Spaces with Lusternik-Schnirelmann category n and cone length n + 1
Donald Stanley
June 25, 1998
Abstract
We construct a series of spaces, X(n), for each n > 0, such *
*that
cat(X(n)) = n and cl(X(n)) = n + 1. We show that the Hopf in-
variants determine whether the category of a space goes up when
attaching a cell of top dimension. We give a new proof of count*
*erex-
amples to a conjecture of Ganea. Also we introduce some techniq*
*ues
for manipulating cone decompostions.
1 Introduction
Let X be a space. The Lusternik-Schnirelmann category, LS categ*
*ory
denoted cat(X), was intoduced in the 20's as an invariant which*
* gave a
lower bound for the number of critical points of a function on *
*a closed
manifold [14]. The original definition was essentially the min*
*imum
size of a covering of the manifold by open sets which are contr*
*actible
in the manifold. An equivalent, for CW complexes, definition w*
*as
given by Whitehead in terms of the minimum fat wedge (Definition
2.7) through which the iterated diagonal factors. The definitio*
*n which
we will use (Definition 2.6) is due to Ganea [9] and is again e*
*quivalent
to the other two (except for the renormalization of subtracting*
* one)
for CW complexes.
The strong category, denoted Cat(X), was introduced by Fox [*
*8].
In this same paper he gave an example to show that the notion w*
*as not
homotopy invariant. This problem is overcome by taking the mini*
*mum
over all spaces of the same homotopy type. He also showed that
cat(X) Cat(X). The cone length (Definition 2.9) was introduced
by Ganea [9] where he at the same time proved that cone length *
*and
strong category are equal. A few years later, Takens [21], and *
*Ganea
too, showed that cone length and LS category differed by at mos*
*t one.
However the only examples where the two invariants where known *
*to
differ were coH spaces that were not suspensions, that is cat(X*
*) = 1,
cl(X) = 2. (The first example actually appeared in [2]). Almost*
* 30
1
years later Dupont constructed an example of a space of category three
and cone length four.
The reason that the case of cl(X) = 1 is easier is that we have
the characterisation, cl(X) = 1 if and only if X = Y . To see that
X is not of some higher cone length we have to somehow look at all
possible cone decompositions of it. To solve this problem we develop
techniques for handling general cone decompositions. We introduce
the notion of an r-cone receiver (Section 5). We construct an r-cone
receiver for our particular space, X, of interest (Section 6). We then
take any r-cone decompostion of X and map it into its r-cone receiver.
This allows us to derive from any r-cone decompostion one of a very
specific form (Section 7). We then find an obstruction to such an r-
cone decompostion (Section 8). The methods of this section are in
the same spirit as some basic ideas of single loop space decompostion
theory. I feel endebted to Paul Selick for introducing me to such
ideas. The obstruction is essentially the attaching map of a cell in a
loop space. For X to have an r-cone decompostion this map must be
trivial since the corresponding homology class needs to be in the image
of Hurewicz. To show cat(X) = r we use a variant of a technique of
Iwase [12]. The proof depends on homotopy classes which have non
trivial Hopf invariants which become trivial after n suspensions. Taken
together we reach our main goal.
Theorem 1.1 9.3 For every r > 0 there exist space X such that
cat(X) = r and cl(X) = r + 1.
There are a number of other interesting results in the paper. We
show that the Hopf invariant (Definition 3.2) of Bernstein-Hilton [2]
characterises when attaching cells of top dimension cause the LS cat-
egory to go up (Theorem 3.4). This fact was known to them in case
dimension (X) (cat(X) + 1)connectivity(X) - 2. Now, however,
Hopf invariants can be used to determine the category of any finite
complex. We offer some cone decompositions of spaces of the form
Q x Sn (Theorem 4.2). This gives an alternative (to [12]) proof of
counterexamples to a conjecture of Ganea. Also we give some general
lemmas for constructing r-cone receivers out of other ones (Lemmas
5.4, 5.5, 5.6 and 5.7). These may prove usefull in the study of cone
length of other spaces.
I would like to thank Hans Scheerer for suggesting that I try to
compute the cone length of Iwase's example and Manfred Stelzer for
bringing the paper of Brayton Gray to my attention. I would also like
2
to thank Pascal Lambrechts and the University of Artois for inviting
me as a visitor. It was during my stay there that my interest in LS
category was kindled and many of the techniques used in this paper
were learnt. I again thank Pascal Lambrechts for supplying me with
inexpensive housing during my stay.
2 Notation and Background
In this section we introduce some notation and definitions that will be
used throughout the paper. A few facts about CW complexes are also
presented. We make no claim to the originality of any of the results
in this section.
For the rest of the paper we assume, unless otherwise stated, that
all our spaces are simply connected, pointed and have the homotopy
type of a CW complex. For any spapce we will let * denote the
basepoint.
Such spaces and maps between them for a category. In general
whenever we refer to a map we will assume unless we state otherwise
that it is a map in this category. For this section let X be a space.
We let Xn denote the nth Cartesian product of X with itself. When
we say that X is a CW complex we mean that X is a space with a
particular CW structure. If we are working p-locally Sn and en both
denote the p-local versions. We also let * denote the binary operation
of the join. It is a fact that for spaces X; Y , X * Y ' X ^ Y .
Lemma 2.1 For i = 1; 2 let f(i) : A(i) ! B(i) be maps. Then there
is a cofibration sequence
A(1)*A(2) ! C(f(1))xB(2)[B(1)xB(2)B(1)xC(f(2)) ! C(f(1))xC(f(2)):
This sequence is natural in both variable. In other words if for i = 1; 2
we have commuting diagrams
f(i)
A(i)_____//B(i)
| |
| |
fflffl|f0(fflffl|i)
A0(i)____//_B0(i)
3
then we get a commuting diagram
A(1) * A(2)______//_C(f(1)) x B(2) [B(1)xB(2)B(1) x C(f(2))____//C(f(1)) x C(f*
*(2))
| | |
| | |
fflffl| fflffl| fflffl|
A0(1) * A0(2)___//C(f0(1)) x B0(2) [B0(1)xB0(2)B0(1) x C(f0(2))//_C(f0(1)) x C*
*(f0(2)):
Proof: Standard.
The last lemma implies that if X and Y are CW complexes then
X xY is also canonically a CW complex (once we have fixed a homeo-
morphism Sn+r+1 ~=Sn *Sr). It is clear that X _Y is also canonically
a CW complex. Also for every map f : Sn ! X, C(f) is canonically
a CW complex. Any skeleton of a CW complex is canonically a CW
complex. We will call all such CW structures the canonical ones. For
Sn we all its minimal cell structure the canonical one. That is the cell
structure with one 0-cell and one n-cell. All this implies that we have
a canonical cell structure on (Sn)r and therefore on T r(Sn).
We let con(X) denote the connectivity of X. For a map f : X ! Y ,
con(f) denotes the connectivity of f. So con(f) = n if and only
if ss*(f) is an isomorphism for * < n and a surjection for * = n.
Throughout the paper we make use of the Serre exact sequence. For
a statement and proof of it we refer the reader to Whitehead [23]. Let
dim(X) denote the dimension of X. This means the dimension of the
top non-zero cohomology class of X. If X is a CW complex we let Xn
denote the n skeleton of X, that is the subcomplex with all cells of
dimension less than or equal to n.
For a map f : Sr ! X we let fa : Sr-1 ! X denote the adjoint.
Let f : X ! Y be a map. Let C(f) denote the reduced cone on
f. Explicitly
C(f) = (Y [ X x I)=((x; 1) = f(x); (x; 0) = (*; t) = *):
C(f) is also refered to as the homotopy cofibre of f. There is a canon-
ical map (inclusion) i(f) : Y ! C(f). A sequence X __f_//_Y_g__//Z
is called a cofibration sequence if Z ' C(f) in a way compatible with
g and the canonical inclusion. In other words such that the obvi-
ous diagram commutes up to homotopy. If we have a commutative
diagram
f
X _____//Y
| |
| |
fflffl|ffflffl|0
X0 ____//_Y 0
4
then we get a canonical map C(f) ! C(f0) in the obvious way. Dually
let F ib(f) denote the path space on f. Explicitly
F ib(f) = {(OE; x) 2 Y Ix X|OE(1) = x; OE(0) = *}:
F ib(f) is also refered to as the homotopy fibre of f. There is a canon-
ical map (evaluation) F ib(f) ! X. A sequence Z _g__//_X_f_//_Yis
called a fibration sequence if Z ' F ib(f) in a way compatible with g
and the canonical evaluation. Again this means we require the obvi-
ous diagram to commute up to homotopy. If we have a commutative
diagram
f
X _____//Y
| |
| |
fflffl|ffflffl|0
X0 ____//_Y 0
then we get a canonical map F ib(f) ! F ib(f0) in the obvious way.
It is obvious that taking the homotopy cofibre keeps us in the
category of spaces having the homotopy type of CW complexes. It
follows from [16] that taking homotopy fibres does as well. In fact it
follows directly if f is a cofibration and in general from the fact that
the cateogry of spaces having the homotopy type of CW complexes is
closed under retracts.
If X is a CW complex then there is a cellular chain complex C*(X)
with a canonical basis induced by the cell structure.
Lemma 2.2 Let X be a CW complex. Then for every change of
basis of C*(X) there exists a CW complex X0 and a cellular homotopy
equivalence f : X ! X0 such that C*(f) induces the change of basis.
If X is a suspension we can assume that X0 and f are suspensions
too.
Proof: Straightforward.
Lemma 2.3 Let x 2 Hn(X) be indivisible and in the image of Hure-
witz. Then there exists a CW complex homotopy equivalent to X with
a cell "xthat has trivial attaching map and represents x in cellular
homology.
Proof: Straightforward.
The following theorem is often refered to as the cube axiom. A
more general theorem appeared in [19].
5
Theorem 2.4 [15] Consider the following cubical diagram.
A ____________//_BBBB
| BBB | BBBB
| BB | BB
| B | B!!
| C _______|_____//D
| | | |
| | | |
| | | |
fflffl| | fflffl||
A0 ______|_____//_B0 |
AA | BB |
AAA | BBB |
AA | BB |
A fflffl| B fflffl|
C0 ____________//_D0
Assume that the bottom face is a homotopy pushout and that all sides
are homotopy pullbacks. Then the top face is a homotopy pushout.
Definition 2.5 Let X be a space. We define fibration sequences.
pn
Fn(X) _in_//_Gn(X)____//X
Let G0(X) = * and p0 the inclusion. Let Fn = F ib(pn) and Gn+1 =
C(in). We get the extension pn+1 by mapping Fn x I to *.
Our definition differs from Ganea's original one [9] in that we do not
turn our pn's into fibrations. This procedure is implicit in taking
homotopy fibres. Because of this diference our next definition also
differs from Ganea's in that we only require homotopy sections.
Definition 2.6 We say a space X has category less than or equal to
n, cat(X) n, if and only if pn has a homotopy section.
Definition 2.7 Let
T n(X) = {x1; : :;:xn 2 Xn|xi= * for somei}:
T n(X) is called the fat wedge.
An alternative definition of category, due to Whitehead, uses the fat
wedge. In his definition X has category less than or equal to n if
and only if the iterated diagonal X ! Xn+1 factors throguh T n+1(X)
up to homotopy. The two definitions concur on a subcategory of the
category of topological spaces containing the category of spaces having
the homotopy type of a CW complex.
We state a lemma proved by Doeraene [5] in the context of J cate-
gories. What follows is an application to T op* where it was originally
proved by Gilbert [10].
6
Lemma 2.8 [5] For every n and every X the following diagram is a
homotopy pull back.
q
Gn(X) ____//_T n+1(X)
|pn| ||
fflffl| fflffl|
X ________//_Xn+1
In particular for every Y [Y; Gn(X)] = [Y; X] x[Y;Xn+1][Y; T n+1(X)]
where the isomorphism is induced by [Y; pn] and [Y; q] and is therefore
natural in Y .
Proof: The proof of the first statement is in [5]. The second statement
follows from the definition of homotopy pullback.
Notice that the Lemma directly implies that 2.6 gives the same
definiton of cat as Whitehead's.
Let X be a space. Let X(i); W (i) be sequences of spaces such that
X(0) = *, X(n) = X and such that there exist cofibration sequences
W (i) ! X(i) ! X(i + 1):
We then call {X(i)} an n-cone decomposition of X.
Definition 2.9 Let cl(X) denote the cone length of X. That is the
minimal n such that X has an n-cone decomposition.
It has been shown by Cornea [3] that if X has an n-cone decomposition
then it has one where each W (i) is an i fold suspension.
3 cat
In this section are a number of results about cat. We first define a Hopf
invariant equivalent to that of [2]. We show that this Hopf invariant
determines when attaching cells of top dimension causes the category
to go up. We also show that if X 6' * then cat(Xn) cat(X). At the
end of the section we prove a geometric result which has as a corollary
a relative version of the fact that cup length is a lower bound for cone
length. We begin with a lemma that is trivial to prove but crutial in
the proof of 3.4.
7
Lemma 3.1 In the category of R modules let a commuative diagram
f0
M B____//_BX
BBB p|
f BBB fflffl||
Y
and a map r : Y ! X be given. Assume that pr = id, pf0 = f,
im(rf) im(f0) and that f is injective. Then rf = f0.
Proof: For any a 2 M. There is a b such that rf(a) = f0(a) + f0(b).
So prf(a) = pf0(a) + pf0(b). So pf0(b) = 0. So b = 0.
Definition 3.2 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)]
It follows from Lemma 2.8 that when we take the map X ! T n+1X
coresponding to out r : X ! GnX our definition of Hopf invariant is
the same as the one in [2].
Some version of the following theorem was previously know to
Halperin, Iwase and perhaps others. It improves on a result of Cornea
[4] which states that cat(Xn) cat(X) + 1. He conjectured that this
improvement could be made. It follows that the main result of the
same paper can be improved by one in the infinite case.
Theorem 3.3 Let X be connected but not necessarily simply con-
nected, cat(X) = n > 0 and r : X ! GnX be a section. Then
there are compatible sections rs : Xs ! GnXs of the Ganea fibration
ps : GnXs ! Xs.
Proof: Assume that we have already defined the compatible sections
for s < l.
Then we have the following commutative solid arrow diagram
rl-1
Xl-1 _____//GnXl-1
| |
| |
fflffl|rl fflffl|
Xl _ _ _ _//GnXl
|il| Gnil||
fflffl|r fflffl|
X _______//GnX:
8
We wish to have a section rl such that the diagram still commutes.
We can get a map rl extending rl-1 since Gnil is a ssl-1 isomor-
phism. We can adjust rl so that the bottom square of the above dia-
gram commutes since Gnilis a sslsurjection. Finally we can adjust rl
so that it is a section since plinduces a surjection kersslGnil! kersslil.
This completes the induction step and thus the theorem.
The following theorem gives a characterization of cat in terms of
Hopf invariants. It was already known in the case dim(X) (cat(X)+
1)con(X) - 2 [2].
Theorem 3.4 Let X be a space that is connected but not necessarily
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)) = *.
Proof: The if direction was done in [2]. So we will assume that
cat(Y ) n. Then by the previous theorem we have a commutative
diagram
r(X)
X ____//_GnX
|i| |Gni|
fflffl|r(fflffl|Y )
Y ____//_GnY
such that r(X) and r(Y ) are sections for the corresponding Ganea
fibrations p(X) : GnX ! X and p(Y ) : GnY ! Y .
The existence of an extension of r(X) to r(Y ) implies that im((_f(i))r)l
ker((Gni)l) = im(((_f(i))a)l). Observe that since suspension and
adjoint are group homomorphisms so is their composition ssl(X) !
ssl((X)). Therefore we can extend ((_f(i))a)* to ssl(_Sl)=ker(_f(i)).
This gives us a commutative diagram
((_f(i))a)*
ssl(_Sl)=ker(_f(i))*_________//_Ussl(GnX)
UUUU(_f(i))*UUUU |
UUUUU |p(X)*
UUUU** fflffl|
ssl(X):
Together with the map r(X)* we then have the setup of Lemma 3.1.
So we can conclude that (_f(i))l = ((_f(i))a)l and hence also the
theorem.
9
Lemma 3.5 Let fi : Sl ! Sr be a coH map, f : Sr ! X any map
and s : X ! Gn(X) a section of the Ganea fibration. Then Hs(ffi) =
Hs(f)fi.
Proof: The lemma follows since (ffi)a = fafi if fi is coH.
Lemma 3.6 Let fi : Sr ! Sn be a coH map. Let f : Sn ! X be any
map. Then X _Sn [fier+1 ' (X _Sn)[fi+ffier+1. Therefore if X 6' *
cat(X _ Sn) [fi+ffier+1 = catX and if clX 2 then cl(X _ Sr [fi+ffi
er+1) clX
Proof: Clear.
Note that if fi is not a coH map then the previous two lemmas will
generally not be true. For an example take fi = [; ] : S15 ! S8 and
f = [1; 2; 3] : S8 ! T 3(S3).
Lemma 3.7 Let A _f__//_B___//_Cbe a cofibration sequence. Then
: C ! C x C factors
0
C ____//_NNNC _ C [ B x B
NNNNN |
NN |i
NN&&fflffl|
C x C:
Proof: Consider the following solid arrow diagram where the first
three spaces in both rows form cofibration sequnces
f
A _________//B____________//COPPP
PPP
|| O0O PPPPPP
f_f fflffl| fflffl i PP((
A _ A ____//_B x B____//C _ C [ B x B____//_C x C:
There exists a dashed extension 0making the square commute since
the obstruction in [A; B] maps to 0 in C. Also - 0i is an element
of [A; Y x Y ] a lift of which can be added to 0to make the triangle
commute since [A; C _ C [ B x B] ! [A; C x C] is a surjection.
The following lemma can be interpreted as a relative version of the
fact that cone length is at least as big as cup length.
Corollary 3.8 Let A ____//_B_g_//_Cbe a cofibration and k a field.
Then if fffi 2 H*(C; k) and ff; fi 62 H0(C) then g*(ff) 6= 0.
10
Proof: From Lemma 3.7 we have a factorization
*
C*(C) C*(C) __i__//C*(B) C*(B) + C*(C) C*(C)
VV
VVVVVV*VVV 0*|
VVVVV |
VVVV++V fflffl|
C*(C):
Therefore for ff; fi 62 H0(C) *(fffi) = 0*(g*(ff)g*(fi)). The result
follows easily.
We also have the dual result which is a relative version of cocup
length being a lower bound for category.
Corollary 3.9 Let A _____//B_g__//_Cbe a cofibration sequence. Then
__
if x 2 H*(C) is such that x = ix0i x00ithen x0i= g(y).
Proof: Use the dual of the proof of the last corollary.
4 r-Cone Decompositions
In this section we construct spaces Q such that cat(Q) = r and cl(Qx
Sn) = r for large n. Since cl is an upper bound for cat this also
gives an alternative (to [12]) proof that there are counterexamples to
Ganea's conjecture.
For this section let fi : Ss-1 ! Srl-1 such that s > rl be any
map and let w : Srl-1! T r(Sl) denote the higher order Whitehead
product. In other words the attaching map of the top cell in the
product. Let Q = T r(Sl) [wfies.
Lemma 4.1 Let s; l 2. Let f : Ss-1 ! Sl be a coH map such that
n-1f ' *. Then nC(f) is homotopic to a wedge of spheres.
Proof: Since f is coH, C(f) splits off of (Sl[fa es-1) and we see
there exists a space X and a wedge of spheres W such that C(f)_W '
X. From James [13] (see [23]) nX ' _i>0nX^i. It is enough
to see that each wedge summand is a wedge of spheres since then
nC(f) would be a retract of a wedge of spheres and so a wedge
of spheres. We use induction. Assume that nX^(i-1)is an n - 1
connected wedge of spheres. Then clearly nX^i is since n-1Y is.
Thinking of X^0 as S0 we have completed the induction.
Theorem 4.2 If nfia ' * then cl(Q x Sn) = r.
11
Proof: Let Z = T r(Sl) [ T r(Sl)(r-1)lx Sn. Since Z _ (Srl-1[fia
es-1) has cone length r - 1, it is enough to construct a cofibration
sequence (+)
W ! Z _ (Srl-1[fiaes-1) ! Z0
such that Z0' Q x Sn.
Let a : Srl-1! Srl-1[fies _ Sn and b : Sn ! Srl-1[fies _ Sn
denote the inclusions. First we construct a cofibration sequence (*).
[a;b]+h rl-1 s np rl-1 s n
Srl+n-2_ Ss+n-1 __________//_S [fie __S___//S [fie x S
F ib(p) = (Srl-1[fies) * Sn and so by Lemma 4.1 it is a wedge of
spheres. We have a cofibration sequence.
___h0//_rl-1 s n rl+n-1p0_//_rl-1 s n
Ss+n-1 S [fie _ S [[a;b]e S [fie x S
The induced map OE : F ib(p) ! F ib(p0) is an H* surjection. Therefore
F ib(p0) is a wedge of spheres and OE is a ss* surjection. So since h0fac-
tors through F ib(p0) it factors through F ib(p) and we get a cofibration
sequence of the form (*).
We can now construct our sequence (+). Let U be a wedge of
spheres and f : U ! Z a map such that C(f) ' T r(Sl) x Sn. Let
f0 : Srl-1! Z _ (Srl-1[fiaes-1) be the map that is id - w on
the bottom sphere and id on the other spheres. Let f00: Srl-1[fies-1_
Sn ! Z _ (Srl-1[fiaes-1) be the inclusion on Sn and a section of
the map (Srl-1[fiaes-1) ! Srl-1[fies-1 on Srl-1[fies-1. Let
= f + f0+ hf00. We check that there is a map C() ! Q x Sn that
is an H* equivalence. This is easy to see except on the top cell. There
it is enough to look at the commutative diagram.
Srl-1[fies _ Sn _______//_Srl-1[fies x Sn
| |
| |
fflffl| fflffl|
Z _ (Srl-1[fiaes-1) __________//C()
There is a canonical map Srl-1[fies ! Q extending w and the product
of this with the identity. So we get an extension C() ! QxSn which
is a Hs+n isomorphism.
It is in fact the breakdown of the proof of the previous theorem
when nfia 6' * that points us in the direction of the examples we are
12
seeking. Essentially we will use the fact that we have no cofibration
sequence of the form (*) in the proof of the theorem to show we cannot
have any r-cone decomposition of Q x Sn. Since nfia 6' * a certain
homology class is not in the image of Hurewicz. But any n-cone de-
composition can be adjusted to show that that element must be in the
image of Hurewicz.
Theorem 4.3 Let fi be a coH map. Then catQ = r if and only if
fi 6' * otherwise catQ = r - 1. If nfia ' * then catQ x Sn = r
Proof: Let i : Sl ! Ml(p) be the inclusion. Let s : T r(Sl) !
Gr-1(T r(Sl)) and s0 : T r(Ml(p)) ! Gr-1(T r(Ml(p))) be sections of
the corresponding Ganea fibrations. By [2] Proposition 2.5 together
with Lemma 2.8 they are unique. A direct proof of this fact using
connectivity arguements is also straightforward. Since T r(i) is a map
of n-cones the sections are compatible with T r(i) and Gr-1(T r(i)).
So Hs0(T r(i)w) = *r-1i(Hs(w)). Now we see using cup length for
H*(_; Z=p) that cat(T r(Ml(p))[Tr(i)werl) = r. Therefore Hs0(T r(i)w) 6=
*. But since con(*r-1T r(Ml(p))) = lr - 2, Hs(w) is nontrivial and
in fact is not p divisible. The fact that T r(Sl) is a product of spheres
together with Lemma 3.5 then imply the first statement in the theo-
rem.
To prove the second part we use a variant of the method of Iwase.
We have a strictly commuting diagram such that the rows are cofibra-
tion sequences.
_____i_// r l s-1_______//
Ss-1 T (S ) [fiwCylS Q
| |
fi|| f|| |id|
fflffl|j fflffl| fflffl|
Slr-1 ____//_Slr-1_ T r(Sl) [(j-w)fies//_Q
Where i is the inclusion into the end of the cylinder and j and f are
the inclusions. The map on CylSs-1 is the homotopy between wfi and
jfi given by the s cell in the image space.
So using Lemma 2.1 we get another strictly commuting solid arrow
diagram such that the rows are cofibrations.
p // n
Ss-1 * Sn-1______//_(T rSl[ CylSs-1) x Sn [_Q____Qjx S
j j
|fi*Sn-1| |fxSn[id|j jj =||
fflffl| fflffl|uujjh p0 fflffl|
Slr-1* Sn-1_____//(Slr-1_ T r(Sl) [ es) x Sn_[_Q_//Q x Sn
13
Since fi * Sn-1 = nfi ' * there exists h such that hp ' f x Sn [ id.
By an easy H* calculation p0h is a weak equivalence. Now it follows
from Lemma 3.6 that cl((Slr-1_T r(Sl)[es)xSn [Q) r. Therefore
cat(Q x Sn) r.
Examples: We can now use the results of Gray [11] (Corollary 9.2)
to see that at every prime p > 2 and every r,n there exists Q such
that cat(Q) = r, cat(Q) x Sn = r and cl(Q x Sn+1) = r. In the next
few sections we will build up the machinery to show that in may cases
cl(Q x Sn) = r + 1.
5 Ganea Sequences and n-Cone Re-
ceivers
In this section we introduce n-cone receivers (Definition 5.1). They
are the main tool we use to transform general n-cone decompositions
into ones of a more manageable form. We prove some lemmas for
constructing n-cone receivers out of other ones. The notion of Ganea
sequence is an example of an LS sequence of Scheerer-Tanre [20].
Fix 0 n 1. Let X be a space. For 0 i n let Y (i)
be spaces and k(i) : Y (i) ! Y (i + 1) and p(i) be maps such that
p(i) = p(i + 1)k(i). We call Y = (Y (i); k(i); p(i)) an n-sequence over
X. In other words it is just a functor from the category of the ordinal
n to spaces over X. For 0 l 1.
Consider the following properties:
1) Y (0) = *
2)l k(i) is attaching a cone on a space of dimension at most l.
2')lk(i) is attaching a cone on a suspension of a space of dimension
at most l - 1.
3)l For every W of dimension less than or equal to l and i 1,
[W; p(i)] is surjective or equivalently for every i 1, p(i) splits up
to Xl In other words there exists a commutative diagram.
(X)l
uuu |
uuu |
zzuuup(ifflffl|)
Y (i) _____//_X
4)l For every W of dimension less than or equal to l and i 1,
ker[W; k(i)] = ker[W; p(i)] or equivalently for every i 1, the induced
14
map F ib(p(i))l! Y (i + 1) is trivial.
4')lFor every W of dimension less than l and i 1, ker[W; k(i)] =
ker[W; p(i)] or equivalently for every i 1, the induced map (F ib(p(i)))l-1!
Y (i + 1) is trivial.
Definition 5.1 We call an n-sequence over X an l-dimensional n-
cone over X if it satisfies 1)l and 2)l and an l-dimensional strong
n-cone over X if it satisfies 1)l and 2')l. We call an n-sequence over
X an l-dimensional n-cone receiver for X if it satisfies 3)l and 4)l
and an l-dimensional strong n-cone receiver for X if it satisfies 3)l
and 4')l. We call an n-sequence over X an n-Ganea sequence for X
if it satisfies 1)1 , 2)1 , 3)1 and 4)1 and a weak n-Ganea sequence
for X if it satisfies 1)1 , 2)1 , 3)1 and 4')1 .
In all of these cases we will (ab)use the notation (Y (i))in if the maps
are clear from the context. Also if l = 1 we will often drop the prefix
l-dimensional from our definitions. Let Y 0= (Y 0(i); k0(i); p0(i)) also
be a n-sequence over X. Then a map of n-sequences over X, f : Y !
Y 0, is just a natural transformation between the two corresponding
functors. In other words a set of maps f(i) : Y (i) ! Y 0(i) such that
f(i + 1)k(i) = k0(i + 1)f(i) and p(i) = p0(i)f(i).
Examples: The constant diagram with value X is always an n-cone
receiver for X. The Ganea spaces (Gi(X))in form a (weak) n-Ganea
sequence for X. Also notice that any strong n-cone is an n-cone, any
n-cone receiver is a strong n-cone receiver and any n-Ganea sequence
is a weak n-Ganea sequence.
The name n-cone receiver is justified by the following lemma.
Lemma 5.2 Let Y 0= (Y 0(i); k0(i); p0(i)) be an n-sequence over X if
Y 0is a (strong) l-dimensional n-cone receiver for X then for every
(strong) l-dimensional n-cone over X, Y , there exists a map f : Y !
Y 0of n-sequences over X. If Y 0is in fact a strong l-dimensional
n-cone receiver then the converse is true as well.
Proof: We use the numbering notation of Definition 5.1. We prove the
first statement. We make a proof by induction. From condition 1) we
have a map f(0) : Y (0) ! Y 0(0) compatible with the p(i), p0(i), k(i)
and k0(i). Assume we have already constructed f(r) : Y (r) ! Y 0(r)
compatible with the p(i), p0(i), k(i) and k0(i). Let W ! Y (r) !
Y (r + 1) be a cofibration sequence. Then W ! X is trivial so by 4)
we get an extension of f(r) to f(r + 1) : Y (r + 1) ! Y 0(r + 1) and
15
then by 3) we can adjust it using the coaction to make it compatible
with p(r + 1) and p0(r + 1). The other compatibilities are automatic
since f(r + 1) extends f(r). The converse is similarly straightforward.
The previous lemma has as a corollary that any n-Ganea sequence
can be used to define LS category. Essentially the same lemma was
proved by Sheerer-Tanre [20].
Corollary 5.3 Let (Y (i); k(i); p(i)) be an n-Ganea sequence for X.
Then cat(X) n if and only if p(n) has a section.
Proof: Compare the sequence to the Ganea sequence.
Next we give a few lemma concerned with constructing cone re-
ceivers out of other ones.
Lemma 5.4 Let Y = (Y (i); k(i); p(i)) be a (strong) l-dimensional n-
cone receiver for X. Then for any W , Y x W = (Y (i) x W; k(i) x
id; p(i) x id) is a (strong) l-dimensional n-cone receiver for X x W .
Proof: Clear.
We do not use the following lemma but include it nonetheless.
Lemma 5.5 Let Y be a strong n-cone receiver for X. Then for every
W , Y _ W = (Y (i) _ W; k(i) _ id; p(i) _ id) is a strong n-cone receiver
for X _ W .
Proof: Consider the following diagram where the rows and collumns
are fibration sequences and j and j0 are the inclusions.
F 0(i)____//Y (i) * W ____//_X * W
| | ||
| | |
fflffl| fflffl|p(i)_id fflffl|
F 00(i)____//Y (i) _ W______//_X _ W
| | | 0
| j| |j
fflffl| fflffl|p(i)xid fflffl|
F (i)______//_Y (i) x_W_____//X x W
Since p(i) has a section we know that for some F (i), Y (i) '
X x F (i). So (*)
Y (i) * W ' X * W _ F (i) ^ W _ X ^ F (i) ^ W:
Since the map F (i) ! F (i+1) induced by Y (i) ! Y (i+1) is trivial, we
see that the induced map on the second and third factors of the wedge
16
decomposition (*) is trivial also. This implies that the induced map
F 0(i) ! F 0(i + 1) is trivial. By assumption the map F (i) ! F (i + 1)
is trivial after looping. j and j0 have compatible sections. So
F 00(i) ' F (i) x F 0(i) and the map F 00(i) ! F 00(i + 1) must be
trivial after looping.
Lemma 5.6 Let f : Sr ! X be a map and let g : X ! C(f) denote
the inclusion. Assume that g has a section. Let Y be an n-sequence
over X such that (Y (i); k(i); gp(i))is a strong n-cone receiver for C(f).
Then Y _Sn = (Y (i)_Sn; k(i)_id; p(i)+f) is a strong n-cone receiver
for X.
Proof: First observe that by using the cube axiom it is easy to see
that X ' (X [f er+1 _ Sr). Consider the map between fibration
sequences
F 0____//_(Y (1) _ Sr)___//(Y (1))
| p(1)_f| |p(1)
| | |
fflffl| fflffl| fflffl|
F _________//X _________//(C(f)):
Since F 0' Sr_Sr^Y (1), we see in fact that a homotopy equivalence
is induced by the sequence
p(1)+f
(C(f) _ Sr) __h__//(Y (1) _ Sr)___//_X
where h is some section of (gp(1) _ id). So we have induced weak
equivalences F ib(gp(i)_id) ! F ib(p(i)+f) such that the following
diagram commutes.
F ib(gp(i) _ id)________//F ib(p(i) + f)
| |
| |
fflffl| fflffl|
F ib(gp(i + 1) _ id)___//F ib(p(i + 1) + f)
Since the left hand arrow is trivial, the right hand one is as well.
The map f in the last lemma would appropriately be called a
homotopy inert map. The lemma then says that we can construct n-
cone receivers for spaces with inert homotopy classes by constructing
n-cone receivers for the cone on the homotopy inert map.
The next lemma says that if we have any n-cone receiver for any
space, we can construct an l-dimension n-cone receiver for that space
with another cell attached.
17
Lemma 5.7 Let (Y (i); k(i); p(i)) be a (strong) n-cone receiver for X
and assume that Y (i) and X are r-connected. Let s : X ! Y (1) be
a section of p(1). Let f : Sl ! X be a map. Denote g(1) = sf and
g(i) = k(i)g(i - 1). Let Y 0(0) = Y and for i > 0 Y 0(i) = C(g(i)). Let
k0(i) and p0(i) be the canonical extensions. Then (Y 0(i); k0(i); p0(i)) is
a (strong) l + r - 1 dimensional n-cone receiver for C(f).
Proof: Use the Serre exact sequence to show that F ib(p(i)) ! F ib(p0(i))
is an l + r - 1 equivalence. It is also easy to construct a section of
p0(i) up to dimension l + r - 1 which extends s.
It seems likely that the number l + r - 1 in the last lemma can be
improved to l + 2r - 1.
6 Eilenberg-Moore Spectral Sequence
The main object of this section is to construct an n-cone receiver
for a product of spheres (Lemma 6.9). To do this we calculate the
fibre of the inclusion of a skeleton into the product and the induced
map between such fibres. The method is to calculate the homology of
the fibres and the induced maps on homologyusing the relative cobar
construction. The primary reference for this section is [7].
Let DGC denote the category of differential graded coalgebras.
We will also refer to the objects(maps) in DGC as DGC's(maps of
DGC's).
For a map A ! B in DGC recall the definition of (A; B) from
[7].
Lemma 6.1 [7] Let A; A0; B; B0 be projective as R modules. In DGC
assume we have a commutative diagram
p
A _____//B
f|| g||
fflffl|pfflffl|0
A0_____//B0
such that f and g are weak equivalences. Then the induced map
(A; B) ! (A0; B0) is a weak equivalence.
The last lemma leads naturally to the next definition.
18
Definition 6.2 Let f : A ! B be a map in DGC. We call f formal
if we have a commuatative diagram
H*(f)
H*(A) _____//H*(B)
i(A)|| |i(B)|
fflffl|f fflffl|
A _________//B
such that i(A) and i(B) are weak equivalences.
Lemma 6.3 [17] The Eilenberg-Zilber map, EZ : C*(A) C*(B) !
C*(A x B), is a map in DGC.
Lemma 6.4 Let
f
A ____//_B
| |
| |
fflffl|fflffl|g
C ____//_D
be a homotopy pushout in DGC. Then if f is formal so is g.
Proof: Clear.
Lemma 6.5 The inclusion C*(Sl)rnl! C*(Sl)rslis formal for every
n s.
Proof: It follows from the last two.
Lemma 6.6 Let
A ____//_B
| |
| |
fflffl|fflffl|
C ____//_D
be a diagram in DGC with all maps formal. Then the induced map
H*(A; B) ! H*(C; D) is the same as the induced map (interpreted
as composed with the isomorphisms of Lemma 6.1) H*(H*A; H*B) !
H*(H*C; H*D). If all the dgc's are of finite type and connected
this map is the same as the dual of the induced maps of E2 terms
of EMSS's.
p+q=nT orp;qH*D(Z=p; H*C) ! p+q=nT orp;qH*B(Z=p; H*A)
19
Proof: The first equality is straightforward. The second one follows
since ((A; B))* = B(A*; B*) when A and B are finite type.
For the next three lemmas let F (r) denote the fibre of the inclusion
((Sl)n)lr! (Sl)n.
Lemma 6.7 Let r 1. Let Es+1;*1= "H*((Sl)n)s H*(((Sl)n)lr) be
the E1 term of the EMSS converging to H*(F (r)). Then in Es+1;*2
we can represent all elements by sums of elements of the form a(1)
: : :a(s) fi. Where a(i) 2 "H*((Sl)n) are indecomposable and fi 2
H*(((Sl)n)lr) has product length r.
Proof: Let ff 2 Z*(Es+1;*1). Grade Es+1;*1left lexicographically by
cohomological product length and assume that ff is in the minimal
gradation. There are two cases. In the first case some of the elements
in "H*((Sl)n) are decomposable and in the second case all are indecom-
posable. We show that the first case cannot occur and then derive the
lemma in the second case.
Case 1)
In this case there exists and integers t and q, indecomposables
a(k; i; j) 2 H*((Sl)n), decomposable fi(i; j) 2 H*((Sl)n), a lower order
term ffi and linearly independent fl(j) 2 Eq;*1such that
ff = i;ja(1; i; j) : : :a(t; i; j) fi(i; j) fl(j) + ffi:
Observe that dff = 0 modulo terms of gradation lower than a(1; i; j)
: : :a(n; i; j)fi(i; j). So in fact for every j
ia(1; i; j) : : :a(n; i; j) fi(i; j)
is a cycle. But if fi(i; j) is not linear then it is a boundary also.
So ff is homologous to an element in a lower gradation which is a
contradiction. So we must have:
Case 2)
There exists fi(i; j) 2 H*(((Sl)n)lr) of product length j r and
indecomposables a(k; i; j) 2 "H*((Sl)n) such that
ff = i;ja(1; i; j) : : :a(s; i; j) fi(i; j):
But then observe that
u = i;j s + n - 2 and i < r or
if * = s + n - 2 and i < r - 1. In particular this implies that Y (i) is
an s + n - 2 dimensional r - 1 cone over Q x Sn.
Proof: Using Lemma 7.3 we see that Y (i)s+n-2 is an r-cone decom-
position for (Q x Sn)s+n-2. We also get a cofibration sequence (*).
f n
U0s+n_____//Y (r - 1)s+n__//_Q x S
We look at two cases.
Case 1):
Y (r - 1)s+n ! Q x Sn is an Hs+n surjection. In this case there is
a subcomplex Y 0of Y (r - 1)s+n such that Hs+n(Y 0) = Z and Y 0!
Q x Sn is an Hs+n sujection. By changing basis we can assume that
Y 0has a unique s + n cell. Then we have a cofibration sequence:
g // //
Ss+n-1 _____Y (r - 1)s+n-1_____Y 0:
Since Hs+n-1)f) = 0, Lemma 7.2 tell s us that g lifts to g0: Ss+n-1 !
Y (r - 1)s+n-2. Then in fact
f0_g0 n
U0s+n-2_ Ss+n-1 ____//_Y (r - 1)s+n-2__//Q x S
is a cofibration sequence.
Case 2):
23
Y (r-1)s+n ! QxSn is not an Hs+n surjection. Let g : Us+n-1 !
U0s+n! Y (r - 1) be the inclusion composed with f from (*). Then
the canonical map C(g) ! Q x Sn is an Hs+n surjection. Again,
as above, we can, by changing basis if necessary, get a subcomplex
j : V ! Us+n-1 such that V = U0s+n-2[ es+n-1, Hs+n-1(V ) = Z
and C(jg) ! Q x Sn is an Hs+n surjection. This implies we can use
Lemma 7.2 to factorize h through, h0 : V ! Y (r - 1)s+n-2. Then
the map C(h0) ! Q x S2 is an H* equivalence and so a homotopy
equivalence.
Lemma 7.6 Assume that l n + 2 and cl(Q x Sn) r. Then there
exists spaces U,Y with dim(U) s and dim(Y ) s a cofibration
sequence U [ es+n-1 ! Y ! Q x Sn be any cofibration such that and
a map f : Y ! X(r - 1) such that
Y ________//Q x Sn
f|| =||
fflffl| fflffl|
X(r - 1)_____//Q x Sn
commutes.
Proof: From [3] we may assume that the cone decomposition of QxSn
is a strong r-cone over Q. The lemma then follows from Lemmas 7.5,
7.1 and 5.2.
Lemma 7.7 Assume that l n + 2 and 2p > n + 2. Let g : U [
es+n-1 ! Y be any map such that Q x Sn ' C(g), dim(U) s and
dim(Y ) s and f : Y ! X(r - 1) any map such that
Y ________//Q x Sn
f|| =||
fflffl| fflffl|
X(r - 1)_____//Q x Sn
commutes.
Then there exists a cofibration sequence (*)
U0_ Ss+n-1 ! (T r(Sl) x Sn _ U0) [fles ! Q x Sn
such that dim(U0) s+n-2 and con(U0) lr-2 and a map OE : U0 !
Srl-1such that the compostion T r(Sl) x Sn _ U0project//_U0OE//_Srl-1
takes fl to a unit times fi.
24
Proof: First of all we can assume that U ! Y is a cofibration.
Consider the following diagram
Ulr-2K
| KKKKK
| KKK
fflffl| K%%K p
U [ es+n-1_____//Y_____//Q x Sn
| | |
| | =|
fflffl| fflffl|p0 fflffl|
U0[ es+n-1 ____//_Y_0__//Q x Sn
We define U0 = U=Ulr-2and Y 0to be the pushout of the left hand
square. Then p0 is determined and since U ! Y is a cofibration the
bottom row is a cofibration sequence. Note that p0 is now lr - 2
connected.
By Lemma 3.8 we see that Hs(p0) is a surjection. So we can write
Y 0' Z [fles such that the composition Z ! Y 0! QxSn is trivial on
Hs. Observe that the map X(n - 1) ! Q x Sn is lr - 2 connected and
a surjection in sslr-1. So we also get a map f0 : Z [fles ! X(n - 1)
which we can adjust by [Ulr-2; X(n - 1)] using the coaction so that
0
Z [fles__p__//_Q x Sn
f0|| =||
fflffl| fflffl|
X(n - 1)_____//Q x Sn
commutes. Since p0 is lr - 2 connected and dim(T r(Sl) x Sn)
lr - 2, we can lift the inclusion T r(Sl) x Sn ! Q x Sn to a map
T r(Sl)xSn ! Z. So we see using homology that Z ' T r(Sl)xSn _U0
and U0[ es+n-1 ! Z [fles ! Q x Sn is of the form (*) of the lemma.
Note that the attaching map of the s + n - 1 cell must be trivial since
by the Serre exact sequence U0 ! U0[es is always injective on sss+n-2
if con(U0) > n and primes less than (n + 3)=2 have been inverted.
Let q : X(n-1) ! Slr-1[fies be the, homotopy unique, projection
map. Since the map X(n - 1) ! Q x Sn is an Hs isomorphism
Hs(f0|Z ) is trivial and Hs(f0) is surjective. Therefore by Lemma 7.2
the inclusion composed with qf0, U0 ! Srl-1[fies factors through
a map OE : U0 ! Srl-1. So by quotienting out T r(Sl) x Sn in both
25
Z [fles and X(n - 1) we get a commutative diagram
OE
U0 __________//Srl-1
| |
| |
fflffl| |fflffl
U0[fl0es ____//_Srl-1[fies
such that the map on the bottom is an Hs surjection. The second
part of the lemma follows.
8 Looking at Loops
The goal of this section is to find obstructions to the existence of the
cofibration sequences produced in the last section (Lemma 7.7). The
method is to look at the implications of such cofibration sequences on
the level of Adams-Hilton models. Again we work at a prime p. We
let Q and fi be as in the last section. Assume fi is a coH map.
For a simply connected CW complex X we let AH(X) denote the
Adams-Hilton model, [1], of X. For a cellular map of CW complexes
f : X ! Y let f* : AH(X) ! AH(Y ) denote the induced map.
For our purposes the relevent properties of AH(X) is that it is a
differential graded algebra of the form T (V ), the tensor algebra on V .
There is a weak equivalence AH(X) ! C*(X) which is compatible
with cellular maps between CW complexes. V is a free Z(p)module
with one basis element of dimension n - 1 for every n cell of X. If f :
Sn ! Xn is the attaching map of an n+1 cell in X then the differential
of the corresponding basis element, a, has the property that d(a) = z
where z is the image in AH(X) of some lift of fa*(n-1) 2 H*(Xn)
to the chains and then to AH(Xn).
Lemma 8.1 Let
f // r l n sg // n
U _ Ss+n-1 _____(T (S ) x S _ U) [fle_____Q x S
be a cofibration sequence with dim(U) s + n - 2 and U simply
connected. Let U have any cell structure with no cells of dimension >
s+n-2 and let the other spaces have their canonical cell structures (see
2.1). Let a 2 AH(U _ Ss+1) be the generator corresponding to Ss+n-1
and b; c 2 AH((T r(Sl)xSn _U)[es) the the generators corresponding
to the Sn and the es respectively. Then f*(a) = u[b; c] + ff + ffi where
26
g*(ff) = 0 in AH(Q x Sn), ffi 2 AH((T r(Sl) x Sn _ U) [ es) is a
boundary, ff and ffi have no linear part and u is a unit.
Proof: The fact that ff and ffi have no linear part follows since
AH(T r(Sl) x Sn _ U) [ es) has nothing linear in dimension s + n - 2
(remember dim(U) s + n - 2).
Next consider the diagram
f // r l n sg // n
Ss+n-1 _ U _____T (S ) x S _ U [fi0e____ Q x S
project|| h|| =||
fflffl| f0 fflffl| fflffl|
Ss+n-1 _______________//_Y_____________//Q x Sn
where the left hand square is a pushout and the rows are cofibration
sequences. Then since fi is coH, H*(fia) = 0. So f0*(a) = u[h*b; h*c]+ffi,
where ffi is a boundary. But kerg* = kerh* and h* is surjective so the
lemma follows.
Lemma 8.2 Let us again be given a cofibration sequence as in the
last lemma. Consider the map
h : (T r(Sl) x Sn _ U) [fles ! (Sn _ Slr-1) [fl0es
that is the projection on T r(Sl)xSn, the map OE from Lemma 7.7 on U
and the canonical extension on es where fl0is the image of fl. Let f and
a be as in Lemma 8.1, b,c and e the generators in AH((Sn _ Slr-1) [
es) corresponding to Sn,es and Slr-1 respectively. Then (hf)*(a) =
u[b; c] + ff where ff 2 T (b; e) and u is a unit.
Proof: (hf)*(a) 2 T (b; c; e). Since lr - 1 > n the lemma follows from
8.1 by looking at dimensions.
Lemma 8.3 Let f : Sl ! Sr be a coH map and s 2. Then the
following diagram commutes
s-1f//
Sl+r-1_____ Sr+s-1
| |
| |
fflffl|f_id fflffl|
Sl_ Ss ____//_Sr _ Ss
where the vertical maps are the universal Whitehead products.
Proof: [23].
27
Lemma 8.4 Assume 2p-3 > n and n-1fi 6' *. Let ss : Sn _Slr-1!
Slr-1be the projection. Let fl0: Ss-1 ! Sn _ Slr-1be a map such that
ssfl = fi. Then in the notation of Lemma 8.2, the image of [b; c] + ff
in H*((S2 _ Slr-1[fles)) is not in the image of Hurewitz for any
ff 2 T (b; e).
Proof: Assume g : Ss+n-1 ! (Sn _ Slr-1) [fl0es is a map such
that H*(ga) = [b; c] + ff. From this statement we plan to derive a
contradiction. Let b : Sn ! Sn _ Slr-1, e : Slr-1! Sn _ Slr-1and
i : Sn _ Slr-1! Sn _ Slr-1[fl0es denote the inclusions. Consider the
following pushout square.(*)
____ss_____//lr-1
Sn _ Slr-1 S
|i| ||
fflffl| 0 fflffl|
(Sn _ Slr-1) [fl0esss//_Slr-1[fies
and extend it to a diagram (+)
F (4)____ffi_//_Sn _ Slr-1_ss_____//Slr-1
|q| |i| i0||
fflffl|ffi0 fflffl| ss0 fflffl|
F (5)____//_(Sn _ Slr-1) [fl0es//_Slr-1[fies
where the rows are fibration sequences.
Consider the following six statements:
1)[fl0; b] 2 ss*(Sn _ Slr-1) lifts non-trivially to F (4).
2) H*((fl0)a) = 0.
3) ffi-1q[fl0; b] = 0.
4) We may assume g lifts to F (5).
5) Hs+n-1(F (5)=F (4)) = Z(p)and F (5)=F (4) is s+n-2 connected.
6) pr O ffi0-1g : Ss+n-1 ! F (5)=F (4) is an Hs+n-1 isomorphism.
Next observe that the truth of these six statements are enough to
prove the lemma. In fact 1), 3) and 5) imply that in a minimal cell
structure for F (5) the unique cell in dimension s + n - 1 not comming
from F (4) has a nontrivial attaching map. On the other hand 6)
together with Lemma 2.3 imply that that cell has a trivial attaching
map. This is a contradiction which means that g cannot be of the
form assumed and the lemma must be true.
We proceed to prove the six statements.
28
1) F (4) ' Sn ^ (Slr-1_ S0) and there is a homotopy eqivalence
OE : _S ! F (4) given by iterating Whitehead products of e with
b. Consider the map : F (4) ! Slr that is OE-1 composed with
projection. Write fl0 = efi + j. Since ssfl0 = fi, j lifts to F (4).
Therefore [j; b] = [j; b] = 0. By Lemma 8.3 [fi; b] = [c; b]n-1fi.
So [fl0; b] = ([fi; b] + [j; b]) = [c; b]n-1fi = n-1fi 6' *.
2) If it were not true nothing of the form [b; c] + ff would be a cycle
in AH(((S2 _ Slr-1) [fl0es).
4) i0: Slr-1! Slr-1[ es induces an isomorphism in sss+n-1 (re-
member 2p - 3 > n and con(Slr-1) > n). So we can replace g by
g - ess0g and therefore may assume that p0g ' *. It follows that g
lifts to F (5).
3) Follows since i[fl0; b] = 0 and since p0 is a sss+n-1 surjection
(since i0ss is one) the long exact sequence of a fibration tells us that
q[fl0; b] = 0.
5) Follows since H*(fla) = 0, F (5)=F (4) is s + n - 3 connected
and Hs+n-2(F (5)=F (4)) = Z(p).
6) Follows since the extra homology class in Hs+n-1(F (5)) corre-
sponds to the bracket [b; c].
9 Existence of maps with required prop-
erties
In our final section we quote some results of Brayton Gray (Theorem
9.1). They show that there exists unstable elements in the homotopy
groups of spheres with the properties we desire. We also state and
prove our main theorem. Again we fix a prime p > 2.
Theorem 9.1 [11] Let t; k be positive integers. Write t = m such
that (m; p) = 1. Let s = t - k. If s > then there exists Xk;s2
sst(2(p-1))+2k-1(S2k+1) such that:
1) S2+1 Xk;s= * and
2) S2 Xk;s6= 0
For our purposes we only need the following corollary of the theorem.
Corollary 9.2 For every pair of integers l; r such that lr is even there
exists s and fi 2 sss-1(Slr-1) such that:
1) fi is a suspension
2) 2n-1fi 6' *
3) 2nfi ' *.
29
Our main theorem follows immediately.
Theorem 9.3 For every pair of positive integers t; r such that r > 1
and 2t < 2p - 3 there exist p-local spaces Q such that cat(Q x Sn) =
catQ = r for n 2t, cl(QxSn) = clQ = r for n > 2t and cl(QxS2t) =
r + 1
Proof: We mearly have to chose l l + 2 so that we have an r-1-cone
receiver in high enough dimensions. The conclusions of the theorem
then follow from 7.6, 7.7, 8.1, 8.2, 8.4, 4.3, 4.2 and 9.2.
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