On the LusternikSchnirelmann category of maps
Donald Stanley
November 25, 1998
Abstract
We give conditions when cat(f x g) < cat(f) + cat(g). We apply
our result to show that under suitable conditions for rational maps f,
mcat(f) < cat(f) is equivalent to cat(f) = cat(f xidSn). Many exam
ples with mcat(f) < cat(f) satisfying our conditions are constructed.
We also resolve one open case of Ganea's conjecture by constructing
a space X such that cat(X x S1) = cat(X) = 2. In fact for every
Y 6' *, cat(X x Y ) cat(Y ) + 1 < cat(Y ) + cat(X). We show that
this same X has the property cat(X) = cat(X x X) = cl(X x X) = 2.
Finally we give an example of a CW complex Z such that cat(Z) = 2
but every skeleton of Z is of category 1.
1 Introduction
The LusternikSchnirelmann category of a space, cat(X), (Definition
2.11) was introduced in the early 1930's [22], [21]. The category of
a map, cat(f), (Definition 2.12) was first defined by Fox [10] and
seriously studied by Berstein and Ganea [2]. The notion of category
of a map is strictly more general since we have that cat(X) = cat(idX ).
For an overview of the history of LS category we suggest the two survey
articles of James [19] [20].
In this paper we study the relationship between cat(f), cat(g) and
cat(f xg). It is well known that cat(f xg) cat(f)+cat(g). Although
examples where inequality holds have been known for a long time, it
was thought that morally equality should hold. In fact no rational
examples of inequality were known and actually if f and g are identity
maps then Felix, Halperin and Lemaire [8] proved that equality holds.
The counterexample of Iwase [17] to the long standing conjecture of
Ganea that cat(X x Sn) = cat(X) + 1 changes our perspective. We
study the implication of this change on our knowledge of cat(f x g).
We prove:
1
Theorem 1.1 3.4 Let us be given a strictly commutative diagram
A ____i___//_B______//B=A
h g f
fflffl fflfflpn fflffl
Fn(X) ____//_Gn(X)_____//X
where i is a cofibration and the bottom row is the nth Ganea fibration
for X (Definition 2.10). Assume cat(f) = n + 1 and rh ' *. Then
for every map g such that cat(g) r > 0, cat(f x g) n + r. In
particular cat(f) = cat(f x idSr) = n + 1.
Our interest in the theorem is due to two applications. The first area of
application is in rational homotopy theory. In [28] Hans Scheerer and
the author constructed an example of a rational map such that cat(f x
idSn) = cat(f) = 2. The proof was a direct calculation with Sullivan
models. Here we show that many such examples can be constructed;
for every r we construct maps f such that for every n, cat(f xidSn) =
cat(f) = r. The reason for the occurence of such counterexamples
is essentially the same as the reason there are counterexamples to
Ganea's conjecture: the instability of certain Hopf invariants. This
same phenomenon also gives rise to examples of f such that mcat(f) <
cat(f) (see Definition 4.1). In fact we show:
Theorem 1.2 4.9 Let W ____//_X_i_//_Ybe a cofibration sequence
of rational spaces and f : Y ! Z a map of rational spaces . Assume
that cat(f) > cat(fi). Assume dimension(X) 2cat(fi)(connectivityZ+
1)2. Then for any n, mcat(f) < cat(f) if and only if cat(f xidSn) =
cat(f).
The second application is to solve a case of Ganea's conjecture which
was left open by Iwase (see [17] pg.2).
Theorem 1.3 5.1 There exists a space X such that cat(X) = cat(Xx
S1) = 2
In fact we show that for the X of the theorem and every Y 6' *,
cat(X x Y ) cat(Y ) + 1 < cat(Y ) + cat(X). In other words the cat
of a nontrivial product with X is always less than what is "expected"
using the product formula. This same X has another very interesting
property.
2
Theorem 1.4 Let X be the space of Theorem 5.1. Then cat(X) =
cat(X x X) = cl(X x X) = 2.
This is the first example of two plocal spaces whose product has LS
category two less than the sum of their categories.
2 Notation and Background
This section contains some general results and definitions. After fixing
some notation we prove a proposition (2.9) which describes a cone
decomposition of a product in terms of the cone decomposition of the
pieces. Next we define the LS category of spaces and maps (Definitions
2.11 and 2.12). Results which tell us if cat goes up when attaching a
cone are then given (2.19, 2.20). This is determined by Hopf invariants
(Definition 2.18) in the space case and by simple obstruction theory
in the map case.
Let CG* denote the category of pointed compactly generated Haus
dorf spaces. For definitions and basic properties of CG* see [32]. All
of our spaces will be assumed to be in CG*. All homotopies will be
pointed and [X; Y ] denotes pointed homotopy classes of pointed maps.
Except where we specifically say we are working in CG* we will also
assume that all our spaces have the homotopy type of pointed CW
complexes [24]. For our purposes the two categories are compatible
since the Kellification functor (k in Definition 3.1 of [32]) does not
effect CWcomplexes and because none of our constructions produce
nonHausdorff spaces from Hausdorff ones. We choose CG* over the
categories of Vogt [38] because it is more familiar to a greater number
of homotopy theorists. In CG* let x denote the weak product. The
only reason we use CG* instead of all topological spaces is because
we want our results to be general enough to handle products of spaces
which are not locally compact. We also assume that the category we
are working in is our category of spaces. This means that all objects,
maps and diagrams will be of spaces unless otherwise indicated.
For a map f : X ! Y in CG*, we let Y [f CylX denote the
reduced mapping cylinder on f. Explicitly
Y [f CylX = (Y [ X x I)=(* x I = *; (x; 1) = f(x))x2X
We let C(f) denote the reduced cone on f. Explicitly
C(f) = (Y [f CylX)=((x; 0) = *)x2X
3
We call X __f__//Y_g__//Z together with a homeomorphism Z ~=
C(f) compatible with g and the inclusion Y ! C(f) a cofibration
sequence. Often we will not explicitly give the homeomorphism. This
is consistent with standard practice. (For example it is usually ignored
that pushouts are only defined up to isomorphism. This is because
the isomorphism is canonical.) We call F ____//_E_p_//_Ba fibration
sequence if p is a fibration and F = p1(*). Observe that with our
definitions fibration sequences and cofibration sequences are not quite
dual notions.
For convenience we work localized at a prime or rationally. Sn and
Dn refer to localized spheres and disks of dimension n. Let A be an
object in any category.
In any category a diagram C is a functor from some small cat
egory into C. This is sometimes refered to as a strictly commuting
diagram. A diagram that commutes up to homotopy is a functor into
the homotopy category of C.
Lemma 2.1 Let f : X ! Y be a map. Then the following two con
ditions are equivalent:
1) For every W , f induces a surjection [W; f] : [W; X] !
[W; Y ]
2) f has a homotopy section.
Proof: The lemma follows since and are adjoint and preserve
homotopies.
The following lemma will be used a number of times. It's proof is
an application of the coaction. (See [34] for example.)
Lemma 2.2 Let f : X ! Y be a map satisfying the equivalent con
ditions of the last lemma. Let g : U ! A be any map. Let us also be
given a (strictly commutative) solid arrow diagram:
g i
U ____//_A____//C(g)
OEzz
h  z h0
fflffl"fflffl"z
X __f___//Y:
Assume that hg ' *. Then there exists OE making the upper left tri
angle strictly commute and the bottom right triangle commute up to
4
homotopy. In particular if there exists a OE making the upper trian
gle strictly commute then there exists one making the upper triangle
commute and the bottom triangle commute up to homotopy.
Proof: The fact that hg ' * implies there exists OE0: C(g) ! X such
that OE0i = h. Next we use [34] Proposition 2.48 i) and its notation. Let
2 [U; Y ] be a map such that fOE0= h0. (fOE0 denotes the action
of on fOE0 via the coation.) Since [U; f] is surjective there exists
02 [U; Y ] such that f = 0. Then by naturality of the coaction any
representative of OE = 0OE0 2 [C(g); X] makes the diagram commute
up to homotopy.
Lemma 2.3 For any map f : W ! X there is a cofibration se
quence
W ! X _ W ! X [f CylW:
Proof: Looking at I x I we see that there is a cofibration sequence
S1 ! S1 _ S1 ! S1 [idCylS1:
Since smashing with W preserves cofibration sequences we get a cofi
bration sequence
W ! W _ W ! W [idCylW:
The lemma follows by taking a pushout.
Lemma 2.4 Let a homotopy commutative diagram
f
W ____//_X___//_C(f)
h k g
fflffljfflfflp fflffl
F _____//_E_____//B
be given. Assume that the bottom row is a fibration sequence and that
p splits. Then there exists a (strictly commutative) diagram
W __i__//X [f CylW ____//_C(f) (+)
h k[H g
fflfflj fflfflp fflffl
F ___________//E___________//_B
where i is the inclusion into the free end of the cylinder and H : jh '
kf is a homotopy.
5
Proof: Start with the diagram (+) with any H. Then the left square
stricly commutes but we know nothing about the right square. Since
p splits we can use Lemma 2.2 on the cofibration sequence
W ! X _ W ! X [f CylW
which exists by Lemma 2.3. This gives us a diagram (+) in which the
left square strictly commutes and the right square commutes up to a
homotopy that fixes X _ W . Since p is a fibration and X _ W !
X [f CylW is a cofibration we can adjust H not changing the ends
of the cylinder so that both squares in diagram (+) commute exactly.
The next three lemmas are preparation for Proposition 2.9.
Lemma 2.5 Consider the following diagram in any category.
A _____//C____//E
  
  
fflfflfflfflfflffl
B ____//_D____//F
Assume that the left hand square is a pushout. Then the right hand
square is a pushout if and only if the outside rectangle is a pushout.
Proof: Follows directly from the definition of pushout.
Lemma 2.6 In CG* let the following diagram be a pushout.
A ____//_B
 
 
fflfflfflffl
C ____//_D
Then for every X
A x X ____//_B x X
 
 
fflffl fflffl
C x X ____//_D x X
is also a pushout.
Proof: [32].
6
Definition 2.7 For i = 1; 2, let f(i) : A(i) ! B(i) be maps in CG*.
Then define (Cf(1) x Cf(2))o by letting the following diagram be a
pushout.
B(1) x B(2)_______//B(1) x Cf(2)
 
 
fflffl fflffl
Cf(1) x B(2) ____//_(Cf(1) x Cf(2))o
Notice that when B(i) = A(i) and f(i) = id then we get (Cf(1) x
Cf(2))o = A(1) * A(2), the join of the A(i).
Lemma 2.8 Let A(i) 2 CG*. Then there exists a homeomorphism OE
such that
j0
A(1) * A(2)_____//C(A(1) * A(2))
QQQQ 
QQQQ OE
j QQQQ((Q fflffl
CA(1) x CA(2)
commutes. Where the two maps j and j0 in the diagram are the usual
inclusions. OE is natural in the A(i). In other words for i = 1; 2, given
maps f(i) : A(i) ! B(i) in CG* the following diagram commutes.
C(f(1))*C(f(2))
C(A(1) * A(2))______________//C(B(1) * B(2))
OE OE
fflffl C(f(1))xC(f(2)) fflffl
C(A(1)) x C(A(2)) __________//_C(B(1)) x C(B(2))
Proof:
C(A(1) * A(2)) = {(a(1); a(2); s; t)a(i) 2 A(i); s; t 2 [0; 1]}
[{(a(1); a(2); s0; t)a(i) 2 A(i); s0; t 2 [0; 1]}= ~
Where ~ is some equivalence relation. In particular (a(1); a(2); s; t) ~
(a(1); a(2); s0; t) if s = s0= 0. Also
CA(1) x CA(2) = {(a(1); a(2); t(1); t(2))a(i) 2 A(i); t(i) 2 [0; 1]}= ~0
Where ~0is some other equivalence relation. We then define OE to be
the map induced by
(a(1); a(2); s; t) 7! (a(1); a(2); t + s(1  t); t)
7
and
(a(1); a(2); s0; t) 7! (a(1); a(2); t; t + s0(1  t)):
It is straightforward to check that OE is compatible with ~ and ~0and
is a homeomorphism. The naturality of OE is clear from the definition.
If B(i) ' * then the following proposition is well known. Our
proposition is stronger than that of Baues [1] since we have homeo
morphisms where he has homotopy equivalences.
Proposition 2.9 For i = 1; 2 let f(i) : A(i) ! B(i) be maps in CG*.
Then there is a cofibration sequence
A(1) * A(2) ! (C(f(1)) x C(f(2)))o ! C(f(1)) x C(f(2)):
This sequence is natural in both variables. In other words if for i = 1; 2
we have diagrams
f(i)
A(i) _____//B(i)
g(i) 
fflfflf0(fflffli)
A0(i)_____//B0(i)
then we get a diagram
A(1) * A(2)______//(C(f(1)) x C(f(2)))o___//_C(f(1)) x C(f(2))
  
  
fflffl fflffl fflffl
A0(1) * A0(2)___//_(C(f0(1)) x C(f0(2)))o_//C(f0(1)) x C(f0(2))
where the maps C(f(i)) ! C(f0(i)) are the canonical extensions over
the cone induced by g x id : A(i) x I ! A0(i) x I.
Proof: In the following diagram
A(1) x B(2)______//B(1) x B(2)______//B(1) x Cf(2)
  
  
fflffl fflffl fflffl
CA(1) x B(2) _____//Cf(1) x B(2)___//_(Cf(1) x Cf(2))o
the left hand square is a pushout by Lemma 2.6 and the right hand
square is by defintion. Therefore Lemma 2.5 implies that the outside
square is a pushout.
8
Next consider the diagram
A(1) x B(2)________//A(1) x Cf(2)_______//_B(1) x Cf(2)
  
  
fflffl fflffl fflffl
CA(1) x B(2) _____//(CA(1) x Cf(2))o___//_(Cf(1) x Cf(2))o
 
 
fflffl fflffl
CA(1) x Cf(2) ______//_Cf(1) x Cf(2):
The upper left square is a pushout by definition and we have just
seen that the upper rectangle is a pushout. Therefore the upper right
square is a pushout. Also the right rectangle is a pushout by Lemma
2.6. Therefore the bottom right square is a pushout. Using the same
arguement again in the second variable we see that:
(CA(1) x CA(2))o ____//_(Cf(1) x Cf(2))o (*)
j 
fflffl fflffl
CA(1) x CA(2) _______//Cf(1) x Cf(2)
is a pushout. But (CA(1) x CA(2))o = A(1) * A(2) and by Lemma 2.8
j0= OE1j. So replacing the map j by j0: A(1)*A(2) ! C(A(1)*A(2))
in diagram (*) gives the same pushout. This is the first statement of
the lemma. Naturality follows from the naturality of Lemma 2.8.
Observe that we get the pushout (*) in any category where Lemma
2.6 holds. Therefore the proposition will hold in many model cate
gories with monoidal structures.
We define a sequence of spaces using the fibrecofibre construction
of Ganea [11]. In this case the spaces coincide,up to homotopy, with
the stages En(X) of Milnor's classifying space construction for X.
The spaces are used to define category.
Definition 2.10 Let X be a 0connected space. We define fibration
sequences.
pn
Fn(X) _in_//_Gn(X)____//X
Let G00(X) = * and p00the inclusion. Let G0n(X) ____//_Gn(X)pn__//_X
be a (functorial) factorization of p0ninto an acyclic cofibration followed
by a fibration. (This is also refered to as turning p0ninto a fibration.)
9
Let Fn(X) = F ib(pn) and G0n+1(X) = C(in). We get the extension
p0n+1by mapping Fn x I to *. Gn(X) is often refered to as the nth
Ganea space and pn as the nth Ganea fibration.
Notice that the fact that we are using a functorial factorization (as
we get from the standard construction of turning a map into a fibra
tion) means that the above construction is functorial. [24] Theorem
3 and [32] imply that these constructions keep us inside our category
of spaces. It is shown in [11] Theorem 1.1 that Fn(X) ' *nX, the
nfold join of X with itself (this has n+1 copies of X in it).
Definition 2.11 We say a space 0connected X has category n, cat(X) =
n, if n is the least integer such that pn has a section. If there does not
exist such an n then we say cat(X) = 1.
We can also define category for maps [10] [2].
Definition 2.12 We say a map f : Y ! X of 0connected spaces has
category n, cat(f) = n, if n is the least integer such that there exists
g : Y ! Gn(X) such that png = f. If there does not exist such an n
then we say that cat(X) = 1.
Observe that cat(idX ) = cat(X). Therefore the category of a map is
strictly more general than the category of a space. It follows directly
from the definitions and the homotopy invariance of the fibrecofibre
construction that cat(f) and cat(X) are homotopy invariant. The
following concept was introduced by ScheererTanre [30].
Definition 2.13 Let f : E ! X be a fibration of 0connected spaces.
Assume there exists maps r : E ! Gn(X) and s : Gn(X) ! E such
that pnr = f and fs = pn. Then we call f an nLS fibration.
Lemma 2.14 Let n > 0 and f : E ! X an nLS fibration. Then f
has a section. In particular pn has a section.
Proof: For pn the lemma follows from [11] Proposition 1.5. For a gen
eral nLS fibration it follows from the result for pn and the definition
of nLS fibration.
We remark that, as observed below in Definition 2.17, G1(X) '
X. Also e and p1 are compatible with this equivalence. It follows
that the splitting of e gives a splitting of p1 and by composition of
pn. This is another way to prove the result of the lemma.
The following proposition follows directly from the definition.
10
Proposition 2.15 [30] Let f : E ! X be an nLS fibration. Then
cat(X) n if and only if f has a section.
At times it can be more convenient to have some nLS fibration rather
than the Ganea fibration. One reason is because the nLS fibration
may be considerably smaller. For example it was shown in [31] that
(Sn)lsn! (Sn)l (that is the inclusion of the sn skeleton into (Sn)l)
turned into a fibration is an sLS fibration. The following well known
facts about the category of maps are generalizations of the correspond
ing facts about the category of spaces.
Proposition 2.16 Let f and g be maps between 0conected spaces.
Then
1) cat(f x g) cat(f) + cat(g)
2) If f and g are composable then cat(gf) min{cat(g); cat(f)}
3) If f is a homotopy equivalence then cat(gf) = cat(g).
4) Let h : X ! Y be any map and f : Y ! C(h) be the inclusion.
Then cat(g) cat(gf) + 1. Also cat(C(h)) cat(Y ) + 1.
Proof: See [2] for a proof of 2). 1) follows from 2) and the product
formula for spaces [10] Theorem 9. 3) is trivial. 4) follows from [2]
Proposition 1.7.
Definition 2.17 There is a homotopy equivalence OE : X = CX[X
CX ! G1(X) induced by choosing a homotopy H : i0 ' *. Since
G0(X) is contractible the homotopy class of OE is independent of H.
Let f : W ! X be any map and fa : W ! X denote its adjoint.
Then fa is a map W ! X. We also let fa denote (the ho
motopy class of) the map OEfa and all further compositions with the
inclusions into Gn(X) for every n.
We next define a kind of Hopf invariant. Our definition is equivalent
to that of BernsteinHilton [3].
Definition 2.18 Let X be a 0connected space such that cat(X)
n 6= 0 and r : X ! GnX a section. Let f 2 [W; X]. Define the Hopf
invariant of f by Hr(f) = rf  fa. Since pnfa ' f ' pnrf we
can, and will, consider Hr(f) to be in [W; Fn(X)]. Observe that this
homotopy lift is unique since the fibration pn has a section.
The next theorem gives a characterization of cat in terms of Hopf
invariants. It is true both locally and integrally. In the theorem
dim(X) refers to the dimension of X. We define dim(X) to be the
dimension of the highest nontrivial cohomology class of X.
11
Theorem 2.19 [31] Let X be a space that is simply connected. As
sume that dim(X) l > 1 and cat(X) = n > 0. Let _f(i) : _i2ISl !
X be a map. Then cat(C(_f(i))) n if and only if there exists a
section r : X ! Gn(X) such that for every i, Hr(f(i)) = *.
We can also characterize when extending over a cone causes the cate
gory of the map to go up. Since we are mapping into a fixed fibration
the proof is easier then in the absolute case and follows directly from
obstruction theory.
Proposition 2.20 Let f : W ! Y be a map of 0connected spaces.
Let i : Y ! C(f) denote the inclusion. Let p : E ! X be an nLS
fibration and F be the fibre of p. Let g : C(f) ! X be a map. Then
cat(g) n > 0 if and only there exists a map h : Y ! E such that
gi ' ph and such that the map hf : W ! E is null homotopic. If
W = W 0then cat(g) n if and only if there exists h : Y ! E such
that gi ' ph and such that the map W 0! F induced by h is null
homotopic.
Proof: Let us assume there exists h as in the proposition such that
hf ' *. Then Lemmas 2.2 and 2.14 imply that there exists a map
OE : C(f) ! E such that pOE ' g. Therefore cat(g) n. The other
direction of the first statement is trivial.
The second part follows since Lemma 2.14 implies that when W is
a suspension the induced map is uniquely determined and is inessential
if and only if hf is.
In the statement of the last proposition the condition gi ' ph could
be equivalently replaced by the condition gi = ph. This is because p is
a fibration. Observe that the homotopic version allows us to replace
C(f) by any homotopy equivalent space. We would replace i by a
corresponding map.
To demonstrate the deceptive power of this simple proposition we
offer an example. The proposition will also we used for the results
of the later sections. For a CW complex X let Xn denote the n
skeleton of X. This is the first example of a CW complex X such that
cat(Xn) 1 for every n but cat(X) = 2. Remember that a phantom
map is an essential map that when precomposed with any map from
a finite complex becomes trivial.
Theorem 2.21 Let f : CP 1 ! S3 be any phantom map. (See [13]
or [40] Theorem D for some examples.) Let j : S3 ! S2 denote the
Hopf map. Then cat(C(jf)) = 2 but cat(C(jfCPn )) = 1 for every n.
12
Proof: That C(jfCPn ) = 1 is clear since f being phantom implies
that fCPn ' * and so C(jfCPn ) ' S2 _ CP n. Observe that
cat(C(jf)) = 1 or2 since it can be represented as a twocone. Also ob
serve that jf 6' * since if it were f would factor through S1 = H(Z; 1).
This can not happen since H1(CP 1) = 0.
Let g : C(jf) ! CP 1 be a map which represents a generator of
H2(C(jf)) = Z. Consider the following solid arrow diagram.
___h__//_ 1 2
S2 G1(CP77) ' S
 o oo 
 oo p1
fflffloOE fflffl
C(jf) ____g____//_CP 1
We can see by looking at cohomology that the only possible homotopy
classes h making the diagram commute are id and id. But then there
does not exist OE making the diagram commute since h is a homotopy
equivalence and jf 6' *. Therefore by Proposition 2.20 cat(g) 2.
Therefore cat(C(jf)) 2. So cat(C(jf)) = 2.
Similarly by attaching with phantom maps to T n(Sl) we could
construct CW complexes such that cat(Xr) n  1 for every r but
cat(X) = n.
3 cat of Products of Maps
This section gives conditions when cat(f x g) < cat(f) + cat(g). We
first prove a a general form of [17] Proposition 5.8. This is used to
give conditions when maps have the property that for every g with
cat(g) r, cat(f x g) < cat(f) + cat(g) (Theorem 3.3). The next
theorem (3.6) shows that spaces have a similar property whenever
they are the cone on a map with an unstable Hopf invariant. The
theorems will be applied in Sections 4 and 5.
For this section let us be given (strictly commuting) diagrams of
0connected spaces of the following form for i = 1; 2.
k(i)
W (i)____//_Y (i)__//C(k(i))
l0(i) l(i) f(i)
fflfflj(ifflffl)p(ifflffl)
F (i)____//_E(i)____//B(i)
13
Assume that the top row is a cofibration sequence and the bottom
row is a fibration sequence. Let g(i) : C(k(i)) ! C(j(i)) and p0(i) :
C(j(i)) ! B(i) denote the canonical extensions over the cone of l(i)
and p(i) respectively. The former induced by l0(i)xid and the latter in
duced by mapping the cone to *. We also assume that f(i) = p0(i)g(i).
In other words that f(i) is the trivial extension over the cone of
p(i)l(i).
The proof of the following lemma uses a method of Iwase [17].
The arguement illustrates the phenomenon which gives rise to exam
ples where cat(f x g) < cat(f) + cat(g). The same phenomenon is
responsible for counterexamples to Ganea's conjecture.
Lemma 3.1 Assume that l0(1)*l0(2) ' *. Then f(1)xf(2) : C(k(1))x
C(k(2)) ! B(1) x B(2) factors up to homotopy through (C(j(1)) x
C(j(2)))o.
Proof: From Lemma 2.9 we get a solid arrow diagram
OE
W (1) * W (2)____//(C(k(1) x C(k(2)))o___//C(k(1)) x C(k(2))
j j
l0(1)*l0(2)(g(1)xg(2))o j hjjj g(1)xg(2)
fflffl fflfflttjjOE0 fflffl
F (1) * F (2)___//(C(j(1)) x C(j(2)))o__//C(j(1)) x C(j(2)):
Since l0(1) * l0(2) ' * we get a map h such that hOE ' (g(1) x g(2))o.
Since OE0 splits after looping we can use Lemma 2.2 and assume that
OE0h ' g(1) x g(2). Since we also have a commutative diagram
C(k(1)) x C(k(2))U
UUUUf(1)xf(2)UUU
g(1)xg(2) UUUUUU
fflffl UU**U
C(j(1)) x C(j(2))p0(1)xp0(2)//_B(1) x B(2);
the lemma follows easily.
Lemma 3.2 Assume again that l0(1)*l0(2) ' * and also that cat(E(i)) =
n(i). Then cat(f(1) x f(2)) n(1) + n(2) + 1.
Proof: If cat(E(i)) = n(i) then [35] Section 5 implies that there
exists W (i) such that Cat(E(i) _ W (i) = n(i). Let Z = ((C(j(1)) _
W (1))x(C(j(2))_W (2)))o. It follows that Cat(Z) n(1)+n(2)+
1. f(1) x f(2) factors through (C(j(1)) x C(j(2)))o and therefore
through Z. So by 2.16 2) and since cat Cat the lemma follows.
14
The last lemma in conjunction with Proposition 2.20 could easily
be used to construct examples where cat(f xg) < cat(f)+cat(g). Next
we use it to prove a theorem which is designed for the applications of
the next two sections.
Theorem 3.3 Assume cat(f(1)) = n + 1 and E(1) ! B(1) is an n
LS fibration for B(1) (for example the nth Ganea fibration for B(1)).
Assume that r(l0(1)) ' *. Then cat(f(1)) = cat(f(1) x idSr). Also
for every map g such that cat(g) r > 0, cat(f(1) x g) n + r.
Proof: From the definition of nLS fibration there is a commutative
diagram.
F (1)____//Fn(B(1))
 
 
fflffl fflffl
E(1) ____//_Gn(B(1))
 
 
fflffl= fflffl
B(1) ______//_B(1)
Therefore we can assume that E(1) ! B(1) is pn : Gn(B(1)) !
B(1). First we show that for the Ganea fibration pr : Gr(X) ! X
cat(f(1) x pr) n + r. There exists a commutative diagram
Fr1(X) ____//_Gr1(X)____//G0r(X)
= = pr
fflffl fflfflpr1 fflffl
Fr1(X) ____//_Gr1(X)______//X
where the top row is a cofibration sequence. Fr1(X) ' r1W .
Therefore l0(1) * Fr1(X) ' rl0(1) ^ W 0' * by assumption. So since
cat(Gn(B(1)) n and cat(Gr1(X)) r  1 we can apply Lemma
3.2 to get that cat(f(1) x X) n + r.
Now let g be any map. Since cat(g) r there exists a factorization
of g as g0pr : Y ! Gr(X) ! X. But then Proposition 2.16 says that
cat(f(1) x g) cat(f(1) x pr) n + r.
F1(Sr) ' Sr ' Sr ^ Z. So the same arguements show that
cat(f(1)) cat(f(1) x idSr) cat(f(1)). .
15
Corollary 3.4 Let us be given a (strictly commutative) diagram of
0connected spaces
k(1)
W (1)_____//Y (1)__//_Y (1)=W (1)
l0(1) l(1) f
fflffl fflffl fflffl
F (1)_____//E(1)_p(1)_//_X(1)
where k(1) is a cofibration and the bottom row is an nLS fibration
for X(1). Assume cat(f) = n + 1 and rl0(1) ' *. Then for every g
such that cat(g) r > 0, cat(f x g) n + r. In particular cat(f) =
cat(f x idSn) = n + 1.
Proof: When we let f(1) be the trivial extension over the cone then
we have the setup of the last theorem. Therefore for every map g such
that cat(g) r > 0, cat(f(1) x g) n + r. We have a commutative
diagram
C(k(1))_ss__//B=A
uu
f(1)uuuuu
fflfflfzzuu
X(1)
Since k(1) is a cofibration [39] chapter I section 5 implies that ss is a
homotopy equivalence. Therefore the theorem and Proposition 2.16
3) imply the corollary.
We also give a more homotopic version of the above corollary. We
do not use it but include it since it could be more convenient to apply
in some situations.
Theorem 3.5 Let n > 0. Let us be given a homotopy commutative
diagram of 0conected spaces
f
W ____//_X___//_C(f)
h k g
fflffl fflffl fflffl
F __j__//_E_p___//B
such that the bottom row is an nLS fibration (for example the nth
Ganea fibration). Assume that rh ' *. Then cat(g) = cat(g x Sr).
Also for every map g0 such that cat(g0) r > 0, cat(g x g0) n + r.
16
Proof: By Lemma 2.14 we can replace the diagram in the theorem
by a strictly commuting one as in Lemma 2.4
W __i__//X [f CylW _____//C(f)
h k[H g
fflffl fflffl fflffl
F _____j_____//E_____p_____//B:
We are now in the situation of Corollary 3.4 and so the theorem fol
lows.
Next we prove a similar theorem but one which is sometimes more
convenient to apply. We use it in Section 5 to construct examples.
Theorem 3.6 Let f : W ! X be a map where X is 0connected.
Assume cat(X) = n > 0 and cat(C(f)) = n+1. Assume there exists a
section of the Ganea fibration s : X ! Gn(X) such that rHs(f) ' *.
Then for every g such that cat(g) = r, cat(idC(f)x g) n + r <
cat(C(f)) + cat(g).
Proof: Since Gn(i)(fa) ' * the following solid arrow diagram com
mutes up to homotopy even though adding the dashed arrow may
cause commutativity to be lost.
f i
W __________//_X_______//C(f)

Hs(f) s 
fflffl fflffl 
Fn(X) _ _ _ _//Gn(X) =

  
Fn(i) Gn(i) 
fflfflin fflffl fflffl
Fn(C(f))_____//Gn(C(f))____//_C(f)
The theorem then follows directly from Theorem 3.5.
4 Applications to Rational Homotopy
In this section we apply the results of the last section to rational
homotopy theory. First we define mcat and cat in the rational con
text. We prove a result of ScheererStelzer that mcat is determined
by the existence of a certain CDGA map. We show how mcat of a
map is determined by obstruction theory (Proposition 4.7). Next we
17
prove Theorems 4.8 and 4.9 which demonstrate a connection between
the statements mcat(f) < cat(f) and cat(f) = cat(f x idSn). This
includes an equivalence of the two statements under certain hypothe
ses. Finally we construct some examples where mcat(f) < cat(f) and
cat(f) = cat(f x idSn) both hold.
We work in the rational homotopy category represented by commu
tative differential graded algebras, CDGA's. For more information on
CDGA's and rational homotopy theory we refer the reader to [14], [33]
and [36]. V refers to a CDGA which is free as a graded commutative
algebra over some graded rational vector space V . V=>n V denotes
V modulo the ideal generated by all products of length greater than
n. For this section all of our CDGA's and spaces will be simply con
nected and of finite type unless stated otherwise. A space is called
rational if "H*(X; Z) is a rational vector space. There is a rationaliza
tion functor from spaces to rational spaces. (See [4] for more details).
There are functors F : CDGA ! CG* and A : CG* ! CDGA
which induce equivalences of rational homotopy category. (See [5]).
The composition F A is equivalent to rationalization. (Actually the
functors are into and from simplicial sets and not CG*. We compose
those functors with the singular simplices and realization functors to
get the F and A above.)
For this section let
j p
A _____//X _____//Y
be a fibration sequence in CDGA. In other words p is a surjection and
A = kerp. Also let
V K____//_KV W
KKK 
KKK ss
KK%%fflffl
V=>n V
be a diagram such that ss is a weak equivalence. Finally for this section
we let the following be a diagram in CDGA.
V ____//_V W ____//_W
f g h
fflffl fflffl fflffl
A ________//_X _______//_Y:
Unless otherwise specified the diagram is assumed to be strictly com
mutative. V W has a differential such that d(V ) V and W
has the induced differential on the quotient.
18
The definition of LS category of CDGA's was made by Felix
Halperin in their pivital paper [7]. The definition of mcat is due to
HalperinLemaire [15].
Definition 4.1 [7][15] cat(f) n if and only if there exists a CDGA
map h making the following diagram commute.
V ____//_LV W
LL
LLL h
f LLLL%%Lfflffl
A
If no such n exists then cat(f) = 1.
Similarly mcat(f) n if and only if there exists a V module
map h making the above diagram commute. If no such n exists then
mcat(f) = 1. If f is a map of spaces then mcat(f) means the
mcat(A(f)).
The equivalence of the algebraic and topological definitions of LS
category for rational spaces was also shown in [7].
Theorem 4.2 [7] cat(f) = cat(F (f)).
We review the algebraic fibrewise Sp1 construction of Scheerer
Stelzer_[29]. Let (V W; d) be considered as a free V module._Let_
W denote the kernel of the augmentation W ! Q. Consider_W
as a graded vector space. Define M(V W ) to be V (W ) as
an algebra with differential defined by the Leibniz law. Another way
to describe the differential is as the unique one such that
V ! M(V W )
is a KS extension and
V ______//_NNV W
NNN 
NNNN i
N&&Nfflffl
M(V W )
is a diagram of V modules. Clearly M(V W ) is a CDGA.
Proposition 4.3 [29] For every map f : V W ! U of V
modules there exists a unique map f0 : M(V W ) ! U of CDGAs
such that f0i = f.
19
Proof: Straightforward.
Applying the proposition to id : V W ! V W we get
that there exists a unique CDGA map r : M(V W ) ! V W
such that ri = id. We apply the proposition to prove a result of
ScheererStelzer.
Theorem 4.4 [29] Let f : V ! A be a map. Then mcat(f) n if
and only if there exists a commutative diagram in CDGA
V ____//_NNNM(V W )
NNNN 
f NNNNN&&Nfflffl
A:
Proof: Follows directly from Proposition 4.3.
Next we describe a relationship between these ideas and the ideas
of determining category by Hopf invariants. Let us translate a cou
ple of results from the previous section into the language of Sullivan
models. The translation of Theorem 3.3 gives us:
Theorem 4.5 Assume cat(f) = n + 1 and H*(h) is trivial. Then for
every r, cat(f) = cat(F (f) x idSr). Also for every map g such that
cat(g) > 0, cat(f x g) (n + r) < cat(f) + cat(g).
Proof: Apply F to the diagram of the section to get a strictly com
muting diagram of spaces.
F(p)
F (Y ) ________//F (X) ________//_F (A)
F(f) F(g) F(h)
fflffl fflffl fflffl
F (W ) ____//_F (V W ) ____//_F (V )
Replace F (X) by the mapping cylinder of F (p) and F (A) by C(F (p))
with the maps being the canonical extensions. Similarly replace F (V
W ) ! F (V ) by a fibration and F (W ) by its fibre. The new maps
are the canonical liftings. [7] Theorem 4.7 implies that the topological
realization of V ! V=>n V is an nLS fibration. Also H*(h) is
trivial if and only if h ' * So we can use Theorem 3.3 to prove the
theorem.
An alternative proof of the above theorem would be given by a
translation of the proof of Theorem 3.3. The translation of Proposition
2.20 says:
20
Proposition 4.6 cat(f) n > 0 if and only if there exists a g that
makes our diagram commute up to homotopy and such that pg ' *. If
F (Y ) is a wedge of spheres then cat(f) n > 0 if and only if there
exists a g such that the induced map h ' *.
There is also a version for mcat.
Proposition 4.7 Consider homotopy commutative diagrams of the
following form with f fixed.
____
V ____//_M(V W ) _____//(W )
f g0 h0
fflfflj fflfflp fflffl
A __________//_X___________//_Y
Then mcat(f) n > 0 if and only if there exists g0making the diagram
homotopy commute such that the pg0 ' *. If F (Y ) is a wedge of
spheres then mcat(f) n > 0 if and only if there exist a g0 such that
the induced map h0' *.
Proof: Assume mcat(f) n. Then by Theorem 4.4 there exists
OE : M(V W ) ! A such that
V NNN_//_M(V W )
NNNN 
f NNNNN 
N&&fflffl
A
commutes. Then define g0 = jOE and let h0 be any extension. Then
pg0= pjOE = * and h0= *.
Now assume there exists g0 such that pg0 ' *. Then there exists
OE : M(V W ) ! A such that jOE ' g0. Notice that V !
M(V W ) is injective on the dual of homotopy (in other words it
models a map that is surjective on homotopy). So OE can be adjusted
using the action so that
V NNN_//_M(V W )
NNNN OE
f NNNNN 
N&&fflffl
A
commutes up to homotopy. (The action exists for fibrations in any
model category [27] Chapter I Section 3. To get the diagram to com
mute up to homotopy using more explicit methods of rational homo
topy theory is also possible. A third way to get commutativity is to
21
translate the problem to spaces, use the coaction (Lemma 2.2) and
translate back to CDGAs.) OE can then be adjusted to make the dia
gram commute exactly since V ! M(V W ) is a KS extension.
Theorem 4.4 then shows mcat(f) n. ____
The sequence F (V ) ! F (M(V W )) ! F ((W )) splits
after looping. So the statements when F (Y ) is a wedge of spheres
follow since in that case pg0' * if and only if h0' *.
We can also put the last two diagrams together:
____
V _____//M(V W ) ____//_(W ) (+)
= r r
fflffl fflffl fflffl
V _______//V W ________//W
f g h
fflffl fflffl fflffl
A ___________//X ___________//_Y
and get the following:
Theorem 4.8 Let f : V ! A be a map such that cat(f) > cat(jf) =
n > 0. Assume we have a commutative diagram_as throughout the
section such that the composition hr : (W ) ! W ! Y is null
homotopic. Equivalently we can assume that F (h) ' *. Then the
following four statements hold:
1) cat(F (f) x idSr) = cat(F (f)) for some r > 0,
2) cat(F (f) x idSr) = cat(F (f)) for all r > 0,
3) cat(f g) cat(f) + cat(g)  1 for all maps g,
4) cat(f idA ) cat(f) + cat(A)  1 for some CDGA A.
It F (Y ) is a wedge of spheres then also:
5) cat(f) > mcat(f)
Proof: 5) follows directly from Proposition_4.7. 1), 2), and 4) are
special cases of 3). Observe that F ((W )) ' 1 1 F (W ) and
that under this equivalence E1__:_F (W ) ! 1 1 F (W ) is equiv
alent to F (r) : F (W ) ! F ((W )). Also rationally for any map,
g ' * if and only if 1 g ' *. Therefore hr being null is equivalent
to F (h) ' *. So 3) follows from Theorem 4.5.
The next theorem says that in a range all the five statements of the
last theorem are equivalent. In the theorem dim(X) is the dimension
of X. This is the dimension of the highest nontrivial homology class
of X.
22
Theorem 4.9 Let f : V ! A be a map such that cat(f) > cat(jf) =
n > 0. Assume that dim(X) 2n(con(V ) + 1)  2. Also assume
that F (Y ) is a wedge of spheres. Then the following five statements
are equivalent:
1) cat(F (f) x Sr) = cat(F (f)) for some r > 0,
2) cat(F (f) x Sr) = cat(F (f)) for all r > 0,
3) cat(f x g) cat(f) + cat(g)  1 for all maps g,
4) cat(f x A) cat(f) + cat(A)  1 for some CDGA A,
5) cat(f) > mcat(f).
Proof: Clearly 3) implies 1), 2) and 4). Since for every n, mcat(Sn) =
1, 1) implies 5) follows directly from the result of Parent [25] that for
every f; g, mcat(f g) = mcat(f) + mcat(g). So we just need to show
that 5) implies 3). Assume 5) holds. Then by Proposition 4.7 there
exists a diagram:
____
V _____//M(V W ) _____//(W ) (*)
f g0 h0
fflffl fflffl fflffl
A __________//_X___________//_Y
such that h0' *. ____
con(W ) n(con(V ) + 1)  2. So we see that r : (W ) ! W
and hence r : M(V W ) ! V W induces an isomorphism on
H*. (indecomposables) of the minimal models in dimensions less than
2n(con(V ) + 1)  3. Therefore since dim(X) 2(con(V ) + 1)  2
we can extend (*) to get a diagram of the form (+). We keep f fixed
and changed g0by a homotopy that fixes V . Since F (Y ) is a wedge
of spheres the new indced h0 is homotopic to the old one. Therefore
in the extended diagram hr = h0is null homotopic. Therefore we can
apply Theorem 4.8 to get 3).
We beleive that the five statements of the corollary are equivalent
for any map f. In particular we beleive that for any f, mcat(f) <
cat(f) if and only if cat(f) = cat(f x idSr).
Examples: Let n 2. Let T n(Sl) = {(x1; : :;:xn) 2 (Sl)n for some ixi=
*} denote the fat wedge. Let i : T n(Sl) ! (Sl)n denote the inclusion.
Let F denote the fibre of i. Porter [26] shows that F is a wedge of
spheres (In this case the result also follows easily from the cube the
orem of Mather [23]. Also rationally it follows by direct calculation.)
Let g0 : Ss ! F be any Whitehead product of the inclusions of two
23
different spheres into F . Let g : Ss ! T n(Sl) be the composition of
g0 into T n(Sl). f : C(g) ! (Sl)n denote any extension of i. Then f
satisfies all the hypothesis of Theorem 4.8 and Corollary 3.4. In par
ticular for every r > 0, cat(f xSr) = cat(f) = n and mcat(f) n1.
(Remember mcat(f) means mcat of a model of the rationalization of
f.)
Proof of Examples: We wish to apply Proposition 2.20 to show that
cat(f) > n  1. Consider the diagram.
Ss___k_//T n(Sl) [g CylSsss//_C(g)
 
g0 h f
fflffl fflffl"i fflffl
F _________//_"T(nSl)______//(Sl)n
Let the bottom row be a fibration sequence and the map "iis the
inclusion i : T n(Sl) ! (Sl)n turned into a fibration. Let k denote
the inclusion into the free end of the cylinder. Let h be an extension
lift of i. This implies that h is a homotopy equivalence since l 2
(remember we are assuming that our spaces are simply connected)
implies that the connectivity of i is greater than the dimension of
T n(Sl). g0 is the lift of hk to the fibre. By [31] Lemma 6.9 "iis an
(n1)LS fibration. Also since "isplits after looping the inclusion of F
into "T(nSl) is injective on ss*. So g 6' * and Proposition 2.20 implies
that cat(f) > n  1. But since cat(T n(Sl)) = n  1, cat(C(g)) n.
Therefore cat(f) n and so cat(f) = n.
g0 ' * since it is a Whitehead product. We can then apply
Theorem 3.4 to see that for every r, cat(f x idSr) = cat(f) = n and
Theorem 4.8 to see that mcat(f) n  1.
Notice that for our example we could have picked g to be any
nontrivial homotopy class such that i(g) ' * and such that the lift of
g to F suspends to a null homotopic map.
5 An application to Ganea's conjec
ture
We give a counterexample to Ganea's conjecture (Theorem 5.1) for
a case left open by [17]. Our example X is also interesting since
cat(X) = cat(XxX) = cl(XxX) = 2 (Corollary 5.4). It is interesting
24
to compare our example to the one of Fernandez [9]. Working at
the prime 3 she shows a certain space Z has the property cl(Z) =
cl(Z x Z) = 2. However her Z has cat(Z) = 1.
For this section fix a prime p > 2. Let fi 2 ss4p3(S3) Z(p)be
a generator. (In fact fi = ff2 but we will not use this.) Let X =
(S2 _ S3) [[2;3]fie4p1. Let Y = S1 _ S2 _ S3 [fie4p2. n always
denotes the inclusion of a sphere of dimension n into a space.
Theorem 5.1 cat(X) = 2 and for every Z 6' *, cat(X x Z) <
cat(Z) + cat(X). In particular cat(X x S1) = cat(X).
Proof: The only facts we use about fi are that fi 6' * and 2fi ' *.
The first fact follow since S3 is an H space. The existence of fi and
the fact that 2fi ' * were proved by Toda [37]. We must verify the
hypotheses of Theorem 3.6.
To show cat(X) = 2 we use Proposition 2.20. Let f : X ! S2 x
S3 be any extension of the identity. Assume cat(f) 1. Then by
Proposition 2.20 there exists a diagram
f0// 2 3
S2 _ S3_____ G1(S x S )
 p1
 
fflfflf fflffl
X _________//S2 x S3
such that f0[2; 3]fi ' *. G1(S2 x S3) ' (S2 x S3) ' S2 _ S3 _
S3 _ higher spheres. (See [39] Chapter VII Section 2 for a proof of the
second equivalence.) Since the diagram above commutes we see that
f0 is injective on ss*. This gives us a contradiction and so cat(f) > 1.
Therefore cat(X) > 1 and so cat(X) = 2 by Propositon 2.16 since X
can be represented as a two cone.
On the other hand 2fi ' * so for every s
Hs([2; 3]fi) = Hs([2; 3])2fi ' *:
So the hypotheses of Theorem 3.6 have been verified.
One of the ingredients needed to make this example work was an
unstable element in the homotopy groups of spheres. Since there are
many unstable elements in the homotopy groups of spheres we could
have chosen many other examples. We chose our example partially to
demonstrate how easy Theorem 2.19 can be to use, even when there
are many sections.
25
We procede to show another interesting property of the space X.
We will show cat(X) = cat(X x X). First we need a preliminary
lemma.
Lemma 5.2 Y * Y is a wedge of spheres.
Proof:
Y * Y ' Y ^ Y
' (S2 ^ (S1 _ S2 _ S3 [fie4p2))
_(S3 ^ (S1 _ S2 _ S3 [fie4p2))
_(S4 [fi e4p1^ (S1 _ S2))
_((S3 [fie4p2) ^ (S3 [fie4p2))
Since 2fi ' * all the pieces in the wedge decompoistion except
(S3 [fie4p2) ^ (S3 [fie4p2) are esily seen to be wedges of spheres.
Again since 2fi ' * there is some f such that (S3[fie4p2)^(S3[fi
e4p2) ' (S6 _ S4p+1 _ S4p+1 [f e8p4). f must be an element
in ss8p4(S7 x S4p+2 x S4p+2). Also 2f ' * since 2fi ' *. So
f ' * since S7 is an H space (hence induces an injection on ss*)
and ss8p4(S4p+1) is already in the stable range. So Y * Y is a wedge
of spheres.
Theorem 5.3 There exists a wedge of spheres W , a space U ' X xX
and a cofibration sequence.
W ! Y _ Y ! U
Proof: From Lemma 2.9 we have a cofibration sequence.
f
Y * Y ____//_Y _ Y ____//_Y x Y
Let p : Y ! X denote a map that sends S4 to [2; 3], is the identity
on S2 _ S3 and is the canonical extension over the 4p  1 cell. Clearly
H*(p) is surjective.
Let r : H*(X ^ X ! H*(Y ^ Y ) be a splitting of H*(p ^ p).
Let Z be a wedge of spheres and iZ ! Y * Y be a map such that
there exists a homotopy equivalence OE : Z ! X ^ X and such that
H*(i) = rH*(OE). That there exists such an i follows from Lemma
5.2.
26
Next consider the following diagram.
fi
Z ______//_Y _ Y ______//_C(fi)
i  g
fflfflf fflffl fflffl
Y * Y ____//_Y _ Y ____//_Y x Y
p_p pxp
fflffl fflffl
X _ X _______//X x X
where g is the induced map between cofibres. Using the long exact
sequence on homology and the fact that p ^ pi : Z ! X ^ X is
a homology equivalence we see that H*((p x p)g) is surjective. Let
h : S4 _ S4 ! Y _ Y denote (4 [2; 3]) _ (4 [2; 3]). Then since
(p _ p)h ' * we get a diagram
fi+h // //
Z _ S4 _ S4 _____Y _ Y _____C(fi + h)
p_p OE
fflffl fflffl
X _ X _______//_X x X
where OE is an extension of (p x p)g. OE is easily seen to be an H*
isomorphism and therefore a homotopy equivalence since X x X is a
CW complex and all spaces are simply connected.
Recall that cl(X) denotes the cone length of X. (See [31] Definition
2.9 for a definition.)
Corollary 5.4 cat(X) = cat(X x X) = cl(X) = 2
Proof: cat(X) = 2 and so cl(X) 2. But we have realized a space
U ' X x X as a two cone. Therefore cl(X) 2 and so cl(X) = 2.
More generally we beleive for every n there exists a simply con
nected space Z such that cat(Z) = cat(Zn) = n. Perhaps an eas
ier thing to construct would be an example of a space Z such that
cat(Z) = cat(Z x (Sr)n1) = n. Simpler still would be to construct a
space Z with torsion free homology such that cat(Z) = n but, for Z0
denoting the rationalization of Z, cat(Z0) = 1.
References
[1]H.J. Baues, Iterierte JoinKonstructionen, Math. Zeit. 131 (1973),
7784.
27
[2]I. Berstein and T. Ganea, The category of a map and of a coho
mology class, Fund. Math. 50 (1961), 265279.
[3]I.Berstein and P.J. Hilton, Category and Generalized Hopf Invari
ants, Illinois J. Math. 4 (1960) 437451.
[4]A.K. Bousfield, Localization of spaces with respect to homology,
Topology 14 (1975) 133150.
[5]A.K. Bousfield and V.K.A.M. Gugenheim, On PL De Rham theory
and rational homotopy type, Memoirs of the A.M.S. 179 (1976).
[6]Y. Felix, La dichotomie elliptiquehyperbolique en homotopie ra
tionnelle, Soc. Math. de France, Asterisque 176 (1989).
[7]Y. Felix and S. Halperin, Rational L.S. category and its applica
tions, Trans. Amer. Math. Soc. 273 (1982), 137.
[8]Y. Felix, S. Halperin and J.M. Lemaire, The rational LS category
of products and Poincare duality complexes, Topology 37 (1998),
749756.
[9]L. Fernandez, Thesis, Univerite de Lille 1, (1998).
[10]R.H. Fox, On the LusternikSchnirelmann Category, Ann. Math.
42 (1941) 333370.
[11]T. Ganea, A generalization of the homology and homotopy sus
pension, Comm. Math. Helv. 39 (1965) 295322.
[12]T. Ganea, LusternikSchnirelmann Category and Strong Cate
gory, Illinois J. Math. 11 (1967) 417427.
[13]B.Gray, Spaces of the same ntype, for all n, Topology 5 (1966)
241243.
[14]S. Halperin, Lectures on minimal models, Memoire de la Societe
Math de France (N.S.), 9/10 (1983).
[15]S. Halperin and J.M. Lemaire, Notions of Category in Differ
ential Algebra, Lecture Notes in Mathematics, SpringerVerlag,
1318 (1988), 138154.
[16]E. Idrissi, Un exemple ou Mcat(f) 6= Acat(f), C.R. Acad. Sci.
Paris (1990) 599602.
[17]N. Iwase, Ganea's conjecture on LusternikSchnirelmann cate
gory, to appear in Bull. London Math. Soc.
[18]N. Iwase, A1 method in LusternikSchnirelmann category,
preprint, (1997).
28
[19]I.M. James, On category in the sense of LusternikSchnirelmann,
Topology, 17 (1978), 331348.
[20]I.M. James, LusternikSchnirelmann category, Handbook of alge
braic topology, NorthHoland (1995), 12931310.
[21]L. Lusternick and L. Schnirelmann, Methodes Topologiques dans
les Problemes Variationnels. Inst. for Math. and Mechanics (1930)
Moscow. (in Russian)
[22]L. Lusternick and L. Schnirelmann, Methodes Topologiques dans
les Problemes Variationnels. Herman, Paris (1934).
[23]M. Mather, Pullbacks in Homotopy Theory, Can. J. Math. 28
(1976) 225263.
[24]J. Milnor, On Spaces Having the Homotopy Type of a CW
comples, Trans. Amer. Math. Soc. 90, 272280 (1959).
[25]P.E. Parent, L.S. Category: Product Formulas, preprint (1998).
[26]G.J. Porter, The Homotopy Groups of Wedges of Suspensions,
Amer. J. Math. 88, 655663 (1966).
[27]D. Quillen, Homotopical algebra, Springer Lecture Notes in
Mathematics 43 (1967).
[28]H. Scheerer and D. Stanley, On the rational LScategory of a
cartesian product of maps, preprint (1998).
[29]H. Scheerer and M. Stelzer, Fibrewise infinite symmetric products
and M  cat, preprint (1998).
[30]H. Scheerer and D. Tanre, Fibrationsa la Ganea, Bull. Belg.
Math. Soc. 4 (1997), 333353.
[31]D. Stanley, Spaces with LusternikSchnirelmann category n and
cone length n+1, to appear in Topology.
[32]N. Steenrod, A convenient category of topological spaces, Mich.
Math. Jour. 14 (1967) 133152.
[33]D. Sullivan, Infinitesimal computations in topology, Pulications
Mathematiques de l'Institute des Hauts Etudes Scientifiques, 47
(1977), 269331.
[34]R.M. Switzer, Algebraic Topology  Homotopy and Homology,
SpringerVerlag (1975).
[35]F. Takens, The LusternickSchnirelmann Categories of a Product
Space, Compositio Math. 22 (1970), 175180.
29
[36]D. Tanre, Homotopie Rationnelle: Modeles de Chen, Quillen,
Sullivan. Lecture Notes in Math. 1025, (1983), SpringerVerlag.
[37]H. Toda, Composition Methods in Homotopy Groups of Spheres,
Princeton University Press (1962).
[38]R.M. Vogt, Convenient categories of topological spaces for homo
topy theory, Arch. Math. 22 (1971) 545555.
[39]G.W. Whitehead, Elements of Homotopy Theory, Springer
Verlag (1978).
[40]A. Zabrodsky, On phantom maps and a theorem of H. Miller,
Israel Jour. Math. 58 (1987) 129143.
30