Nearly Trivial Homotopy Classes Between
Finite Complexes
Martin Arkowitz and Jeffrey Strom
Abstract We construct examples of essential maps of finite complexes f :*
* X -! Y
which are trivial of order n. This latter condition implies that for a*
*ny space
K with cone length n, the induced map f* = 0 : [K, X] -! [K, Y ]. The
main result establishes a connection between the skeleta of the infinite *
*dimen-
sional domains of essential phantom maps and the finite dimensional domai*
*ns of
maps which are trivial of order n. In particular, there are essential *
*maps f :
2i(CPt=S2) -! M(Z=ps, 2l + 3) which are trivial of order n.
2000 MSC: Primary 55P99; Secondary 55M30, 55P60
Keywords: cone length, killing length, weak category, phantom maps.
1 Introduction
Let f : X -! Y be a map and consider the class K(f) of all spaces K such that
f O h is homotopic to the constant map * for every map h : K -! X. The larger *
*the
class K(f), the more nearly trivial we consider the map f to be. We are interes*
*ted in
finding essential maps f : X -! Y such that the class K(f) is large. For examp*
*le, if
K(f) contains all finite dimensional complexes, then f is a phantom map. Howeve*
*r,
if f is an essential phantom map, then the domain X must be infinite dimensiona*
*l.
In this note we study analogs of phantom maps between finite complexes - that i*
*s,
we study essential maps f : X -! Y of finite complexes for which K(f) contain*
*s a
large class of finite dimensional spaces.
A natural first step is to search for maps f : X -! Y for which K(f) contai*
*ns Sn
for each n 0, or equivalently, such that ß*(f) = 0 : ß*(X) -! ß*(Y ). We say *
*that
such a map is trivial of order at least 1. There are numerous examples of maps*
* of
this kind - for example, the canonical quotient map X x X -! X ^ X. To extend
this notion, we will define what it means for a map to be trivial of order at l*
*east n.
A (spherical) cone length decomposition of length n for a connected space K *
*is a
sequence of cofibrations
Si-! Li-! Li+1, 0 i < n,
where Si is a wedge of spheres, L0 * (i.e., L0 is contractible) and Ln K. *
*The
(spherical) cone length of K, denoted cl(K), is defined as follows: if K is con*
*tractible,
set cl(K) = 0; otherwise, cl(K) is the smallest integer n such that there exist*
*s a cone
length decomposition of K with length n. If instead we require that L0 K, Ln *
* *
and that each Sibe connected, then we have a (spherical) killing length decompo*
*sition
1
of length n. The (spherical) killing length of K, written kl(K), is defined ana*
*logously.
It is shown in [2] that kl(K) cl(K) for any space K. Furthermore, any cellul*
*ar
decomposition of an n-dimensional complex K is a cone length decomposition of
length at most n. Therefore, if K is an n-dimensional complex, then kl(K) cl(*
*K)
n.
These definitions suggest two numerical homotopy invariants of maps f : X -!*
* Y .
We write Tc(f) n if f O h ' * for every map h : K -! X with cl(K) n. Simil*
*arly,
we write Tk(f) n if f O h ' * for every map h : K -! X with kl(K) n, and
say that f is trivial of order at least n. This includes the particular case me*
*ntioned
above: f is trivial of order at least 1 in this latter sense if and only if ß**
*(f) = 0.
Since kl(K) cl(K) for all K, it follows that Tk(f) Tc(f). Moreover, if f is*
* trivial
of order at least n, then f O h ' * for any h : K -! X with K an n-dimensional
complex. It follows that if X is an n-dimensional complex, then no essential m*
*ap
f : X -! Y can be trivial of order at least n.
Killing length is also related to the weak category of a space. The reduced *
*(n+1)-
fold diagonal map
z__n+1_factors"_______-
dn+1 : X -! X ^ X ^ . .^.X = X(n+1)
is the composite of the diagonal X -! Xn+1 with the projection Xn+1 -! X(n+1)
onto the smash product. We say wcat(K) n if dn+1 ' *, i.e., if dn+1 is homoto*
*pic
to *. We have the following commutative diagram for any map h : Sn -! X
_______h_______//
Sn X
d2|| |d2|
fflffl|h^h fflffl|
Sn ^ Sn _________//_X ^ X.
Since wcat(Sn) = 1 for each n 1, it follows that Tc(d2) Tk(d2) 1. We will*
* show
how this example can be greatly generalized.
The inequalities wcat(K) cat(K) cl(K) [6] and kl(K) cl(K) do not direc*
*tly
imply any relation between killing length and weak category, but our first resu*
*lt shows
that such a relation does exist.
Proposition 1 If K is a space with kl(K) = m, then wcat(K) < 2m .
This allows us to produce a large family of examples of essential maps betwe*
*en
finite complexes which are trivial of order at least n.
n)
Example 2 Let X be a finite complex with wcat(X) 2n. Then d2n : X -! X(2
is essential and it follows from the naturality of the reduced diagonal that it*
* is trivial
n
of order at least n. Since wcat(CPt) = t, we can take X = CP2 .
2
In these examples, f : X -! Y is an essential map with Tk(f) n, and wcat(*
*X)
2n. Furthermore, if f : X -! Y is essential and Tk(f) n, then we must have
kl(X) > n. In view of Proposition 1, it is reasonable ask whether wcat(X) 2n
whenever X is a finite complex that is the domain of an essential map which is
trivial of order at least n. In the special case n = 1, the answer is no: in *
*[3] we
constructed essential maps f : 2(CPt=S2) -! S5 which are trivial of order at l*
*east
1, and wcat( 2(CPt=S2)) = 1 because it is a suspension. There remains the quest*
*ion:
is there an upper bound on the triviality of an essential map of finite complex*
*es whose
domain is a suspension?
In this note we show that the answer to this more general question is again *
*no.
We refine and expand the examples of [3] by constructing, for each n, maps f wi*
*th
Tk(f) n and domain a suspension of a finite complex. Since every map with Tk *
* n
is trivial of order at least 1, this provides many more examples of the type co*
*nsidered
in [3].
These new examples are closely related to phantom maps. Our main result (The-
orem 3) forges a link between the infinite dimensional domains of phantom maps
and finite complexes which are domains of maps with Tk n. In fact, these fin*
*ite
complexes are quite common - they lurk among the skeleta of most familiar infin*
*ite
dimensional spaces.
Let M(G, m) denote the Moore space with homology G in dimension m. We write
Xk for the k-skeleton of the CW complex X and X(p)for the p-localization of the
space X.
Theorem 3 Let X be a 1-connected CW complex of finite type, let p > 3 be a pr*
*ime
and let n 1. Assume that there is an essential phantom map 2lX -! S2(k+l)+*
*1(p)
where k, l 1. Then, for each i < l, there are positive integers t = t(n), s *
*= s(n)
and a map
f : Xt=X2k- ! M(Z=ps, 2k + 1)
such that 2if : 2i(Xt=X2k) -! M(Z=ps, 2(k +i)+1) is essential and trivial of *
*order
at least n.
To show that the theorem is not vacuous, and to give the desired examples, we r*
*ecall
the following basic result from the theory of phantom maps (see [19, Thm. D] and
[10, Thms. 5.2 and 5.4]). Let bYpdenote the Sullivan p-completion of the space *
*Y [16],
and write Ph(X, Y ) [X, Y ] for the set of phantom maps from X to Y .
Theorem Let X be a 1-connected CW complex of finite type. If the pointed mappi*
*ng
space map *(X, bS2(k+l)+1p) is weakly contractible, then
Ph(X, S2(k+l)+1(p)) = [X, S2(k+l)+1(p)] ~=H2(k+l)(X; R).
3
The mapping space condition holds for any infinite loop space or for any sus*
*pension
of such a space [11, Thm. 2]. Thus Theorem 3 applies, for example, when X = CP1*
* ,
and we conclude that there are essential maps
2l(CPt=S2) -! M (Z=ps, 2l + 3)
which are trivial of order at least n. Other examples can be obtained by apply*
*ing
Theorem 3 to the classifying space BG, where G is any simply connected Lie group
[10, Thm. 5.6]. In particular, it follows that for any k, l, m, n 1 there are*
* essential
maps f : BU(m)t=BU(m)2k- ! S2k+1 such that 2if is essential and trivial of ord*
*er
at least n for each i < l.
For information about the cone length and killing length of spaces in a more
general context, we refer the reader to [5, 2, 3]. The invariants Tc and Tk int*
*roduced
in this paper are closely related to the the essential category weight of a map*
* (also
known as the strict category weight), as studied in [15, 13]. In fact, a map f *
*: X -! Y
has essential category weight at least n, written E(f) n, if f O h ' * for an*
*y map
h : K -! X with cat(K) n. Since cat(K) cl(K), E(f) is a lower bound for
Tc(f).
We would like to thank Don Stanley for the statement and the proof of Lemma *
*4.
2 Proofs
In this section, we prove Proposition 1 and Theorem 3. In x2.1 we establish Pro*
*po-
sition 1. We then give some definitions and lemmas which are used in x2.3 to pr*
*ove
Theorem 3.
We only consider based spaces of the homotopy type of 1-connected CW com-
plexes. We use localization techniques, and write ~ : X -! X(p)for the natural*
* map
from X to its localization at the prime p. We refer to [9] for the standard pro*
*perties
of localization.
2.1 Proof of Proposition 1
We proceed by induction. If kl(K) = 0, then K is contractible and the result fo*
*llows.
Assume that the result is known for all spaces with killing length less than m *
*and
j k
that kl(K) = m. Write S -! K -! L for the first step in a minimal killing leng*
*th
decomposition for K, so kl(L) = m - 1. Since d2 O j ' * : S -! K ^ K, there is a
map ffi : L -! K ^ K such that d2 ' ffi O k. Thus we have the homotopy commutat*
*ive
4
diagram
d2 d2m-1 m
K _________//_OOKO^_K_____________//K(2O)OO
OOO
OOOO ffi| || (2m-1)
kOOOOOO | |ffi
O''|______d2m-1'*___//_ m-1
L L(2 ).
m-*
*1)
Since kl(L) = m - 1, the inductive hypothesis shows that d2m-1 ' * : L -! L(2 *
* ,
m)
and therefore d2m = d2m-1d2 ' * : K -! K(2 . *
*||
2.2 Lemmas
We begin with a lemma which will allow us to modify a given killing length deco*
*m-
position.
Lemma 4 [14] Let
j0 k0
S0 _________//_L0________//L1
j1 k1
S1 _________//_L1________//L2,
be two cofibrations in which S0 and S1 are wedges of spheres, S0 is (n - 1)-con*
*nected,
L0 and S1 are n-connected and L2 is (n + 1)-connected. Then there is another pa*
*ir
of cofibrations
__ _j0 _k0 __
S 0__________//L0________//_L1
__ _j1 __ _k1
S 1__________//L1________//_L2
__ __ __ __ __
where S 0and S 1are wedges of spheres, S 0is n-connected and S 1and L 1are both
(n + 1)-connected.
Proof
Write S0 = T _ U where T is the subwedge consisting of all n-spheres in S0 and *
*U is
the complementary subwedge. Since j0|T ' *, L1 ' T _ C where C is the cofiber *
*of
the map j0|U. Notice that C is n-connected.
Write S1 = V _ W , where V is the subwedge of all (n + 1)-spheres and W is t*
*he
complementary subwedge. Applying Hn+1 to the second cofibration, we obtain the
exact sequence
Hn+1(V )_________//Hn+1( T _ C)________//Hn+1(L2) = 0.
5
Thus j = j1|V : V -! T _C is surjective on Hn+1 and hence on ßn+1. Let b1, . *
*.,.bk 2
ßn+1( T ) be the standard generators, and choose a1, . .,.ak 2 ßn+1(V ) such th*
*at
j*(ai) = bi for each i. Then the map s = (a1, . .,.ak) : T -! V satisfies j O*
* s = i T,
the inclusion of T into T _C. The long homology exact sequence of the cofibra*
*tion
_j
__________________________________________________*
*_____________________________________________________________________@
""_________________________________________________*
*___________
T ____s____//V_______//_A,
_
where j = p T O j and p T projects T _ C onto T , induces the split short exa*
*ct
sequence
_j
__*________________________________________________*
*_____________________________________________________________
________________________________________________________*
*_____________________________________________________________________@
zz_______________________________________________________*
*__________________________________________________
0____//_Hn+1( T_)_s*___//_Hn+1(V_)______//_Hn+1(A)___//_0.
Thus A has the homotopy type of a wedge of (n + 1)-spheres and there is a homot*
*opy
equivalence (s, t) : T _ A -! V for some map t : A -! V .
With these identifications, the following diagram commutes
j1
V _OWO ____________//_ T _ C
| ||
(s,t)_id| ||
| (i T,g) ||
T _ A _ W __________// T _ C,
where g : A _ W -! T _ C is some map. Thus we have a square of cofibrations
T __________________ T_____________//*
i|| |i|T ||
fflffl| (i T,g) fflffl| fflffl|
T _ (A _ W ) _________//_ T _ C_________//L2
| | |
| | |
fflffl| pCOg fflffl| fflffl|
A _ W ________________//C____________//_L2.
We have now constructed the following pair of cofibrations
j0|U
U ____________//L0__________//C
pCOg
A _ W _________//_C_________//L2.
It remains to move the A term in the second cofibration to the first cofibratio*
*n. By
definition, U is n-connected so the map L0- ! C is surjective in ßn+1. Since A *
*is a
wedge of (n + 1)-spheres, the map pC O g|A : A -! C lifts through l : A -! L0.
6
__
_ Finally, the desired_pair of cofibrations_is obtained_as follows:_let S0 = U*
* _ A,
j0= (j0|U, l) and let L1 be the cofiber of j0; let S1 = W and let j1be the comp*
*osite
pCOg|W __ _
W -! C ,! L1. It is a simple matter to verify that the cofiber of j1 is homot*
*opy
equivalent to L2. ||
Proposition 5 Let c 2 and let K be a (c-1)-connected CW complex with kl(K) =
m. Then
(a)K has a minimal killing length decomposition in which each Si and Li is (c*
* -
1 + i)-connected;
(b) for each q 0, kl(Kq) m + 2;
(c)for each q 0, kl(K=Kq) 2m + 2.
Proof
ji ki
Let Si-! Li-! Li+1, 0 i < m, be a minimal killing length decomposition for K
- that is, L0 K, Lm *, and Si is a wedge of spheres for each i.
We observe that, to prove (a), it is enough to show that K has a minimal kil*
*ling
length decomposition in which S0 is (c - 1)-connected and Siand Liare c-connect*
*ed
for i > 0. Then (a) follows on applying this to the resulting L1 and its minima*
*l killing
length decomposition of length m-1, and then applying it to the resulting L2 an*
*d its
minimal killing length decomposition of length m-2, and so on. Let k be the gre*
*atest
integer for which S0 is (k - 1)-connected and Siand Liare k-connected for i > 0*
*. We
want to show there is a killing length decomposition of K with k c. Assume th*
*at
k < c. Let i denote the greatest index for which Si or Li is not (k + 1)-connec*
*ted. If
i > 0 then we have a pair of cofibrations
Si-1__________//Li-1_________//_Li
Si____________//Li_________//_Li+1
in which Si-1is (k - 1)-connected, Si and Li-1are k-connected, and Li+1is (k + *
*1)-
connected. By Lemma 4, these cofibrations can be replaced by the cofibrations
__ __
Si-1__________//Li-1________//Li
__ __
Si___________//_Li_________//_L2
__ __ __
in which Si-1is k-connected and Li and Siare (k + 1)-connected. Continuing in t*
*his
way, we eventually obtain a spherical killing length decomposition in which eac*
*h Si
7
and Li, i > 0, is (k + 1)-connected. Applying Lemma 4 to the first two cofibrat*
*ions
in this decomposition shows that there is a killing length decomposition of K w*
*ith
S0 k-connected and Si and Li (k + 1)-connected for all i > 0. This shows that t*
*here
is a minimal killing length decomposition for K in which S0 is (c - 1)-connecte*
*d and
Si and Li are c-connected for_i > 0.
Now we prove (b). Let_Si_be the subwedge_of_Si consisting_of the spheres wi*
*th
dimension at most q, let L0 = Kq and_let j0: S0- ! L 0be a_lift of j0|_S0, whic*
*h exists
by cellular approximation. Define L1 to be the cofiber of j0. Thus, we have a d*
*iagram
of cofibration sequences
__ __ __
S 0__________//L0________//_L1
| | |
| | |
fflffl| fflffl| fflffl|
S0 __________//L0________//_L1
__
In this diagram, L1 is a subcomplex of L1_which_contains the_q-skeleton_of_L1. *
*By cel-
lular approximation, we may lift j1|_S1: S1- !_L1 to_a map_j1: S1- ! L 1. Conti*
*nuing
in this way, we obtain_cofibration sequences Si- ! Li- ! L i+1for each_0_ i < *
*m in
which each inclusion_Li- ! Li is a (q - 1)-equivalence. Since Lm *, Lm is (q *
*- 1)-
connected. Now Lm is (q + 1)-dimensional by construction, which shows that Lm h*
*as
killing length at most 2. Append the two cofibrations of a killing length decom*
*posi-
tion of Lm to the previously constructed sequence of m cofibrations to obtain a*
* killing
length decomposition for Kq with length m + 2.
Finally, we prove (c). The cofibration K -! K=Kq- ! Kq yields the inequali*
*ty
kl(K=Kq) kl(K) + kl( Kq) by [3, Thm. 3.4]. Since kl( Kq) kl(Kq) the result
follows from part (b). ||
If p > 3 is an odd prime, then S2k+1(p)is a homotopy commutative, homotopy
associative H-space [1]. Hence [K, S2k+1(p)] is a p-local abelian group for an*
*y finite
complex K. Our next lemma provides an upper bound on the exponent of this
group, and may be interesting in its own right.
Lemma 6 Let K be a (2k + 1)-connected finite complex (k 1) with kl(K) = m *
*and
let p > 3 be a prime. Then [K, S2k+1(p)] is a finite abelian group with exponen*
*t dividing
pmk.
Proof
We work by induction on kl(K). If kl(K) = 0, then K is contractible so the conc*
*lusion
is obvious. Now assume the result is known for any space with killing length le*
*ss than
m. By Lemma 4 we may find a minimal killing length decomposition for K in which
8
W n
all terms are (2k + 1)-connected. If S i- ! K -! L is the first step in suc*
*h a
decomposition, then kl(L) < m. From the exact sequence
W n 2k+1 2k+1 2k+1
[ S i, S(p)o]o_____[K, S(p) ]oo______[L, S(p) ]
we see that [K, S2k+1(p)] is a finite group with exponent at most the product o*
*f the
W n 2k+1 2k+1
exponents of [ S i, S(p) ] and [L, S(p) ]. By the inductive hypothesis applied*
* to L
and a result of Cohen, Moore and Neisendorer [7], this product is at most pkpk(*
*m-1)=
pmk. ||
The following well-known lemma will be used in the proof of Lemma 8.
j
Lemma 7 Let A -i!B -! C be a cofibration and let f : X -! B be any map. If
j O f ' *, then there is a map s : X -! A such that i O s ' f.
Proof
Since j O f ' *, the composite factors through CX, the cone on X. Thus we may
construct a homotopy commutative ladder of cofibrations
_____id___//
X __________//CX__________// X X
f|| || |s| ||f
fflffl|j fflffl| fflffl| i fflffl|
B ___________//C__________//_ A_________// B.
||
Armed with this lemma, we give a criterion which guarantees that certain maps
f : X -! S2k+1 remain essential when composed with the standard inclusion map
's : S2k+1 ,! M(Z=ps, 2k + 1).
Lemma 8 Let X be a finite complex and let h : X -! S2k+1 be a map such that*
* for
some odd prime p, ~ O 2h 2 [ 2X, S2k+3(p)] is nontrivial and has finite order *
*divisible
by p. Then the composite
's s
X ____h____//S2k+1_______//M(Z=p , 2k + 1)
is essential for sufficiently large s.
9
Proof
Consider the diagram
______~_______//_2k+1
S2k+1 S(p)
| |
ps|| ps|
h fflffl| ~ fflffl|2k+1
X _________//_MMS2k+1__________//_S(p)
MM
MMM | |
'sOhMMMMM's| ('s)(p)|
M&& fflffl| fflffl|
M(Z=ps, 2k + 1) _=__//_M(Z=ps, 2k + 1)
in which the vertical sequences are cofibrations and ps denotes the map with de*
*gree
ps. If 's O h ' *, then ('s)(p)O ~ O h ' *, and so (~ O h) lifts through the *
*map
ps : S2k+2(p)-!S2k+2(p)by Lemma 7. Suspending once more, we obtain the lift ind*
*icated
by the dashed line in the diagram
gg3S2k+3(p)3
g g g
gg g g g |ps|
gg g g fflffl|
g __________//2k+3
2X _____2h__//_S2k+3 ~ S(p) .
The torsion subgroup of [ 2X, S2k+3(p)] is a finite abelian p-group, and so it *
*has an
exponent pe. If s e, then ~ O 2h ' *, which is a contradiction, and so 's O*
* h is
essential. ||
2.3 Proof of Theorem 3
Let G denote the tower {[ ( 2lXt), S2(k+l)+1(p)]}. Since Ph( 2lX, S2(k+l)+1(p)*
*) is natu-
rally isomorphic to lim1G, the tower G cannot be Mittag-Leffler [4]. This means
that the index of Im([ ( 2lXt), S2(k+l)+1(p)]) [ ( 2lX2k), S2(k+l)+1(p)] (whi*
*ch is finite
by [12, Prop. 0]) is unbounded as t increases. Let r be the rank of the group
[ ( 2lX2k), S2(k+l)+1(p)], let T be its torsion subgroup, and write
` '
At= Im [ 2l+1Xt, S2(k+l)+1(p)] -! [ 2l+1X2k, S2(k+l)+1(p)]
and ` '
Zt= Im [ 2l+1X2k, S2(k+l)+1(p)] -! [ 2l(Xt=X2k), S2(k+l)+1(p)] .
Choose t large enough that the index of At [ 2l+1X2k, S2(k+l)+1(p)] is divisi*
*ble by
pr((2n+2)(k+l)+1)|T |. Then the quotient [ 2l+1X2k, S2(k+l)+1(p)]=AtT is an ab*
*elian group
10
which is generated by a set of at most r elements and which has order divisible*
* by
pr((2n+2)(k+l)+1). Since there is a surjection of finite groups
Zt~= [ 2l+1X2k, S2(k+l)+1(p)]=At- ! [ 2l+1X2k, S2(k+l)+1(p)]=AtT
it follows that Zt also contains elements of order divisible by p(2n+2)(k+l)+1.
The commutativity of the diagram
[ Xt, S2k+1(p)]_______//[ X2k, S2k+1(p)]_____//_[Xt=X2k, S2k+1(p)]
| | |
| 2l ~=| 2l |2l
fflffl| fflffl| fflffl|
[ 2l+1Xt, S2(k+l)+1(p)]//_[ 2l+1X2k, S2(k+l)+1(p)]//_[ 2l(Xt=X2k), S2(k+l)*
*+1(p)]
clearly shows that Zt Im( 2l) [ 2l(Xt=X2k), S2(k+l)+1(p)]. Thus there is a *
*map
g : Xt=X2k- ! S2k+1(p)such that 2lg has finite order divisible by p(2n+2)(k+l)*
*+1. Notice
that g itself also must have finite order since it is an element of the finite *
*group Zt,
and so 2ig has finite order divisible by p(2n+2)(k+l)+1for 0 i l.
Since the composition ~O 2l(p(2n+2)(k+l)Og) has finite order divisible by p,*
* Lemma
8, applied to 2(l-1)g, shows that 's O p(2n+2)(k+l)O 2(l-1)g is essential if *
*s is large
enough. Fix such an s and define f = 's O p(2n+2)(k+l)O g. Thus 2if is essenti*
*al for
each 0 i < l.
Finally, we demonstrate that, for 0 i < l - 1, the essential map 2if is t*
*rivial
of order at least n. Let kl(K) n and let h : K -! _2i(Xt=X2k)_be any map. Si*
*nce
2i(Xt=X2k) is 2(k + i)-connected, h factors through h : K=K2(k+i)-! 2iXt=X2k.
Since 2ig has finite order, the induced homomorphism ß2(k+i)+1( 2ig) = 0. Ther*
*efore
__ 2(k+i)+1
2ig O h can be extended to a map eh: K=K2(k+i)+1-! S(p) as in the diagram
________2if_____________________________*
*_____________________________________________________________________@
_________________________________________________*
*_____________________________________________________________________@
____2ig__________________________________((_________*
*______________________________________________________2(k+i)+1p(2n+2)@
K ______h_//o2i(Xt=X2k)___//S(p)77_____________//_S(p) __'s//_M
oo p77p __66____________________*
*__________________
| _hoooo eh ppp _____________________________*
*_____________________________________________________________________@
| oooo pppp ____________________________________*
*_____________________________________________________________________@
fflffl|oo pp _______________*_______________________________*
*_____________________________________________________________________@
K=K2(k+i) ____//_K=K2(k+i)+1__________________________________________
where we have abbreviated M = M(Z=ps, 2(k + i) + 1). By Proposition 5(c) we have
kl(K=K2(k+i)+1) 2n + 2, so p(2n+2)(k+l)O eh' * by Lemma 6. Thus 2if O h ' *,
which shows that 2if is trivial of order at least n and completes the proof. *
* ||
11
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12