Ganea's conjecture on LusternikSchnirelmann category
Norio Iwase*
Permanent Address: Graduate School of Mathematics, Kyushu University, J*
*apan.
Current: Department of Mathematical Sciences, University of Aberdeen, Unit*
*ed Kingdom.
email: n.iwase@maths.abdn.ac.uk
November 17, 1997
Abstract
A series of complexes Qp indexed by all primes p is constructed with ca*
*tQp = 2 and
catQpxSn = 2 for either n 2 or n = 1 and p = 2. This disproves Ganea's c*
*onjecture
on LS category, or LusternikSchnirelmann category.
1 Introduction
Problem 2 paused by Ganea [4], Ganea's conjecture on LS category states the fol*
*lowing: The LS
category of a space is increased by one by taking the product with a sphere. A *
*major advance
in this subject has been made by Hess [6] and Jessup [7] working in the rationa*
*l category: The
rational version of the conjecture is true. Also by Singhof [10] and Rudyak [8]*
*, the conjecture
has been verified for a large class of manifolds.
In this paper, we work in the category of CW complexes with base points and*
* the LS
category is considered as normalized, i.e., catX is the least number n such tha*
*t the diagonal
map : X ! Xn+1 can be compressed into the 'fat wedge' X[n+1]. Hence cat{*} = 0*
*. We
introduce the plocal version of category catpX for a nilpotent space X as the *
*least number
n such that the diagonal map : X ! Xn+1 can be compressed into X[n+1], at the *
*prime p.
This immediately implies that catpX catX for a nilpotent space X.
Let us recall that an A1 space, in the sense of Stasheff [11], is a space *
*with an A1 form.
Stasheff has shown that any given A1 space is homotopy equivalent to the loop *
*space of some
space, which is often called the A1 structure of the given A1 space. Our poin*
*t of view is the
___________________________________
*1991 Mathematics Subject Classification. Primary 55M30, Secondary 55P35, 55*
*Q25, 55R35, 55S36.
Keywords and phrases. LS category, A1 structure, Ganea's conjecture
1
other way around: For a given space, its loop space is an A1 space with the gi*
*ven space as
its A1 structure. More precisely, every space X has a filtration given by the *
*projective spaces
P m(X) of its loop space X. From this point of view, we recover the following f*
*undamental
result due to Ganea (see [2]).
Theorem 1.1 (Ganea) Let X be a connected CW complex. Then catX m if and only*
* if the
canonical inclusion eXm: P m(X) P 1(X) ' X has a right homotopy inverse.
From the product structure of the projective spaces, we have the following *
*wellknown facts:
Theorem 1.2 Let X and Y be connected CW complexes. Then catXx Y m if and on*
*ly if
S
the canonical inclusion a+b=mP a(X)x P b(Y ) P 1(X)x P 1(Y ) ' Xx Y has a ri*
*ght
homotopy inverse.
Corollary 1.2.1 catXx Y catX + catY . Hence catXx Sn is either catX or catX +*
* 1.
Corollary 1.2.2 Let catX = m. Then catXx Sn = m if and only if Xx Sn is dominat*
*ed by
P m(X) [ P m1(X)x Sn.
The proofs of the above results suggest that the obstruction to the existen*
*ce of a compression
of X into P m1X is given by a map to the mfold join of X; its nfold suspensi*
*on gives the
essential obstruction to the existence of a compression of Xx Sn into P mX [ P *
*m1Xx Sn.
This suggests how one might obtain counter examples to Ganea's conjecture and, *
*using Toda's
results on the homotopy groups of spheres, we establish the existence of such e*
*xamples. Al
though some results below are wellknown to the experts, we reprove them in a m*
*anner which
illuminates the computations needed for the above counter examples. The main re*
*sult of this
paper is as follows.
Theorem 1.3 There exists a series of 1connected 2 cell complexes Qp indexed *
*by all primes
p. For an odd prime p, Qp satisfies catQp = catQpx Sn = 2 and catpQp = catpQpx *
*Sn = 2
for n 2. For p = 2, Q2 satisfies catQ2 = catQ2x Sn = 2 and cat2Q2 = cat2Q2x Sn*
*= 2 for
n 1. In addition, when n = 1 and p odd, we have catQpx S1 = catpQpx S1= 3.
These examples are in a sharp contrast to the HessJessup theorem for rational *
*case (see [7]
and [6]), or the SinghofRudyak theorem for manifolds (see [10] and [8]). We r*
*emark that
this construction in the case p = 2 is strongly related to the fact that S7 is *
*a Hopf space
but S15 is not (Toda [13]), and that we could not give examples of Q at odd pri*
*mes p with
catpQx S1 = catpQ. They also suggest the following conjecture.
2
Conjecture 1.4 If catXx Sk = catX for some k, then catXx Sn = catX for all n *
*k.
The author would like to express his gratitude to John Hubbuck, Koyemon Iri*
*e and Yuli
Rudyak for valuable conversations, the University of Aberdeen for its hospitali*
*ty and the mem
bers of Graduate School of Mathematics Kyushu University for allowing me to be *
*away for
a long term, without which this work could not be done. He also thanks the ref*
*eree for the
valuable advice.
2 Pushout pullback lemma
Let (X; A) and (Y; B) be CW pairs with i : A X and j : B Y the inclusions. We*
* denote
by i and j the mapping fibres of i and j. For given f : Z ! X and g : Z ! Y , w*
*e also
define some pullbacks:
i = {(a; `X ) 2 A x L(X)* = `X (0); i(a) = `X (1)} ~={`X 2 L(X)* = `X (0)*
*; `X (1) 2 A};
j = {(b; `Y) 2 B x L(Y )* = `Y(0); j(b) = `Y(1)} ~={`Y 2 L(Y )* = `Y(0); *
*`Y(1) 2 B};
i;f = {(z; `X ) 2 Z x L(X)f(z) = `X (0); `X (1) 2 A};
j;g = {(z; `Y) 2 Z x L(Y )g(z) = `Y(0); `Y(1) 2 B};
where L() denotes the space of free paths on the space . Similarly, for maps *
*ix j : Ax B
Xx Y , k : Xx B[ Ax Y Xx Y and (f; g) = (fx g)Z : Z ! Xx Y , we define
ixj= {(`X ; `Y) 2 L(X) x L(Y )* = `X (0); * = `Y(0); `X (1) 2 A; `Y(1) 2 *
*B} = ixj;
k = {(`X ; `Y) 2 L(X) x L(Y )* = `X (0); * = `Y(0) and(`X (1); `Y(1)) 2 *
*Ax Y [Xx B};
ixj;(f;g)= {(z; `X ; `Y) 2 Z x L(X) x L(Y )f(z) = `X (0); g(z) = `Y(0); (`X *
*; `Y) 2 ixj};
k;(f;g) = {(z; `X ; `Y) 2 Z x L(X) x L(Y )f(z) = `X (0); g(z) = `Y(0); (`X ;*
* `Y) 2 k}:
Then there are natural projections OE : ixj;(f;g)! i;fand : ixj;(f;g)! j;*
*ggiven by
OE(z; `X ; `Y) = (z; `X ); (z; `X ; `Y) = (z; `Y):
We establish the following lemma.
Lemma 2.1 Let (X; A) and (Y; B) be connected CW pairs and Z a connected CW c*
*omplex with
maps f : Z ! X and g : Z ! Y . Then the homotopy pullback k;(f;g)of (f; g) : Z*
* ! Xx Y
and k : Xx B[ Ax Y Xx Y has naturally the homotopy type of the homotopy push*
*out of
OE : ixj;(f;g)! i;fand : ixj;(f;g)! j;g.
3
OE
ixj;(f;g)________wi;f
 y

 HP O 
 
u u
j;g z_________k;(f;g)w____wXx B [ Ax Y
 y
 
 HP B k
 
u (f;g) u
Z ___________XxwY
Proof. We can determine subspaces E1, E2 and E0 in E = k;(f;g)as follows:
E1 ={(z; `X ; `Y) 2 E`Y(1) 2 B} {(z; c(f(z)); `Y) 2 E`Y(0) = g(z); `Y(1) *
*2 B} ~=j;g;
E2 ={(z; `X ; `Y) 2 E`X (1) 2 A} {(z; `X ; c(g(z))) 2 E`X (0) = f(z); `X *
*(1) 2 A} ~=i;f;
E0 ={(z; `X ; `Y) 2 E`X (0) = f(z); `Y(0) = g(z); `X (1) 2 A; `Y(1) 2 B} = *
*ixj;(f;g);
where c(w) denotes the constant path at w. Then we have E = E1[ E2 and E1\ E2 =*
* E0. We
can easily show that j;gand i;fare deformation retracts of E1 and E2, respectiv*
*ely. Also,
the inclusions of E0 in E1 and E2 are, up to homotopy, given by and OE. Hence*
* E has the
homotopy type of the (unreduced) homotopy pushout j;g[ {[0; 1] x ixj;(f;g)} [ *
*i;f. QED.
3 Proof of Theorem 1.1
Let Em+1 be the homotopy fibre of the inclusion X[m+1]! Xm+1 and P m, which is *
*socalled
the Ganea space, be the homotopy pullback of
X[m+1]
y




u
m+1
X _____wXm+1 ;
where X[m+1]= {(x0; :::xm ) 2 Xm+1 xt= * for some t} and m+1 denotes the diago*
*nal.
Let us recall that catX m if and only if the diagonal map m+1 is compressi*
*ble into
X[m+1]. The latter condition is clearly equivalent to the existence of a homoto*
*py crosssection
of the projection P m! X.
Now we take Z = X, Y = Xm , f = 1X , g = m , A = {*}, and B = X[m]and we th*
*en have
4
k;(f;g)= P m, i;f' *, j;g= P m1and the following pullback diagram:
j ______wixj;(f;g)____wi;f

  
 
PB
 
u 
u
j ________wj;g ________Z:w
Since f = 1X and A = {*}, i;fis contractible, and hence ixj;(f;g)is homotop*
*y equivalent
to j the fibre of j;g! Z, in this case. Here j is the inclusion map X[m] Xm , a*
*nd hence j
is Em by definition. Thus we have the following pushout and pullback diagram:
Em ____________wP m1
 y
 
 HP O 
 
u u
{*} z____________Pwm _____wXx X[m][ {*}x Xm
 y
 
 
 HP B k
 
u m+1 u
X ___________wXx Xm
Hence P m has the homotopy type of a (unreduced) mapping cone of the canonical *
*inclusion
Em P m1, m 1.
Similarly using Lemma 2.1, we have the following pushout and pullback dia*
*gram:
pr2
Xx Em ___________wEm
y
 
pr1 HP O 
u u
X z____________Em+1w_____wXx X[m][ {*}x Xm
 y
 HP B k
u u
{*} ____________XxwXm*
Hence Em+1 has the homotopy type of the (unreduced) join of X and Em . This*
* implies
that {(Em+1 ; P m); m 0} gives the A1 structure for X in the sense of Stashef*
*f. Thus P m
has the homotopy type of P m(X) the Xprojective mspace. This implies Theorem *
*1.1.
5
4 Product formulas
Firstly we prove Theorem 1.2. We define a modified A1 structure for Xx Y as fo*
*llows:
[
P^m = P a(X)x P b(Y ) P 1(X)x P 1(Y );
a+b=m
[
E^m+1 = Ea+1(X)x Eb+1(Y ) E1 (X)x E1 (Y ):
a+b=m
Then we immediately obtain that ^Emis contractible in ^Em+1and ^P m+1has the ho*
*motopy type
of the mapping cone of the projection ^Em+1! ^P m. By Stasheff [11], this gives*
* an A1 structure
for Xx Y and the inclusion P m(Xx Y ) P 1(Xx Y ) = P 1(X)x P 1(Y ) can be
deformed into the subspace ^P m P 1(X)x P 1(Y ). Also we know that cat^P m m. T*
*hen
by Theorem 1.1, Theorem 1.2 follows.
Remark 4.1 Since ^P mhas the homotopy type of the mapping cone of ^Em! ^P m1*
*, cat^P m
m for all m 1.
This immediately implies Corollary 1.2.1.
Next we show Corollary 1.2.2: Let X satisfy catXx Sn = m = catX. Then by*
* The
S
orem 1.2, Xx Sn is dominated by a+b=mP a(X)x P b(Sn ) and hence by P m(X)x {**
*} [
P m1(X)x P 1(Sn ) ' P m(X) [ P m1(X)x Sn. This implies the Corollary 1.2.2.
5 Counter Examples to Ganea's conjecture
To show Theorem 1.3, it is sufficient to construct the following
Example 5.1 1) For an odd prime p, let ff be the generator of the pprimary s*
*ummand of
ss4p3(S2) which is isomorphic with Z=pZ and Qp = S2 [ffe4p2. Then catQp = cat*
*pQp = 2
and catQpx S1 = catpQpx S1= 3, but catQpx Sn = catpQpx Sn= 2 for n 2.
2) For the prime 2, let ff be the generator of the direct summand Z=4Z of s*
*s29(S8) ~=Z=4Z
(Z=2Z)3 and Q2 = S8 [2ffe30. Then catQ2 = cat2Q2 = 2, while catQ2x Sn = cat2Q2x*
* Sn= 2
for n 1.
In each case, we know that catqQp = 1 and catqQpx Sn= 2 for 0 q 6= p and n*
* 1.
All the examples in Example 5.1 are obtained by similar methods. We will conce*
*ntrate on
part 1) of Example 5.1. First of all, let us recall that the Hopf map j : S3 ! *
*S2 induces an
6
isomorphism j* : ss*(S3) ! ss*(S2) for * 3. In particular, j* : ss4p3(S3) ! s*
*s4p3(S2) ~=Z=3Z
is an isomorphism. So let ff and fi = ff21(3) be the corresponding generators i*
*n ss4p3(S2) and
ss4p3(S3). Let Qp be the mapping cone of ff. To avoid too much calculation o*
*f homotopy
groups, we consider fi rather than ff. We show the following lemma which is wel*
*lknown for
experts.
Lemma 5.2 The map fi = ff21(3) is not a suspension map but a coHopf map of *
*order p, whose
iterated suspensions tfi are trivial for t 2 but fi 6= 0.
Proof. We can easily obtain the latter part of the lemma by examing Theorem 13*
*.4 in [14].
In fact, ss4p1(S5) has no elements of order p. Thus 2fi is trivial. However we*
* know that the
suspension homomorphism ss*(X) ! ss*+1(X) is a split monomorphism for any Hopf *
*space X
(due to James). Thus ss4p3(S3) ! ss4p2(S4) is a split monomorphism, and hence*
*, fi gives a
nontrivial generator of a direct summand of order p.
Thus it remains to show the first part of the lemma: Since the finite group*
* ss4p4(S2) has no
ptorsion, fi cannot be a suspension. In [9], Saito has extended the results of*
* BersteinHilton [1]
which describes the obstruction for a general map to be a coHopf map using Gan*
*ea's criterion
for a coHopf space: Let f : X ! Y be a map of simply connected coHopf space*
*s. Then
the obstruction to f being a coHopf map is an element H(f) 2 [X; Y *Y ], where*
* H is the
generalised Hopf invariant homomorphism H : [X; Y ] ! [X; Y *Y ]. In our case, *
*H(fi) lies in
ss4p3(S3*S3) ~=ss4p3(S3^ S3)
~=ss4p3(S3^ (S2 [ e4[ ::: [ e4p4[ (higher cells 4p )2))
~=ss4p3(S3^ (S2 _ S4 _ ::: _ S4p4))
~=ss4p3((S2 _ S4 _ ::: _ S4p4)^ (S2 _ S4 _ ::: _ S4p4))
~=ss4p3({S2+2_ S4+2_ S2+4_ ::: _ S4p6+2_ ::: _ S2+4p6})
~=ss4p3({S4 _ S6 _ S6 _ ::: _ S4p4_ ::: _ S4p4})
~=ss4p4(J(S4 _ S6 _ S6 _ ::: _ S4p4_ ::: _ S4p4))
~=ss4p4(J(S4)x J(S6 _ S6 _ ::: _ S4p4_ ::: _ S4p4))
~=ss4p4(J(S4)x J(S6)x J(S6)x :::x J(S4p4)x :::x J(S4p4))
~=ss4p3(S5) ss4p3(S7) ss4p3(S7) ::: ss4p3(S4p3) :::*
* ss4p3(S4p3);
which has no element of order p by [14], where J(X) denotes the James' reduced *
*product space
of X (see Whitehead [15]). Since the order of fi is p, H(fi) is trivial and we*
* obtained the
7
lemma. QE*
*D.
We show the following proposition which was shown by Gilbert [5] working wi*
*th the notion
of wcat.
Proposition 5.3 The map ff = jfi = jff21(3) is not a coHmap and the obstruct*
*ion is described
by the 2nd JamesHopf invariant h2(ff) = fi, which is a generator of the pprim*
*ary summand
of ss4p3(S3) which is isomorphic with Z=pZ:
2ff ' (ff _ ff)4p3+4p3[i1; i2]fi
where we denote by k : Sk ! Sk _ Sk the (unique) coHopf structure of the spher*
*e Sk and by
+k the multiplication induced by the coHopf structure of sphere Sk.
Proof. There is a wellknown formula for the Hopf map j:
2j ' (j _ j)3+3[i1; i2]
in ss3(S2 _ S2) where it: X ! X _ X is the inclusion to the tth factor. Since *
*ff ' jfi, we have
the homotopy relation 2ff ' 2jfi ' {(j _ j)3 +3 [i1; i2]}fi in ss4p3(S2 _ S2).*
* Since fi is a
coHopf map by Lemma 5.2, this is homotopy equivalent to
(j _ j)3fi +4p3[i1; i2]fi ' (jfi _ jfi)4p3+4p3[i1; i2]fi ' (ff _ ff)4p3+4*
*p3[i1; i2]fi:
This implies that h2(ff) ' fi which gives the obstruction to ff being a co*
*Hopf map and
hk(ff) = 0 for k 3. *
*QED.
To determine the LS category of Qp and Qpx Sn, we need to show the followin*
*g lemma.
Lemma 5.4 The following diagram, without the dotted arrow, commutes up to ho*
*motopy.
ff _____________i
S4p3____________S2wz Qpw
fi y i 
u  i 
S1*S1 ij1 i 1Qp
(i*i)(j1*jy1)  i 
u pQp1 u evQpi u
Qp *Qp ________Qpw 0_______i____Qpw'
'') iik[[]
Qp1 [ eQp2
P 2Qp
where i : S2 ! Qp and jt : St ! St+1 give the bottom cell inclusions and pQp1de*
*notes the
Hopf construction of the loop addition of Qp , Qp1: Qp ! P 2Qp denotes the incl*
*usion to
the mapping cone of pQp1and eQpt: P tQp P 1Qp ' Qp denotes the canonical inclu*
*sion.
8
Proof. The commutativity of the right half square of the diagram and the triang*
*le below are
clear. So we concentrate on showing the commutativity of the left half square o*
*f the diagram.
There is the following homotopy commutative diagram due to Ganea:
[i1;i2]
S1*S1 _______S2w_ S2 z______S2wx S2
y y y
  
(i*i)(j1*j1) i_i ixi
  
u qQp u u
Qp *Qp _____wQp1_ Qp z______Qpwx Qp:
By Proposition 5.3, we have
(i _ i)2ff ' (iff _ iff)4p3+4p3(i _ i)[i1; i2]fi ' (i _ i)[i1; i2]fi ' qQp1*
*(i*i)(j1*j1)fi:
Also Ganea showed, for any coHopf space X, that there exists a map (shown *
*as a dotted
arrow) corresponding uniquely to the coHopf structure so that the following di*
*agram commutes
up to homotopy:
X*X ___X*X
 
 
 X  X
XN"hh p1hX q1
N "]h h h 
N u sshjX u
NX _____Xw_ X (*
*5.1)
N  y
1X N  
NevHPXB 
NNP 
u X u
X ______Xx_X:w
2
Since a sphere has a unique coHopf structure, we have 2 ' ssS j1 and hence
2 Qp
(i _ i)2 ' (i _ i)ssS j1' ss ij1:
Thus we get the following relation:
ssQpij1ff ' (i _ i)2ff ' qQp1(i*i)(j1*j1)fi ' ssQppQp1(i*i)(j1*j1)fi:
Here the diagram 5.1 is a pullback diagram. Since qX1 induces a split monomor*
*phism on
homotopy groups, ij1ff is determined, up to homotopy, by the equations
evQpij1ff ' ievS2j1ff ' i1S2ff ' * ' evQppQp1(i*i)(j1*j1)fi;
ssQpij1ff ' ssQppQp1(i*i)(j1*j1)fi:
Therefore we have that ij1ff ' pQp1(i*i)(j1*j1)fi. Q*
*ED.
9
Remark 5.5 There exists a map : Qp ! P 2Qp given by the homotopy deforming i*
*j1ff
in Qp to ff0 = pQp1S1*S1fi and by ^OC(S1*S1)C(fi), where we denote by C the f*
*unctor taking
cones and ^O: (C(Qp *Qp ); Qp *Qp ) ! (P 2Qp ; Qp ) the characteristic map of *
*the
attached cone of the mapping cone space P 2Qp of pQp1.
The following theorem is a special case of a result of BersteinHilton [1],*
* or Gilbert [5].
However we include a proof as it contains the idea used to determine the LS cat*
*egory of
Qpx Sn.
Theorem 5.6 catpQp = catQp = 2 but catqQp = 1 for q 6= p.
Proof. For the prime p, we compute the homotopy group ss4p3(Qp *Qp ), where th*
*e element
(i*i)(j1*j1)fi lies:
ss4p3(Qp *Qp ) ~=ss4p3((S2 [ (higher cells 4p )3)^(S2 [ (higher cells 4p *
*)3))
~=ss4p3(S2^ S2)
~=ss4p3({S1+1_ S2+1_ S1+2_ (higher spheres 4)}):
Hence ss4p3(S1*S1) is a direct summand of ss4p3(Qp *Qp ). As (i*i)(j1*j1) is *
*the bottom
cell inclusion, (i*i)(j1*j1)fi gives a generator of ptorsion subgroup of ss4p*
*3(Qp *Qp ).
By Sugawara [12], the projection pQp1is a quasifibration with the fibre Qp*
* which is con
tractible in the total space Qp *Qp . Thus we have the following (split) short *
*exact sequence:
0 ! sst(Qp *Qp ) ! sst(Qp ) ! sst(Qp) ! 0 (*
*5.2)
Since ij1 is the bottom cell inclusion, it gives a generator of ss2(Qp ) = *
*Z. Hence, if
cat Qp = 1, in other words, if Qp is dominated by Qp , then there is an embeddi*
*ng of Qp in
Qp whose restriction to S2 is given by ij1 and hence ij1ff should be trivial. T*
*his
contradicts the exactness of (5.2) at t = 4p  3, and hence we obtain catpQp = *
*2.
On the other hand, if q 6= p, then fi = 0 and, by Lemma 5.4, the bottom cel*
*l inclusion
ij1 can be extended to a map 01: Qp ! P 2Qp . The difference of 1Qp and 01in Qp*
* is
described by fl012 ss4p2(Qp). By the exactness of (5.2) at t = 4p  2, fl01can*
* be pulled back on
Qp to fl1 2 ss4p2(Qp ). Thus we can obtain the genuine compression 1 of 1Qp to*
* Qp
by adding fl1 to 01. This implies that catqQp = 1 for q 6= p and it completes t*
*he proof of the
theorem. QE*
*D.
10
Qp
Remark 5.7 The difference between the identity 1Qp and the map e2 is given b*
*y an ele
ment evQpfl 2 ss4p2(Qp), where fl 2 ss4p2(Qp), since ss4p2(Qp) ! ss4p2(Qp) *
*is a split
surjection.
Finally we calculate the LS category of Qpx Sn. The attaching map of the to*
*p cell of Qpx Sn
is the map
^ff: S4p2*Sn1 = D4p2xSn1 [ S4p3xDn ! Qpx {*} [ S2x Sn
which is given by
f^fD4p2xSn1= Ox *
f^fS4p3xDn= ffx On
where O : (D4p2; S4p3) ! (Qp; S2) denotes the characteristic map of the top c*
*ell of Qp and
On : (Dn; Sn1) ! (Sn; {*}) denotes the relative homeomorphism. Thus we have th*
*e following
equations for (x {*} [ (ij1)x 1Sn)^ff:
(x {*} [ (ij1)x 1Sn)^ffD4p2xSn1= Ox *
(x {*} [ (ij1)x 1Sn)^ffS4p3xDn = ij1ffx On
As for the space Qpx Sn, the space P 2Qp xSn is also the mapping cone of
^pQp1: (Qp *Qp )*Sn1 = C(Qp *Qp )x Sn1[ (Qp *Qp )x Dn ! P 2Qp x {*} [ Qp xSn
which is given by
p^Qp1C(Qp*Qp)xSn1= ^Ox*
p^Qp1(Qp*Qp)xDn = pQp1xOn
where ^O: (C(Qp *Qp ); Qp *Qp ) ! (P 2Qp ; Qp ) denotes the characteristic map *
*of the
attached cone of the mapping cone P 2Qp . By Remark 5.5, the bottom cell inclus*
*ion ij1
can be extended to : Qp ! P 2Qp which is a compression of the identity. More p*
*recisely, is
the homotopy given by the composition of the homotopy of ij1ff in Qp to ff0= pQ*
*p1S3fi
and the nullhomotopy C(fi) in C(Qp *Qp ). The former part of the homotopy als*
*o gives
the homotopy of ij1ffx On to ff0xOn. Thus we have that (x {*} [ (ij1)x 1Sn)^ff*
*is
homotopic to ^ff0which is given by
^ff0D4p2xSn1= ^OC(S1*S1)C(fi)x * = ^pQp1C(S1*S1)xSn1(C(fi)x 1*
*Sn1);
^ff0S4p3xDn= (pQp1S1*S1fi)x On = ^pQp1(S1*S1)xDn(fix 1Dn):
11
Qp
Thus ^ffis homotopic in P 2Qp x {*} [ Qp xSn to ^p1(S1*S1)*Sn1(fi*1Sn1). Thi*
*s yields the
following proposition.
Proposition 5.8 The following diagram, without the dotted arrow, commutes up t*
*o homotopy.
__________^ff ___________ n
S4p3*Sn1 Qpxw{*} [ S2x Snz QpxwS"
 y 
   "
   "
fi*1Sn1   "
  
   "
u   " 1Qpx1Sn
(S1*S1)*Sn1 x{*}[(ij1)x1Sn x1Sn "
y   "
   "
   "
((i*i)(j1*j1))x1Sn1  
   "
   "]
u ^pQp1 u u eQp2x1Sn
(Qp *Qp )*Sn1 ___Pw2Qp x {*} [ Qp xSn z_____Pw2Qp xSn ______________Qpxw*
*Sn
Since fi*1Sn1' (fi^ 1Sn1) ' nfi, we have established the following result.
Proposition 5.9 1Qpx 1Sn can be compressed into P 2Qp x {*} [ Qp xSn, for n *
*2.
Proof. In the case when n 2, fi*1Sn1 is trivial. Since the inclusion P 2Qp *
* x {*} [
Qp xSn ! P 2Qp x Sn induces a split epimorphism in the homotopy groups, a simil*
*ar
argument to that used in the proof of Theorem 5.6 leads us the conclusion that *
*there is a com
pression ffi of x 1Sn to P 2Qp x {*} [ Qp xSn . Moreover, we may assume the co*
*mpression
homotopy leaves the subspace Qpx {*} [ S2x Sn fixed. By Remark 5.7, the identit*
*y 1Qp is given
from eQp2 by adding an element evQpfl, fl 2 ss4p2(Qp). We define a map ffi2 by
x1Sn 10 n n 10 nffi[(flx12Sn) n
ffi2 : Qpx Sn ! (Qp _ S )x S = Qpx S [ S xS _____wP Qpx {*} [ Qp xS*
* ;
where denotes the coaction of S4p2. Since ffi is homotopic to in P 2Qp x S*
*n with the
subspace {*}x Sn left fixed, ffi2 is homotopic to
x1Sn 10 n n 10 n(x1Sn)[(flx1Sn2) *
* n
( + fl)x 1Sn : Qpx Sn ! (Qp _ S )x S = Qpx S [ S xS __________wP Qpx *
*S ;
in P 2Qp x Sn which is a compression of 1Qpx 1Sn. Thus ffi2 : Qpx Sn ! P 2Qp *
*x {*} [
Qp xSn gives the compression of 1Qpx 1Sn. *
*QED.
Thus we have 2 = catpQp catpQpx Sn catQpx Sn cat(P 2Qp x {*} [ Qp xSn )
2, for n 2, and hence we have established our main theorem.
12
Theorem 5.10 catQpx Sn = catpQpx Sn= 2, for n 2, while catQpx S1 = catpQpx S*
*1= 3.
In the case when n = 1, we have fi 6= 0. Then a similar argument to that u*
*sed in
the proof of Theorem 5.6 leads us the conclusion that catQpx S1 = catpQpx S1 = *
*3 while
cat Qp = catpQp = 2. The details are left to the reader.
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