed version Qn: S ! S of nn: S* ! S* where S (resp.
S*) is the convenient category of compactly generated (resp. well pointed comp*
*actly
generated) spaces. For that we recall first the notion of unpointed suspension:
Definition 5. Let I = [0; 1]. The unreduced suspension of Y 2 S is e(Y ) := (Y *
*xI)= ~,
where (y; 0) ~ (y0; 0) and (y; 1) ~ (y0; 1) for any y; y02 Y . By induction we *
*define the
n-unreduced suspension of Y 2 S by fn (Y ) = en]-1 (Y ).
We will number the coordinates from right to left; i.e. an element of fn (Y )*
* is an
equivalence class denoted by [tn; : :;:t1; y]. Observe that we have a canonica*
*l map
jn: @In ! fn(Y ), (tn; : :;:t1) 7! [tn; : :;:t1; y] (y arbitrary).
FIBREWISE CONSTRUCTION APPLIED TO LUSTERNIK-SCHNIRELMANN CATEGORY 17
Definition 6. Given Y 2 S we define Qn(Y ) as the set of maps ! :In ! fn(Y ) su*
*ch
that !|@In= jn. The map c: Y ! Qn(Y ), y 7! c(y), c(y)(tn; : :;:t1) = [tn; : :*
*;:t1; y] is
a coaugmentation.
There are bonding maps bn: Qn ! Qn+1 compatible with the coaugmentations given
by bn(!)(tn+1; : :;:t1) = [tn+1; !(tn; : :;:t1)] for ! 2 Qn(Y ).
Set Q(Y ) := lim!Qn(Y.)
Note that for X 2 S* the canonical map nf (X) ! n(X) (where n(X) is the
reduced suspension) is a relative homeomorphism (fn (X); fn(*)) ! (n(X); *) and*
* that
fn (*) is contractible. Moreover fn (X) ! n(X) induces a map Qn(X) ! nn(X).
Proposition 7. 1) The canonical map Qn(X) ! nn(X) is a homotopy equivalence.
2) For Y; Z 2 S there is a canonical map Qn(Y ) x Z ! Qn(Y x Z) compatible wi*
*th
the coaugmentations.
3) There is a natural transformation m: QnQn ! Qn such that Qn together with c
and m is a triple.
Proof.1) Note that for all ! 2 Qn(X) the restriction of ! to the boundary @In *
*is
equal to the restriction to @In of In ! fn(*) ! fn(X). Thus dividing @In+1 in t*
*wo
halfs along an equator @In we obtain an element in nnf (X) by ! on one half and
the composite In ! fn(*) ! fn(X) on the other half. This gives an equivalence
Qn(X) ! nnf (X). Composing this map with nnf (X) ! nn(X) we obtain the
announced equivalence. Note that it is compatible with the bonding maps.
2) We define j :Qn(Y ) x Z ! Qn(Y x Z) as follows. For ! 2 Qn(Y ) wr*
*ite
!(tn; : :;:t1) = ["tn; : :;:"t1; "y], then j(!; z)(tn; : :;:t1) = ["tn; : :;:"t*
*1; ("y; z)]. This definition
does not depend on the choice of the representative in the class !(tn; : :;:t1)*
* (because
!|@In is the fixed canonical map jn). One checks immediately that th*
*e map is
compatible with the coaugmentations.
3) We define m: QnQn(Y ) ! Qn(Y ) by the following device. Given ! *
*:In !
fn Qn(Y ) write as above !(tn; : :;:t1) = ["tn; : :;:"t1; "!] with "!2 Qn*
*(Y ). Then set
m(!)(tn; : :;:t1) = "!("tn; : :;:"t1). As above this definition does not *
* depend on the
choice of representative ["tn; : :;:"t1; "!]. A calculation shows that we have*
*_obtained_a
triple. |*
*__|
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Mathematisches Institut, Freie Universit"at Berlin, Arnimallee 1-3, 14*
*185 Berlin,
Germany
E-mail address: scheerer@math.fu-berlin.de
Departement de Mathematiques, UMR 8524, Universite de Lille 1, 59655 Villeneu*
*ve
d'Ascq Cedex, France, and, Mathematisches Institut, Freie Universit"at Berlin, *
*Arni-
mallee 1-3, 14185 Berlin, Germany
E-mail address: Don.Stanley@agat.univ-lille1.fr
Departement de Mathematiques, UMR 8524, Universite de Lille 1, 59655 Villeneu*
*ve
d'Ascq Cedex, France
E-mail address: Daniel.Tanre@agat.univ-lille1.fr