On the LusternikSchnirelmann category of maps
Donald Stanley
August 15, 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 ,
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.
1 Introduction
The LusternikSchnirelmann category of a space, cat(X), (Definition
2.7) was introduced in the early 1930's [18], [17]. The category of a
map, cat(f), (Definition 2.8) was first defined by Fox [8] 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 [15] [16].
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 [6] proved that equality holds.
The counterexample of Iwase [13] 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 3.3 Let us be given a diagram
A _________//B___i___//C
h g f
fflffl fflfflpnfflffl
Fn(X) _____//Gn(X)_____//X
where the top row is a cofibration sequence and the bottom row is the n
th Ganea fibration for X (Definition 2.6). Assume cat(f) = n + 1 and
rh ' *. Then for every g such that cat(g) r > 0, cat(f xg) n+r.
In particular cat(f) = cat(f x Sn) = n + 1.
Our interest in the theorem is due to two applications. The first
area of application is in rational homotopy theory. In [27] the author
constructed an example of a rational map such that cat(f x Sn) =
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 xSn) = 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
and f : Y ! Z a map. 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 x Sn) = cat(f).
The second application is to solve a case of Ganea's conjecture which
was left open by Iwase (see [13] 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 , cat(X x
Y ) cat(Y ) + 1. This same X has another very interesting property.
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
2 Notation and Background
This section contains some general results and definitions. After fixing
some notation we prove a proposition (2.5) 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
2.7 and 2.8). Results which tell us if cat goes up when attaching a
cone are then given (2.13, 2.14). This is determined by Hopf invariants
(Definition 2.12) 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. When we talk about spaces and maps between spaces we
think of them as objects and morphisms in CG*. For definitions and
basic properties of CG* see [28]. For the purposes of this paper we
could have as easily used the category of spaces having the homotopy
type of a CW complex [20]. We choose CG* over the categories of
Vogt [33] because it is more familiar to a greater number of homo
topy theorists. In CG* let x denote the weak product. (CG*; x) is a
monoidal category. For a map f : X ! Y in CG*, we let C(f) denote
the reduced cone on f. Explicitly
C(f) = (Y [ X x I)=(* x I = *; (x; 1) = f(x); (x; 0) = *)x2X
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. Following Milnor and Moore [21] we let A also
denote the identity morphism A ! A.
The next three lemmas are preparation for Proposition 2.5.
3
Lemma 2.1 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 2.2 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: [28].
Definition 2.3 For i = 1; 2, let f(i) : A(i) ! B(i) be maps in CG*.
Then define (Cf(1) x Cf(2))o to be the pushout of the following dia
gram.
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 2.4 Let A(i) 2 CG*. Then there exists a homeomorphism OE
such that
A(1) * A(2)_____//C(A(1) * A(2))
QQ
QQQQ 
QQQQ OE
QQ((Q fflffl
CA(1) x CA(2)
4
commutes. Where the other two maps in the diagram are the usual
inclusions.
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)
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.
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 2.5 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)
 
 
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)):
5
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 2.2 and the right hand
square is by defintion. Therefore Lemma 2.1 implies that the outside
square is a pushout.
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
2.2. 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
i 
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 2.4
i is equivalent to A(1) * A(2) ! C(A(1) * A(2)), so the lemma follows.
Observe that we get the pushout (*) in any category where Lemma
2.2 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 [9]. 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.
6
Definition 2.6 Let X be a 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.)
Let Fn = F ib(pn) and G0n+1= 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 fibration)
means that the above construction is functorial.
Definition 2.7 We say a space 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 [8] [2].
Definition 2.8 We say a map f : Y ! X 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 [25].
Definition 2.9 Let f : E ! X be a fibration. 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.
The following proposition follows directly from the definition.
Proposition 2.10 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
7
may be considerably smaller. For example it was shown in [26] 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 2.11 Let f and g be maps. 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)}
Proof: See [2] for a proof of 2). 1) follows from 2) by the result of
[30].
We next define a kind of Hopf invariant. Our definition is equiva
lent to that of BernsteinHilton [3].
Definition 2.12 Let X be 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) = fa  rf. Since pfa ' f ' prf we can, and will, consider
Hr(f) to be in [W; Fn(X)]
The next theorem gives a characterization of cat in terms of Hopf
invariants. It is true both locally and integrally.
Theorem 2.13 [26] Let X be a space that is connected but not neces
sarily simply connected. Assume 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 2.14 Let f : W ! Y be a map. Let i : Y ! C(f)
denote the inclusion. Let g : C(f) ! X be a map such that cat(gi) =
n. Then cat(g) n if and only there exists a map h : Y ! GnX
such that gi ' pnh and such that the map hf : W ! Gn(X) is
null homotopic. If W = W 0then cat(g) n if and only if there
exists h : Y ! Gn(X) such that gi ' pnh and such that the map
W 0! Fn(X) induced by h is null homotopic.
Proof: Straightforward.
8
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 [13] 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 (3.4) 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 diagrams 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)
Assume that the top row is a cofibration sequence and the bottom row
is a fibration sequence. For some of the proofs we will only use that
p(i)j(i) ' *.
The proof of the following lemma uses a method of Iwase [13].
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) x f(2) : Z(1) x
Z(2) ! B(1) x B(2) factors up to homotopy through (C(j(1)) x
C(j(2)))o.
Proof: From Lemma 2.5 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))
where g(i) is the canonical extension over the cone of l(i). Since
l0(1) * l0(2) ' * we get a map h such that hOE ' (g(1) x g(2))o. Since
OE0 splits after looping it is a [W (1) * W (2); _] surjection. Therefore
we can adjust h using the coaction so that OE0j ' g(1) x g(2). Since
9
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))_p(1)xp(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: cat((C(j(1)) x C(j(2)))o) n(1) + n(2) + 1 and f(1) x f(2)
factors through this space.
The last lemma in conjunction with Proposition 2.14 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 Sr). 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
10
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 Lemma 2.11 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 Sr) cat(f(1)). .
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 3.4 Let f : W ! X be a map. Assume cat(X) = n
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(C(f) x g) n + r.
Proof: Notice that the following solid arrow diagram commutes up
to homotopy even though adding the dashed arrow may cause com
mutativity to be lost.
f
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)
We will show there exists a homotopy H : Gn(i)sf ' inFn(i)Hs(f)
such that
W ______//_X [f CylW ____//_C(f)
Fn(i)Hs(f) (Gn(i)s)[H =
fflfflin fflffl fflffl
Fn(C(f)) ______//_Gn(C(f))pn___//_C(f)
commutes. To see this take any homotopy H. Since Gn(C(f) '
C(f) x Fn(C(f)), [2W; pnC(f)] is surjective. So we can adjust
H using the coaction of the cofibration sequence W ! X _ W !
X [f CylW to make the right square commute up to homotopy
11
without changing the map restricted to W . Since pn is a fibration
we can then adjust H to make the diagram commute exactly. Now
the theorem follows directly from Theorem 3.3.
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 prove Theorems 4.8 and 4.9 which demonstrate a connection be
tween the statements mcat(f) < cat(f) and cat(f) = cat(f x Sn).
Finally we construct some examples where mcat(f) < cat(f) and
cat(f) = cat(f x Sn) 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 [10], [29]
and [31]. 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. 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 weak equivalence. Finally for this section
we fix the following commutative diagram
V ____//_V W ____//_W
f g h
fflffl fflffl fflffl
A ________//_X _______//_Y:
12
The definition of LS category of CDGA's was made by FelixHalperin
in their pivital paper [5]. The definition of mcat is due to Halperin
Lemaire [11].
Definition 4.1 [5] cat(f) n if and only if there exists a CDGA
map h making the following diagram commute.
V ____//_LV W
LLLLfL 
LLL h
LL%%fflffl
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 mcat(f) means the mcat of a
model of the rationalization of f.
The equivalence of the algebraic and topological definitions of LS cat
egory for rational spaces was also shown in [5].
Theorem 4.2 [5] If f represents a rational map f"then cat(f) =
cat(f").
We review the algebraic fibrewise Sp1 construction of ScheererStelzer
[24]._Let (V W; d) be considered as a free V module. 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 [24] 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.
13
Proof: Straightforward.
Applying the proposition to id : V W ! V W we get
that there exists a unique map r : M(V W ) ! V W such
that ri = id. We apply the proposition to prove a result of Scheerer
Steltzer.
Theorem 4.4 [24] 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 Let "fdenote the topological realization of f. Assume
cat(f) = n+1 and H*(h) is trivial. Then for every r cat(f) = cat(f"x
Sn). Also for every map g such that cat(g) > 0, cat(f x g) (n + r).
Proof: The topological realization of V ! V=>n V is an nLS
fibration. Also H*(h) is trivial if and only if h is trivial. So we can
either translate Theorem 3.3 or its proof to prove the theorem.
The direct translation of Proposition 2.14 says:
Proposition 4.6 cat(f) n if and only if there exists a g such that
pg ' *. If Y represents a wedge of spheres then cat(f) n 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 diagrams of the following form with f
fixed.
____
V ____//_M(V W ) _____//(W )
f g0 h0
fflfflj fflfflp fflffl
A __________//_X___________//_Y
14
Then mcat(f) n if and only if there exists g0 such that the pg0' *.
If Y represents a wedge of spheres then cat(f) n 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 ' *.
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. To get the diagram to commute up to homotopy using
more explicit methods of rational homotopy theory is also straightfor
ward. A third way to get commutativity is to translate the prob
lem to spaces, use the coaction and translate back to CDGAs.) OE
can then be adjusted to make the diagram commute exactly since
V ! M(V W ) is a KS extension. Theorem 4.4 then shows
mcat(f) n. ____
The fibration which V ! M(V W ) ! (W ) models splits
after looping. So the statements when Y represent a wedge of spheres
follow since in that case pg0' * if and only if h0' *.
15
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. Let f"and "hdenote the topological realization of f and h respec
tively. 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 "h' *. Then the fol
lowing five statements hold:
1) cat(f) > mcat(f),
2) cat(f"x Sr) = cat(f") for some r > 0,
3) cat(f"x Sr) = cat(f") for all r > 0,
4) cat(f g) cat(f) + cat(g)  1 for all maps g,
5) cat(f A) cat(f) + cat(A)  1 for some CDGA A.
Proof: 1) follows directly from Proposition 4.7. 2), 3), and 5) are
special_cases_of 4). Let Z be a topological realization of W . Then
r : (W ) ! W represents E1 : Z ! 1 1 Z. Also rationally for
any map, g ' * if and only if 1 g ' *. Therefore hr being null is
equivalent to "h' *. So 4) follows from Theorem 4.5.
The next theorem says that in a range all the five statements of
the last theorem are equivalent.
Theorem 4.9 Let f : V ! A be a map such that cat(f) > cat(jf) =
n. Let "fdenote the topological realization of f. Assume that dim(X)
2n(con(V )+1)2. Then the following five statements are equivalent:
1) cat(f) > mcat(f),
2) cat(f"x Sr) = cat(f") for some r > 0,
3) cat(f"x Sr) = cat(f") for all r > 0,
4) cat(f x g) cat(f) + cat(g)  1 for all maps g,
5) cat(f x A) cat(f) + cat(A)  1 for some CDGA A.
16
Proof: Clearly 3), 4) and 5) all imply 2). Since for every n, mcat(Sn) =
1, 2) implies 1) follows directly from the result of Parent [22] that for
every f; g, mcat(f x g) = mcat(f) + mcat(g). So we just need to show
that 1) implies 4). Assume 1) 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 the generators (indecomposables) of the minimal models in dimen
sions 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
(+). In the extended diagram hr = h0 is null homotopic. Therefore
we can apply Theorem 4.8 to get 4).
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 Sn).
Examples: Let T n(Sl) = {(x1; : :;:xn) 2 (Sl)n for some ixi = *}
denote the fat wedge. Let w : Snl1 ! T n(Sl) denote the higher
order Whitehead product. In other words the attaching map of the
top cell in (Sl)n. Let i : T n(Sl) denote the inclusion and (j) : Sl !
T n(Sl) denote the inclusion of the jth Sl. Let g = [[(1); w]; [(2); w]] 2
ss*(T n(Sl)). Let f : C(g) ! (Sl)n denote any extension of i. Then f
satisfies all the hypothesis of Theorems 4.8 and 3.3. In particular for
every r, cat(f x Sr) = cat(f) = n and mcat(f) = n  1. (Remember
mcat(f) means mcat of a model of the rationalization of f.)
Proof of Examples: Consider the diagram
_g__//_n l ____//_
S2(nl+l)5 T (S ) C(g)

 h f
  
fflffl fflffl"i fflffl
F ________//_"T(nSl)__//_(Sl)n
where the bottom row is a fibration sequence and the map "iis the
inclusion i : T n(Sl) ! (Sl)n turned into a fibration. The only maps h :
17
T n(Sl) ! T n(Sl) such that hi ' i are maps such that h ' id. Porter
[23] shows that F is a wedge of spheres (In this case the result also
follows easily from the cube theorem of Mather [19]. Also rationally
it follows by direct calculation.) [(i); w] lifts to F and has nontrivial
Hurewicz image. It follows easily that g 6' *. Since g 6' *, n =
cat(f) > cat(h) = n1. i[(j); w] ' * so both [(1); w] and [(2); w] lift
to the fibre. So g ' * since it is a Whitehead product. We can then
apply Theorem 3.3 to see that for every r, cat(f x Sr) = 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 [13]. Our example X is also interesting since
cat(X) = cat(XxX) = cl(XxX) = 2 (Corollary 5.4). It is interesting
to compare our example to the one of Fernandez [7]. 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. Let X = (S2 _ S3) [[2;3]fie4p1. Let Y = S1 _ S2 _ S3 [fi
e4p2. n always denotes the inclusion of a sphere of dimension n into
a space.
Theorem 5.1 cat(X) = 2 and for every Z, cat(X xZ) cat(Z)+1.
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 [32]. We must verify the hy
potheses of Theorem 3.4. To show cat(X) = 2 we use Theorem 2.13.
So we must show that for every section s : S2 _ S3 ! (S2 _ S3) of
the evaluation map ev : (S2 _ S3) ! S2 _ S3, Hs([2; 3]fi)W6' *.W
FWib(ev) is a wedge of spheres [34]. In fact F ib(ev) ' S2 _ S3 _
S4 _ higher spheres. We see by looking at H* that for some pro
jection ss : F ib(ev) ! S4 that ss([2; 3]a) is an equivalence. But for
18
every s, H4(s[2; 3]) = 0 since the map factors through a three dimen
sional complex. Therefore for every s, ssHs[2; 3] is an equivalence. It
follows easily that since fi 6' *,
Hs([2; 3]fi) = Hs([2; 3]) O fi 6' *:
So by 2.13 cat(X) = 2. On the other hand 2fi ' * so
Hs([2; 3]fi) = Hs([2; 3])2fi ' *:
So the hypotheses of Theorem 3.4 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 2.13 can be to use, even when there
are many sections.
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 [fie4p2) ' (S6 _ S4p+1 _ S4p+1 [f e8p4. f must be an ele
ment 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.
19
Theorem 5.3 There exists a wedge of spheres W and a cofibration
sequence.
W ! Y _ Y ! X x X
Proof: From Lemma 2.5 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.
"H*(Y ) ~=Z(p)(a; b; c; e)
That is the free Z(p)module on generators a = 1, b = 2, c = 3 and
e = 4p  2. Let Z be a wedge of spheres such that
"H*(Z) ~=(Z(p)(a; b; e) Z(p)(a; b; e)):
Let i : Z ! Y * Y be a map which induces the inclusion
(Z(p)(a; b; e) Z(p)(a; b; e)) ! (Z(p)(a; b; c; e) Z(p)(a; b; c; e))
on "H*. This can be done because of Lemma 5.2.
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. A homology calculation
shows that H*((p x p)g) is surjective. Let h : S4 _ S4 ! Y _ Y
denote (4  [2; 3]) _ (4  [2; 3]). Then we get a diagram
fi+h // //
Z _ S4 _ S4 _____Y _ Y _____C(fi + h)
p_p OE
fflffl fflffl
X _ X _______//_X x X
20
where OE is an extension of (p x p)g. Thus OE is easily seen to be an H*
isomorphism.
Recall that cl(X) denotes the cone length of X.
Corollary 5.4 cat(X) = cat(X x X) = cl(X) = 2
More generally we beleive for every n there exists a simply connected
space Z such that cat(Z) = cat(Zn) = n. Perhaps an easier 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.
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