Determination of the J-groups of complex projective and lens spaces

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K-Theory 29: 27-74, 2003.

Determination of the

J

-groups of Complex

Projective and Lens Spaces

I. DIBAG

Department of Mathematics, Bilkent University, 06800 Ankara, Turkey. e-mail: dibag@fen. bilkent. edu. tr

(Received: April 2002)

27

Abstract. We determine completely the I-groups of complex projective and lens spaces by means of a set of generators and a complete set of relations.

Mathematics Subject Classifications (2000): Primary 55R25; secondary 55Q50, 55S25. Key words: sphere bundles, vector bundles, I-morphism, K-Theory operations.

1. Introduction

For a finite-dimensional CW-complex X, let

J

_{(X) denote the finite Abelian group}

of stable fibre homotopy classes of vector bundles over X and for a prime p, l

p

(X),

the p-summand of J(X). For n,kEZ

+

_{, let Pn(C)=S}

2n+I

_{jU(I) and U(p}

k

₎₌

s

2n+I

_/Z

p

k denote the complex projective space of (complex) dimension n and the

associated lens space, respectively. The aim of this paper is to determine l

p

(P

n(C))

and J(L

n

_(p

k

_{)) by means of a set of generators and a complete set of relations.}

Let rn

be the greatest integer such that p

r

_{"� n/(p - 1). Then for O � s �Jn}

_and

0 � j �

r

n -

s,

we define the integer

s

- [ n -p

s

(p

i

- 1)

J

tj -

_ps

+

i(p - l) .

Let r, denote the complex Hopf bundle over Pn(C) and w

=

r(r,)-2 E KR(P

n

(C)),

where r denotes realification and also denote by w the pull-back of w to KR(L

n

(p

k

_{)). It is proved in [10] that the Adams operation if.,� passes to the quotient and}

acts on J(X) and that

if.,�=

1 on l

p

(X) for (k, p)

=

1. It follows from this and the

results of [4] on J(P

n(C)) that lp

(Pn(C)) is generated by {w, iJr;(w), ... ,

if.r{"

(w)}. We seek (rn

+

1)-relations between these generators of the form:

rn-S

'°'

s t� ps+j

La1

p

1

if.,

R

(w)=O (O�s�r

n),

( 1.1)

J=O

where the integers aj are to be determined. Let p� denote the K -theory charac

teristic classes associated to the Adams operation if.,� introduced in [ 1, Section 5]

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28 I. DIBAG

(also denoted by

ek

in [7]). We prove a key result (e.g. Proposition 2.2.1) to the effect that if K-IR.(X) is p-torsion free then lp(x)

=

0 iff p'(x) is integral. Thus, for

p odd or p

=

2 and n ¢.1 (mod4),

if and only if

p'

('f

ajl1lfr{+j

(w))

=

rfi

p,(1/f{+j

(w)t:ip'J is integral.

j=O j=O

We let

s [n/2]

P,(1/f{+j

(w)/j

=

1

+

L

b~jwm.

m=I

m is called a singular s-exponent if and only if b~j <j. Z for some j. We denote the set of singulars-exponents by Mn,s· We define the (s, j)-index of m (m ~ 1)

which is an integer denoted by e~j and prove that m E Mn,s if and only if e~j > 0

for some j E

z+.

If m E Mn,s, then m can be written in the form, m

=

(l /2)

(p - l)ps+i t,(i ~ 1, (L., p)

=

1). We define

<t>n,s

=

{tj: 0 ~ j ~ rn - s}, <t>~.s

=

{tj E <t>n,s: tj

=

0 (mod p).

If tf

=

pv L. E <t>~.s (v ~ 1, (L., p)

=

1) then (tjh+1 ~j ~k+v

=

(pv-l L. - 1,

pv-2 _{t, -} _{1, ... ,}_{t, -} _{1). Thus, if we denote}_Tf

=

_{(t{, tf+i • ... , tf+v) then for}

con-secutive elements t{, tf E <t>~ s (k < l), Tf

n

T/ is either empty or equal to {tt} and

the latter is always the case if p

=

2. A key result is Proposition 4.4.4 which states that there exists a bijection, o-: <t>~.s -+ Mn,s given by

o-(tD

=

0/2)(p - l)ps+ktt.

If

ti= pv L.(v ~ 1, (L., v)

=

1) and m

=

o-(tt)

=

(1/2)(p - l)ps+k+v L.,

then

(1) (e~jh ~ j ~ k+v

=

(v v v - 1 v - 2 · · · 1) (e.g. Lemma 4.4.3).

(2) (e~j) j ~ k-l is a strictly increasing sequence bounded above by v (e.g. Lemma 4.4.7).

and as a consequence of (1) and (2) we have the

OBSERVATION 1.1.1. Given j ~ k such that e~j > 0, then there exists a unique k

+

1 ~ j' ~ k

+

v such that e~j

=

e~j,.

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29

We prove in Section 4.3 a number of integrality theorems as a result of which it follows that if Q P is the sub-ring of rationals which are p-integral then

(1) b~j

=

=f(l/ l-.)(1/ p0_~j)(mod Qp) (e.g. Corollary 4.5.2).

.

,s

fl

rn-S P ps+J asp j ~ m •

(2) If j=O PJR.(V'JR. (w)) 1

=

l

+

~m ~ 1 CmW then Cm E Z If m

</.

Mn,s and if m E Mn,s, then Cm= =f(l/ 6) I:/aj/ p0_~j)_{(mod Qp) if either}_{p -::j:.}_{2 or}_m

is not the least element of Mn,s and 1

'°"'

aj 1 af_ 1

Cm

= - /':,.

L.., 2s~j

+ /':,.

2s!ii-l (mod Qp) O~j~i

Hi-I

if p

=

2 and m is the least element of Mn,s (i.e. Corollary 4.5.3). For O :::;; s ,=:;;

rn, we define as= (aj)o ~j ~ rn-s to be an s-admissible sequence if and only if Lj aj/ p0_~j_E_ZVm _E_Mn,s·_{Except for}p

=

_{2 and}mis _{the least element of}

Mn,s in which case it is required that

s s

'°"'

aj -

'°"'

ai-1 E

z

L.., 2s!ij L.., 2s!ii-l .

O~j~.i

Hi-I

Then it follows from Proposition 2.2.1 and the above that I:

1

n:i ajli(i/J{+j

(w))

=

0 in lp(Pn(CC)) if and only if as= (aj) is ans-admissible sequence. We extend this result to the case p

=

2 and n

=

1 (mod 4) where 2-torsion exists in

KJR.(Pn(CC)). Using the Observation stated above and by ordering the singular

exponents and using induction, two s-admissible sequences, called a- and

/3-sequences are constructed and we obtain the corresponding (rn

+

1)-relations in lp(Pn(CC)). Of these, the a-relation has the advantage that the sequence consists only of O and

±

1. Both of these relations are complete except for

p

=

2 and n

=

3 (mod4) in which case the complete set of relations is given by the corresponding a- and {3-relations for (n - 1) via the isomorphism,

]z(Pn(C))

=

]z(Pn-1 (C)).

The subgroup G(p, n, k) of the I-group of the lens space U(pk) generated by powers of w, or, equivalently, by the images of the Adams operations acting on w,

k

is a quotient of Jp(Pn (CC)) with the relation,

1/J;

(w)

=

0. Hence, a complete set of relations on G(p, n, k), called the fundamental relations, is given by either the

a-k k+I

or {3-relations with

1/J;

(w),

1/J;

(w), ... all omitted.

(1) If pis odd,

J(C(pk))

= {

G(p, n, k) ~f n ¢= 0 (mod4),

G(p, n, k) E9 Z2 If n

=

0 (mod4).

(2) If p

=

2, let ~ be the non-trivial real line bundle over L n (2k) such that

c(~)

=

1/lt-

1 _(TJ) _and

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30 I. DIBAG

We have

2).

=

re().)

=

re(~) - 2

=

r(v/l-l

(TJ)) - 2

=

VJt-·

(r(T})) - 2

2k-l 2k-l

=

VJR

(r(TJ) - 2)

= VJR

(w).

Then J(U(2k)) is generated additively by A and G(p, n, k) with the relation

2).

=

VJik-•

(w). Hence, the fundamental relations together with this relation yields a complete set of relations on J ( L n (2k)).

As a consequence of the a-relations for the complex projective space, we k

prove (e.g. Proposition 6.2.10) that J(p, n, k)

=

VJ'

(lp(Pn(C))) is isomorphic to lp(P[n/pkJ(C)). This furnishes an alternative proof to Proposition 3.3.1 that

k

the Ip-order of

VJ'

(w) in lp(Pn(C)) is the p-component, M[n/pkJ+I, p of the Atiyah-Todd number Min/ pkJ+ 1• As a corollary of Proposition 3.3.1, follows a

very simple proof of Lam's result which is stated in [20] without proof. For n

=

(p - 1) pk+ r (0 ~ r ~ p - 2), the a-relation takes an overwhelmingly simple form (e.g. Proposition 6.2.12) from which we immediately deduce the primary decomposition of lp(Pn(C)) as a direct-sum of cyclic groups (e.g. Proposition 6.2.13). As a demonstration of how simply the relations can be computed, the case of fi(P164(C)) is studied and the a-and ,B-relations are written down explicitly.

Lett' : lp(Pn(C))-+ J(U(ps)) and put J(p, n, s; k)

= t'(J(p, n,

k)), (k < s). As a consequence of the a-relations for the lens space, we obtain (e.g. Defi-nition 7.3.4) that J(p, n, s; k)

=

G(p, [n/ pk], s - k). From this and [17, Theorem

k

1.1] and [16, Theorem 1.2], we compute the order of

VJ'

(w) in J(U(ps)) (k < s) (e.g. Proposition 7.3.7). Also the primary-decomposition of J(U(ps)) is written down when n

=

pk(p - 1)

+

r, (0 ~ r ~ p - 2) (e.g. Proposition 7.3.10).

The method of this paper, e.g. that of singular exponents and indices is new in literature. We believe that this method is absolutely indispensable to obtain the relations in the simple (computable) form that we have obtained. Our strategy is first to tackle the case of the complex projective space whose KR-groups are (almost) free by using the characteristic classes p~ where (as a result of freeness) the computations are straightforward and then to study the J -groups of lens spaces as extensions of quotient groups of those of complex projective spaces. In papers on ]-groups of lens spaces, this approach via the complex projective space is missing. Direct assaults are made on lens spaces where their

KR

groups are torsion groups and this makes the computations unnecessarily complicated and makes it impossible to obtain these relations in this simple (computable) form. Also, sep-arate papers are devoted to the cases, p odd and p

=

2, whereas in this paper, a unified treatment is given to cover both cases simultaneously.

Finally, these groups are looked on from a purely algebraic viewpoint in Section 8, describing, briefly, how they naturally arise in algebra, drawing references to [13]. Let D(a,

,B;

n) be the finite Abelian group of equivalence classes of truncated weak D-series with rational coefficients defined in [13]. Let Dn

=

D(a,

,B;

n) for which aP = p - 1 and

,Bp

= 1. Then one can let the Adams operation VJk act on Dn

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in a natural way. The subgroup of Dn of truncated weak D-series which satisfy the

Adams conjecture is, precisely, JdPn(C)) which coincides with J(Pn(C)) except

for n

=

1 (mod4) in which case, J(Pn(C))

=

JdPn((;,)) EBZ2 • This motivates us to

study the group Dn (or, D(a, {3; n), in general) to which some of our methods may

perhaps be adapted. Of course, the quotient group, Dn/ JdPn (C)) will measure

how far these truncated weak D-series will deviate from satisfying the Adams

con-jecture. It is of interest to see whether the subgroup of D(a, {3; n) which satisfies

the Adams conjecture carries any geometric significance for any a and

f3.

The present paper together with [14] constitute a self-contained solution to the problem. Reference [ 14] starts from scratch and determines the exact sequences

relating the J -groups of lens spaces. In this present paper, technical references

concerning J(U(pk)) are drawn mainly on [14].

Finally, the author would like to thank the referee for his suggestions.

2.

J

p-Order and the Characteristic Class

p'

2.1. RELATION BETWEEN THE RATIONAL ]-INVARIANT Q

AND THE CHARACTERISTIC CLASS

p'

Let for k E

z+,

(Q and (Qk denote the rationals and the subring of rationals whose

denominators are powers of k. Let p be a prime and p:: K-JR.(X) ~ 1 + .KJR.(X)®(Qp

denote the characteristic class attached to the Adams operation

t:

(e.g._ [1, 2]; it

is denoted by E>p in [7]). In [7, Section 16], a rational I-invariant Q: KJR.(X) ~

1 + .KR(X) ® (Q is defined and it is proved in [7, Theorem 16.2] that if x E

KR (

X) is J -trivial then Q (x) is integral. In this section, we shall prove that

Q((VJ: - l)x) =p:(x) in 1 + .KR(X) ® (Q which will enable us to prove a local

converse to [7, Theorem 16.2] for bundles of the form

(VJ: -

l)x, x E .KR(X).

PROPOSITION 2.1.1.. Let p be a prime and KR(X) be free of p-torsion and

x E .KR(X). Then J((t: - l)x)

=

0

if

and only

if

p,(x) is integral.

Proof. We have a commutative diagram,

j 1 +.KR@ (Qp

1 + .KR(X) /

li

~

-k 1 +KR® (Q

Since Q is multiplicative and commutes with

t;,

VJP(Q(x))

-Q((VJ; - l)x)

=

R

=

p:(x) in 1 +KR® (Q

Q(x)

by [7, Theorem 16.1], hence t(p:(x))

=

Q((VJ; - l)x). Suppose that

J ( (

t: -

1 )x)

=

0. Then it follows from [7, Theorem 16.2] that

Q((VJ; - l)x)

=

k(l + u), u E KR(X).t(j(l + u))

=

k(l + u) = Q((VJ; - l)x) = t(p:(x)).

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32 I. DIBAG

Let, for a prime q, Tq(X) denote the q-torsion subgroup of .KR(X). Then p'(x)

and j ( 1

+

u) differ by an element of ker

L = (Eflq,epTq(X) ® Q!p) = (Eflq,epTq(X)) C j(.KR(X)),

i.e.

p'(x) = j(l

+

u)

+

j(v) = j(l

+

u

+

v), v E .KR(X);

i.e.

p'

(x) is integral.

Conversely, let k E

z+.

Then the co-cycle condition (cf. [1, Proposition 5.5], or, [7, Definition 8.1]) states that

p;{(x)

=

p~(x),t,~(p,(x))

=

p'(x),f,,(p~(x)) in 1

+

.KR(X) ® «J!kp·

Since the class p~ is multiplicative and commutes with the Adams operations,

p~((,t,P _ l)x) = P~(,t,,(x)) = t:(p~(x)) = ,t,~(p,(x))

R p~(x) p~(x) p'(x)

(2.1)

in 1

+

.KR(X) ® «J!kp·

The class p'(x) is integral. Since .KR(X) is free of p-torsion, the inclusion-map,

.KR(X) ® «J!k-+ .KR(X) ® «J!kp is 1-1. Hence, Equation (I.I) holds in 1

+

.KR(X) ®

«J!k, i.e. p~ ( ( ,t,' - l)x) = ,t,~ (p' (x)) / p' (x) in 1

+

.KR

(X) ® «J!k It follows from the Adams conjecture (refer to [12] for an elementary proof and [2, Theorem 1.1])

thatJ((,t,,-l)x)=O. D

2.2. lp-TRIVIALITY AND INTEGRALITY OF THE CLASS p:(x)

We know from [12] that the Adams operations pass to the quotient and act on the group J (X). In this section, using [12, Proposition 2.3.3] to the effect that

(,t,: - 1) is an isomorphism on lp(X) and is zero on lq(X) for q -=I= p, we obtain from Proposition 2.1.1 that in the absence of p-torsion, a necessary and sufficient condition for Ip-triviality of x E .KR(X) is integrality of the class p,(x).

PROPOSITION 2.2.1. Let p be a prime and KR(X) be free of p-torsion. Then for

x E .KR(X), lp(x)

=

0

if

and only

if

p'(x) is integral.

Proof Let J(x) = u and lp(x) =up.Since

(1/1,

-1) = 0 on lq(x) for q -=I= p by [12, Proposition 2.3.3], it follows that (,t,' - l)u = (,t,' - l)up. Since (,t,' - 1) is an isomorphism on lp(X) by [12, Proposition 2.3.3], up =0 if and only if

(,t,: - I)up = 0 if and only if (,t,' - l)u = 0 if and only if p,(x) is integral by

Proposition 2.1.1.

o

Remark 2.2.2 . .KR(Pn((C)) is free of p-torsion except for p = 2 and n

=

1 (mod4) and Proposition 2.2.1 reduces the proof of the JP-relations that we seek to checking whether the corresponding p:-classes are integral.

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3. Coefficients of

~(t/t(

(w)) in 1

+

KR(P00(C)) ® Qp 3.1. REVIEW OF KJR>.(Pn(C))

For n ¢=.1 (mod4), KR(Pn(C)) is freely generated by {w, w2 , ... , w,[n/2_1}

sub-ject to w[n/2₁₊1

=

_{0. For}_n

=

_{1 (mod4), KR(Pn(C)) is generated by}_{w,_{u>2, ... ,}

w[n/21+1} subject to the relation, 2w[n/2l+1

=

_w[n/2₁₊2

₌

0. The reader is referred to [4] for details.

3.2. ADAMS OPERATIONS AND THE ASSOCIATED CHARACTERISTIC CLASSES FOR THE COMPLEX PROJECTIVE SPACE

DEFINITION 3.2.1. Let k E

z+.

Define a power-series Tk(x) with integer coeffi-cients by

=

~

k2(k2 - 1) ... (k2 - (q - 1)2) xq

L

q2(q2 _ 1) ... (q2 _ (q _ 1)2) q=I ~ k2(k2 - 1) ... (k2 - (q - 1)2) L - - - x q . q=I (1 /2)(2q) ! PROPOSITION 3.2.2. In KR(P00(C)), 1/l~(w)

=

Tk(w).

Proof This is [10, Theorem 5.2.4]. D

Let for a prime panda rational q, vp(q) denote the exponent of pin the prime factorisation of q.

COROLLARY 3.2.3. Let p be an odd prime. Then in KR(P00(C)),

p-1 1/1:(w)

=

Laiwi

+

wP, i=l where Vp(ai)

= {

r

_!<P

1 ~ i

₊

<

!(P

+

1), 1) ~ i ~ p - 1.

COROLLARY 3.2.4. Let p be a prime. Then in KR(P00(C)),

p-1

1/J:(w)

=

Laiwi

+

wP where a;= 0 (modp) (1 ~ i ~ p - 1).

i=l

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34 where

ai = O(modp2), l~i<!(p+l)pk-1,

a;= O(modp), !(p+l)pk-1_!(i!(pk-l.

I. DIBAG

Proof By induction on k. The case k = 1 follows from Corollary 3.2.3. Let

k > I and assume the induction-hypothesis for (k-1). By the induction-hypothesis,

1/J"{-\w)

=

Li~~

1- 1ci<ii +wPk-1, wherec; =O (modp2), 1

~

i < (l/2)(p+l)pk-z and C;=O (mod p), (1/2)(p + l)pk-Z ~ i ~ pk-I - 1. By Corollary 3.2.3,

1/J';(w)

=

L~::;i

bjwj+wP, where bj = 0 (mod p2 ), 1 ~ j < (l/2)(p+l) and bj =0 (mod p), (l/2)(p + 1) ~ j ~ p - 1. pk-1_1

1/J'l

(w)

=

1/;'; o

1/J'l-

1 (w)

=

L

c;[i/;';(w)i

+

[1/;';(w)]Pk-l i=I pk-1_1

L

ci(b1w + · · · + bp-1Wp-l + wP)i + i=I h+h+-·+jp=j h+2h+··+pjp=i ki +kz+··+kp=pk-1 k1 +2k2+··+pkp=i (3.1) (3.2)

Terms in the first sum for which js > 0 for some s < p contain c j = 0 (mod p)

and

b{'"

= 0 (mod p) and, hence, congruent to O (mod p2 ). _{Terms for which is}

=

₀

Vs<p have j=jp and i=pjp, i.e. i=pj. If 1 ~ i <(l/2)(p + l)pk-l then

1 ~ j < (1 /2)(p + 1) pk-z and c j = 0 (mod p2) and this particular term is con-gruent to O (mod p2 ). Thus d; = 0 (mod p2 ). If ( 1 /2) (p + 1) pk-I ~ i < pk-I, then (1/2)(p + l)pk-2 ~ j <pk-Zand Cj =0 (modp) and this particular term is con-gruent to (mod p ). Thus d; = 0 (mod p ).

In the second sum, terms for which

k1 + k2 + · · · + k(p-1)/2 ~ 1 or k(p+l)/2 + · · · + kp-1 ~ 2

are congruent to O (mod p2 ). The particular term for which ks= 1 for (p

+

1)/2 ~

s ~ p-landks,=OVl ~ s' ~ p-lands'/sisequaltopk-lbs=O(modp2 ).

Then e; =0 (modp2 ). Thus, a; =0 (modp2 ) if 1 ~ i < (1/2) (p + l)pk-l and

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PROPOSITION 3.2.6. Let k, q E

z+

and Uq(k)

=

(k2 - l)(k2 - 32) • · · (k2 - (2q - 1)2). Then in 1

+

.KR(P00(C)) © Qk, k w

=

1

~

uq(k) wq

=

1

~

uq(k) wq , PR< ) +Lu (2q

+

3)

+

L 22q(2q

+

1)! · q=I q q=I In particular,

(i) Ifk is odd then p~(w) is a polynomial of degree (1/2)(k - 1).

(ii) P]i(w)2 ₌_P]i(2w)₌₁

+

_(1/4)w_{and P]i(l/F]i(w))}₌₁

+

_(1/2)w.

Proof This is [10, Theorem 5.2.8]. D

COROLLARY 3.2.7. Let p be an odd prime. Then in 1

+

.KR(P00(C)) © Qp,

p;(w)=l+I:~~;HP-3_>aqwq

+

_(1/p)w<1₁2_Hp-I) _{for aqEZ+ (1::(q::((1/2)}

(p - 3)).

PROPOSITION 3.2.8. Let p be a prime and k E Z such that k ? 0 for p odd and

k? lfor p =2. Then in 1

+

.KR(P00(C)) © Qp,

where ai E Z, (1 ::( i < (1/2)(p - l)pk) and ai =0 (modp)for 1 ::( i ::( (p - 1)

k-1 1

p - .

Proof

(i) For p

=

2, it is proved by induction on k starting with k

=

1 and by noting that P]i(l/F]i(w))

=

1

+

(l/2)w and 1/F]i(w)

=

4w

+

w2 •

(ii) For p

=

3,

Pi(w)

=

1

+

(1/3)w and

Pi(l/Ft

(w))

=

1

+

(1/3)1/Ft (w) and it fol-lows from Proposition 3.2.5. by noting that (1/2)(p

+

l)pk-I

=

(p - l)pk-1•

(iii) For p odd and p

?

5,

(l/2)(p-l)

p;(w)

=

1

+

L

aiwi

+

(1/p)w<1₁2_{Hp-l)' (1/2)(p -} ₁₎_>_1. i=l

Let v,~(w)

=

I:;:~

1 _bjwj

+

_{wPk where by Proposition 3.2.5, bj}

=

_{0 (mod}_{p), (1 ::(} j ::( pk - 1).

(10)

36 I. DIBAG The products in the first sum whose coefficients are possibly not congruent to 0 (modp) are

and those in the second sum, are of the form wi . wPk ... wPk

=

wi+< 112HP-3>P\

(j?:

1) and both ipk, j

+

(l/2)(p - 3)pk

?:

pk> (p - l)pk-l_

o

3.3. J-ORDER OF LINE BUNDLES

In this section, we shall determine the J -order of line bundles over complex

pro-jective spaces and as a corollary, obtain a proof for Lam's result stated in (20] without proof. In what follows, let Mn denote the Aliyah-Todd number of [5] defined by Mn

=

TIP pvp(Mn), where

vp(Mn)

=

sup [r

+

Vp(r)].

I

~r~[;:::\J

k PROPOSITION 3.3.1. Let n, k E

z+

and p be a prime. The order of

i;,;

(w) in lp(Pn(IC)) is M[n/pk]+J, p.

k

Proof It follows from Proposition 3.2.8 that

p;(i;,;

(w)) is a strict D-series of

type (a,

/3)

in the sense of [13, Definition 3.1] where

aP,

=

l

!

pi (p - 1) if p'

=

p'

1 if p' =/=- p,

and /3p'

=

1. As remarked in Section 3.1, for n ¢. 1 (mod4), .KJR.(Pn(C)) is torsion free and is freely generated by w, w 2, ... , w[n/21 and we deduce from (11, Theorem

k -

-3.5] that the order of

p;(v,;

(w)) in (1

+

KJR.(Pn(C)) ® (Q))/(1

+

KJR.(Pn(C))) is k

M[n/pk]+J, p. By Proposition 2.2.1, this is the order of

i;,;

(w) in lp(Pn(C)). For k

n

=

1 (mod4) let n

=

4v

+

1. By naturality, order of

i;,;

(w) in lp(P4v(C))

I

order

k k

of

i;,:

(w) in lp(P4v+I (IC))

I

order of

i;,:

(w) in lp(P4v+2(!C)). By the torsion-free case, the first and the third orders are equal to M[4v/pkJ+b p and M[4v+2/pkJ+J, p, respectively and [4v/ pk] = [4v

+

1/ pk] = [4v

+

2/ pk]. Hence, the order of

k

i;,:

(w) in lp(P4v+1(/C)) is M[4v+l/pk]+J,

p.

D

COROLLARY 3.3.2. Let k

=

flI=i

p~; for distinct primes Pi and integers ai

?:

1.

(11)

Proof.

CXJ a.\·

1/1;1

···p., (w)

a· CXJ a.\·-l a.,._J a1 a.\·-2

(1/J;.,' - 1)1/f;'

···P,-1 (W)

+

(1/J;.,·-I - 1)1/f;I

···Ps-2

+ ... +

a2 cq a 1

+

(1/1;2 - 1)1/f;1

(w)

+

1/f;1

(w).

It follows from the Adams conjecture that the terms on the right-hand side apart

"I

from the last one, e.g.,

1/1;1

(w), are of orders the powers of Ps, Ps-I, ... , P2,

respectively. Hence, the order of

1/J~(w)

in Jp, (Pn((C)) is equal to the order of

"I

1/1;1

(w) in Jp,(Pn((C)) which by Proposition 3.3.1 is equal to M[n/p~'J+I' PI· The

same result is also true for the primes P2, ... , Ps. D

k COROLLARY 3.3.3 (Lam's result). Let p be a prime and n, k

E

z+.

Then

1/J;

(w)

is trivial in lp(Pn((C))for pk > n/(p - 1).

Proof. It follows from the definition of the Atiyah-Todd number that M[n/ pkJ+J, p

=

1 and the result follows from Proposition 3.3.1 . D

4. Singular Exponents and Indices

4.1. THE INTEGERS

tk

Throughout, n will denote a nonnegative integer and p a fixed prime. For a real number x, let [x] denote the greatest integer less than or equal to x. Let rn denote

the greatest integer such that pr" ~ n / (p - 1).

DEFINITION 4.1.1. For O ~ s ~ rn, 0 ~ k ~ rn - s, we define numbers tl

induc-tively by to= [n/ps(p - 1)] and having defined tk-1' define tt by:

if tl_1

=

0 (modp),

if tl_1 ¢ 0 (mod p),

One can, easily, check that tl

=

[(n - ps(pk -1))/ps+k(p -1)]. From now

onwards we put tk

=

tf

= [

(n - pk

+

I)/ pk (p - I)]. The numbers {tt}o,:;; k,:;; rn-s

form a strictly decreasing sequence.

LEMMA 4.1.2. Let O ~ i ~ j ~ k ~ rn. Then tJ-i ~ t;k-i. Proof. We first prove the special case,

rt-

1 ~ ti. Either (i)

k-1 [ n

J

t0

=

k l ¢.O(modp).

(12)

38 I. DIBAG Then by definition, k-l [ 1 [ n ]] [ n ] k

t1

=

P

pk-l(p _ l) = pk(p _

1)

= to. Or, (ii) t t1

=

[pk-I(:-1)]

=O(modp). Again by definition, tk-l -

.!_ [

n ] - 1 - [ n ] - 1 - tk - 1 < tk I - p pk-l(p

-1)

-

pk(p _ 1) - 0 O·

In the general case, let i ~ h < h

+

1 ~ j. It follows from the special case above that tt-h-l ~ tth. t!;t} and tt:} are obtained in the same inductive fashion

f _rom_tk-h-l d k-h . l . (h .) Th k-h-l _., k-h H

1 an t0 , respective y, m -z -steps. us, th+l-i ;:::::, th-i . ence,

k-j _., k-J+l _., k-}+2 _., _., k-i

t .. o:,...t .. 1-1 ~ 1-1-1 ,:,...f . . ~ 1-1-2 ""-···o:,...t~ ~ 0 •

tJ-1 _and_{tti are obtained in the same inductive fashion from tJ=f and tti in}

. H k-J _., k-i

z-steps. ence, t₁ ;:::::, ti . D

DEFINITION 4.1.3. For O ~ s ~ rn, define cl>n,s

=

LJ~'::;

{tt).

cl>~.s

=

{tt E cl>n,s}tt

'FO

and t% =0 (modp).

Put cl>n

=

cl>n,O and cl>~

=

cl>~.o-OBSERVATION 4.1.4.

If

k is the least integer such that tk E cl>~ and O ~ s ~ k ~ rn then

ts_ t . _ [ n ]

j - s+J - ps+J(p

-1)

(O~j ~k-s).

COROLLARY 4.1.5.

If

cl>~

=

<P, then cl>~.s

=

<P (0 ~ s ~ rn) and tj ts+J (0 ~ j ~ rn - s).

OBSERVATION 4.1.6. Let O ~ s ~ rn, 0 ~ j < i ~ rn - s, then

i-j - 1 ( i-j - 1)

i-Jts

+

P ~t~ ~ i-Jf

+

_P_P _ _ _

p '

p-

1 "'1"'P i

p-

1

with equality attained on the left-hand side if and only if t%

=

pt%+ 1

+

1, V j ~ k < i

and on the right-hand side if and only if t%

=

pt%+i

+

p, V j ~ k < i.

If

further t% (/. cl>~.s, V j ~ k < i, then tj ~ pi-i tf

+

pi-i - 1 with equality attained if and only ift%

=

pt%+1

+

p -1, Vj ~k < i.

Proof. The first part follows by induction from the inequality, ptf

+

1 ~ tf_1 ~

(13)

COROLLARY 4.1.7. LetO ~ s ~:.Yn, 0 ~ j < i ~ rn - s. Then pi-ht+ i - j ~ t

1.

COROLLARY 4.1.8. Lets, i E

z+.

Then

ps+i (p _ l)t[

+

ps (pi _ 1) ~ n ~ ps+i (p _ l)t[

+

ps+1(pi _ 1).

If

further tl ~ <1>~.s VO~ k < i then

n ~ ps+i(p - l)t[

+

ps(p - l)(pi - 1).

Proof. Put j

=

0 in Observation 4.1.6 and note that to

=

[n/ PS (p - 1)].

o

DEFINITION 4.1.9. Let O ~ s ~ rn, 0 ~ k ~ rn - s and for tt E <l>~.s• let tt

pvt::,., (v? 1, (t::,., p) = 1). Then it follows from Definition 4.1.1 that t

1

= pk+v-j t::,. - l(k

+

1 ~ j ~ k

+

v). Define

k+v k+v

Tt = LJ{tj} ={pvt::,.} U LJ {pk+v-j l::,. - l}.

j=k j=k+I

OBSERVATION 4.1.10. Let O ~ s ~ rn, 0 ~ k, l ~ rn - s. Then for distinct tt, tt E

<l>~.s, the sets Tt - { tt} and T/ - { tl} are disjoint.

If

fork < l, tt, tt are consecutive

in <l>~,S then Tt

n

T/ is either empty or {tt} and the latter is always the case for

p =2.

DEFINITION 4.1.11. Let O ~ s ~ rn, 0 ~ k ~ rn - s. Define <l>~.s

=

Urte<1>2}Tt

-{tl}) and <l>~.s

=

<l>n,s - <l>~,s·

LEMMA4.1.12. LetO~s~rn,O~k~rn -s. Then

if

S ,+.I

l tk E -vn s•

if

S ,+.II

l tk E -vn,s·

Proof. It is by induction on k. For k

=

0, it follows from Definition 4.1.11 and the fact that t

0

E <l>~.s. Let k > 1 and assume the induction-hypothesis for all k' < k. Suppose tl E <l>~.s• i.e. that tl E T/ - {t!} fort[ E <l>~,s· Let t[

=

pvt::,., (v? 1, (t::,., p) = 1). Then k ~ v

+

i and tt = pv-k+i t::,. - 1. According to the

induction-hypothesis, t~

= [

n ] -E, 1 ps+i(p _ i) where _ { 1 if

tf

E <l>~,s' E - 0 'f s ,+.II 1 ti E -vn,s·

(14)

40 l.DIBAG

r'

k

pv !::,. - 1 = [n/ps+i(p

~

1)] - E - 1 = [[n/p·''+k(p

-i)]]-

1

pk-i pk-, pk-,

[ _{ps+i(p _ i)}n

]-1.

Suppose tf E <l>~.s· Then tf_ 1 (/. <l>~,s and tf

hypothesis, tf-1

=

[ps+k-l~p -

l)J-E,

where _ { 1 if tf_1 E <l>~,s' E - 0 'f s ,+.II 1 tk-1 E -vn,s· If tf _ 1 E <l>~.s then ts= [[n/ps+k-l(p - 1)]]

= [

n ] . k p ps+k(p _ 1)

[tf_if p ]. By the induction

If tf_ 1 E <l>~.s' then let tf_ 1 E T/ - {tt} for some tt E <l>~.s· Suppose that tt

=

pv t::,.(v ~ 1, (t::,., p)

=

1). Since tf (/. <1>~.s' k - 1

=

i

+

v and tf_ 1

=

t::,. - 1, (tf_i, p)

=

_{(tf_ 1}

+

1, p)

=

1 and, thus,

t5

=

[tf_

1 ]

=

[tf_

1

+

1]

=

[[n/ps+k-l(p -

l)]J

= [

n ] . ₀

k p p p ps+k(p _ 1)

COROLLARY 4.1.13. Let p

=

2, 0:::;; s,::;; rn, <I>~ s =I=- 0 and ko, is the smallest integer such that tf0 E <l>~.s·

If

k > ko, then tf

=

[n/25+k] - 1.

Proof If tf E <l>~.s' tf

=

2v t::,. (v ~ 1, t::,. odd) then the last element of

r,:

is

tf+v

=

t::,. - 1. If t::,. =/=- 1 then tf+v E <l>~.s and the sets

r,: -

(tD and Tk+v - (tf+v)

are adjacent. Hence, by definition, <1>~.s

=

LJk>ko tf and the result follows from

Lemma 4.1.12. D

COROLLARY 4.1.14. Let p

=

2, 0:::;; s,::;; rn and <1>~.s =I=- 0. Then k is the largest integer such that tf E <l>~,s'

if

and only

if

tf = 2v for v ~ 1 (i.e. !::,. = 1).

Proof Let tf

=

2v t::,. E <l>~.s (v ~ 1, t::,. odd). Then tf+v

=

t::,. - 1 is even; i.e., either, tf+v

=

0, or, tf+v E <l>~,s· Hence, k is the largest integer such that tf E <1>~.s

if and only if tf+v

=

0 if and only if t::,.

=

1. D

4.2. THE COEFFICIENT b5

1; / AND THE ASSOCIATED SEQUENCES

DEFINITION 4.2.1. Let

O,::;;

s,::;; rn, 0,::;; j,::;; rn - s (s, j) =/=- (0, 0) if p = 2, q =

(15)

, [n/2]

p:

('1/r{+j (w) / 1

=

1

+

Lb~/

Wm. m=I Define for 1 ~ m ~ [n/2],

s:/

{s

=

(so, s1, ... , sq): s0

+

s1 +···+sq ts p j , s1

+

2s2

+ · · · +

qsq

=

m}. Fors

ES!/,

T( ) -

(p1J)!

SJ Sz

Sq-l(l)Sq

s - a1 a2 · · • a 1 - • so !s1 ! · · · sq! q- p Then b~1

=

L

_SEs·'·j _m T(s).

DEFINITION4.2.2. LetO~s~rn,O~j~rn -s.Form

=

O(modq),i.e.m

=

0 (mod (1/2(p - l)ps+i)), let s0(s, j, m) be the unique sequence defined by

s1 =···=Sq-I

=

0, Sq

=

m/q, s0

=

p11 -

m/q. When there is no ambiguity, we

shall, simply, write s0 _{instead of}_s0 _(s,_j,_m)._If_{we express}_m

=

₍₁_{/2)(p - 1)}

ps+i t::,., (j ~ i, (!::,., p)

=

1) then sf

= · · · =

s~_1

=

0, si

=

pi-J t::,., sg

=

p11 -

pi-J t::,.. Then

0 - (

p1J )

(_!_)pi-j

t,

T(s ) - . .

p•-Jt::,. p

If p

=

2 and (s, j)

=

(0, 0) assume without loss of generality that t0

=

n

is even. Then Pi_(w)2'0

=

_{[pj_(w)2]}_{210 - 1•}_{For 1}~ _m~ _n/2,_define_s0_(0,_0,_m)_by s1

=

m, so

=

210 - m and

T(s0)

=

(2:1) ( ~)

m

= (

2

:1)

2~m ·

In this case, b~0

=

_T(s_{0 ).}

DEFINITION 4.2.3. Let O ~ s ~ rn, i ~ 1, 0 ~ j < i and m

=

(1/2)(p- l)ps+i t::,.,

2m ~ n, (t::,., p)

=

1. Define the (s,j)-index of m by r::~ 1

=

pi-J t::,.

+

i - j - tj. Let

vi - c-0,j

vm - vm

LEMMA 4.2.4. v P (T (s0 ))

=

-r::~1.

Proof It follows from [14, Lemma 2.2.10] that

Vp ((pr;!::,.))

=

tj -

(i - j)

and hence vp(T(s0 )) =

tj -

(i - j) - pi-J t::,. -r::~1 _{except for p} = 2 and

(16)

42 I. DIBAG

2.2.10]. If m

=

(1/2)2;

ti=

i-1

ti

(i � 1,

ti

odd), then v2(T(s0₎₎

=

_t

o - 1 - (i - 1) -

i ti =

-(i

ti+

i - to)

=

-c�0. D

4.3. INTEGRALITY THEOREMS ON T(s)

Let O � s � rn, 0 � j, j' � rn - s, 1 � m, m' � [n/2]. In this section, we shall prove

that for s E

s:i:/,

s =I= s0_{(s, j, m), T(s), T(s)T(s}0_{(s, j, m)) and T(s}0_{(s, j, m))}

T(s0_{(s, j', m')) are all integral if (j, m) =I= (j', m').}

LEMMA 4.3.1. Let O � s � rn, 0 � j � rn - s and 1 � m � [n/2]. Ifs E

s:,:/

-{s0_{(s, j, m)} then T(s) E Z.}

p ps+j

'°' .

Proof Let PJR.(V'JR. (w)) = 1

+

L..,; � 1 a;w1 as in Proposition 3.2.8. Let

q

=

(1/2)(p - l)ps+j. Since s =I= s0_,_{s; > 0 for some 1 � i � q - 1. qs}

q � m and thus

sq � [m/q]. Let f3

=

[m/q] - sq� 0. We consider the following four cases:

Case (i): si > 0 for i < (p - l)ps+j-l;

Case (ii):si > Ofor i�(p- l)ps+j-l_

Case (ii-a): f3 � 1. Case (ii-b): tj E

<f>�.s-Case (ii-c): f3

=

_{0 and tj E <f>�.s·}

We number these four cases as 1, 2, 3, 4 and define

l

1 for h

=

1, 2, i.e. in cases (i) and (ii-a), c -h

-O for h

=

3, 4, i.e. in cases (ii-b) and (ii-c).

(h(s)

=

[m/q ]-ch - vp(s;) -sq+ vp(a;1 ···a;�;). We shall prove that in case

h, vp(T(s)) � (h(s) � 0, (1:,;;; h:,;;; 4). T. _ ( rl) I p . ( s1 Sq-I) ( 1 )sq s - _S - al ···a 1 · o !s1 t · · · Sq! p q -By [5, Lemma 6.2], Vp( (p 1i_)! )�tj- i�f Vp(Sj)�tj-Vp(Si) so !s1 ! · · · s_q! o � 1 � q

and by Lemma4.l.12, tj � [njps+j(p - 1)] - ch. Since

n � 2m, tj � [m/q] - ch, (1 � h � 3). Hence,

(17)

J-GROUPS OF COMPLEX PROJECTIVE AND LENS SPACES 43

For h

=

1, i.e. in case ( i), it follows from Proposition 3.2.8 that vp(a;1 • • • a;�;)

� Sj. 81 (s) � ([m/q] -Sq)+ (Si - Vp(si} - 1) � 0.

For h

=

2, ps+i-1(p - l)si �s1

+ · · · +

isi

+ · · · +

( q - l)sq-l

=

m - qsq

=

f3q +r, (O� r < q ).

s-�-_{I '-:}f3 ₂p

+

r < --

(/3

+

1)

� fJ

ps+j-l(p - 1) 2 p '-: p '

i.e. si < p/3 and vp(si) < {3-1. 82(s)

=

[m/q ]-sq-1-vp(si)

=

{3-1-vp(si) � 0

by the above.

For h

=

3, as in the proof of the case for h

=

2 above, 03(s)

=

[m/q] -sq

-vp(si)

=

f3 - vp(si) � 0. This completes the proof for the first three cases.

For h

=

4, tj E <l>�.s· Let tj E Tf - {tt} for some tt

=

put:,. E <1>�.s (v � 1,

(t:,., p)

=

1). m � qsq

+

isi � qsq

+

ps+j-l (p - 1).

n � 2m � ps+j (p - l)sq

+

2ps+i-1(p - 1). (4.1)

Also, put:,.= tt � [n/ps+k(p - 1)] - 1 by Lemma4.l.12, i.e.

n � ps+k+u(p - l)fl

+

ps+k(p - 1)

+

r, (0 � r < ps+k(p - 1)), i.e.,

n < ps+k+u(p - l)fl

+

2ps+k(p - 1). (4.2)

We deduce from inequalities (4.1) and (4.2) that sq < pu+j-k t:,.

equivalently, that [m/q]

=

sq � tj. Since tj � [m/q],

tj

+

1 or,

sq

=

[m/q] and as in the proof of h

=

2, si < p/2 and thus vp(si)

=

0. Hence

84(s) � 0. D

Corollary 4.3.2. Let O � s � rn, 0 � j � rn -sand 1 � m � [n/2]. Ifm ¢ 0 ( mod

( l/2)(p - l)ps+_{i) then b5,;,j E Z.}

Proof. b5,;,1

=

L 5 ... j T(s₎and since m ¢ 0 ( mod ( ( 1/2)(p - l_)ps+i), the

SE m

distinguished sequence s0_{(s, j, m) is not defined and thus T(s}

) E Z, Vs E s5,;,i by

Lemma4.3.l. D

COROLLARY 4.3.3. LetO � s � rn, i � 1 andm

=

( l/2)(p-l)ps+i fl ( 1 � m �

[n/2], ( fl, p)

=

1). Thenfor 0� j � i , b5,;,i _{¢Zif and only ifs5,;,}i _{> 0 i n whic h}

(bs_,j

) s,j

case, Vp m

=

-Em .

Proof. b5,;,i

=

_T(s0₎

+

_L

s/so T(s). The second term on the right�hand side is

(18)

44 I. DIBAG

LEMMA 4.3.4. Let O ~ s ~ r11 , 0 ~ j

"I-

j1 ~ rn - s, 0 ~ j1 ~ i1, i1 ? 1 and m1

=

(1/2)(p - l)ps+i1 tq, (2m 1 ~ n, (61, p)

=

1), s E

s:,;

1 (2m ~ n) not be a distin-guished sequence (i.e. s

"I-

s0_(s,_{j, m)) and s}0 _{= s}0_(s,_{h, mi) be a distinguished}

sequence. Jf 2m

+

2m 1 ~ n then T (s) T (s0 ) E Z.

Proof Let q = (l/2)(p - l)ps+J, q1 = (l/2)(p - l)ps+Ji and Oh(s) be as

defined in case h, (1 ~h ~4), as in the proof of Lemma 4.3.1. We shall prove that in case h, Vp(T(s))

+

vp(T(s0 )) ~01,(s)

+

[m1/q]

+

[m/qi] - U1 - j1

+

1) and that [m 1/q]

+

[m/qi] - (i1 - j 1

+

1)? 0. The result will follow from the

inequality (}h(s)? 0 proved in the proof of Lemma 4.3.1. As in the proof of Lemma 4.3.1, vp(T(s))? tj - inf1 vp(s1) - sq+ vp(af1 _···a;~;)_{and by Lemma 4.2.4,}

Vp(T(s0 ))

=

tj_{1 -} pii-h 61 -(ii - h)? [n/ ps+Ji (p - 1)]-1-pii-Ji 61 -(i1 - j1)

by Lemma 4.1.12 and since n? 2m

+

2m 1,

vp(T(so)) ? [ m ]

+ [

m1

J

(l/2)(p - l)ps+h (l/2)(p - l)ps+h

-- 1--pi1- 1161 - U1 - h)

= [;]

+

([;1

1 ]- / 1_-11_{61)- (i1}_-j1

+

₁₎

m1 1 . .

and since - = -p'I -JI 61

qi 2

[; ]- (i1 -

j1

+

1).

Thus, vp(T(s))

+

vp(T(s0 )) ~ tj - vp(s;) - sq

+

vp(af1 _···a;~;)

+

_[m/qi]

-(i 1 -

h

+

1). Let ch be defined as in the proof of Lemma 4.3.1. For 1 ~ h ~ 3 and

tj? [n/ps+J(p - 1)] - ch? [m/q]

+

[mifq] - ch and

Vp(T(s))

+

Vp(T(s0 )) ?

[;]-Ch -

Vp(s;) - Sq+ Vp(af1 ••• a;~;)+

+ [

:I

J

+ [; ]-

(i1 -

j1

+

1)

(}h(s)

+

[:I

J

+ [;]-

(i1 -

j1

+

1).

For h = 4, let tj E T{ - { tt} as in the proof of Lemma 4.3.1. We have 2m

+

2m 1

in the place of 2m in inequalities ( 4.1) and ( 4.2) holds as it is and we obtain from these two inequalities that [m/q]

+

[m 1/q] ~ tj and [m/q] =sq.Hence

Vp(T(s))

+

Vp(T(s0 )) ;;?: [ ;

J -

Vp(Si) - Sq

+

Vp(af1 ••• a;~;)+

+ [:

1

J

_{+ [; ]-}

_U1

-ji

+

1)

(19)

To prove that [mi/q]

+

[m/qd - U1 - j 1

+

1);? 0 we consider the following two cases:

(i) j ~ ji. Then [m1/q]

=

pii-j L1;? pii-j;? i1 - j

+

1;? i1 - j1

+

1.

(ii) j > ji. Let sq;? 1. Then [m/qd;? [qsq/qd;? [q/qd

=

pj-ji and [mifq]

=

i

1-j L;? pii-j_ Thus, [m/qd

+

[m1/q];? pii-j

+

pj-ji;? U1 - j

+

1)

+

(j -

h

+

1)

=

i 1 - j1

+

2. Since we are in case (ii) of the proof of Lemma

4.3.1, S; > 0 for i;? (p - l)ps+j-l and hence m;? is;;? (p - l)ps+j-l _

(a) Leti1;? j. Thenm/q1;? (p - l)ps+j-l /(1/2)(p - l)ps+h

=

2pj-j1-l;?

2(j - h), [m1/q]

=

pii-j L1;? pii-j;? i1 - j

+

1 and, thus, [m/qd

+

[m1/q]:? U1 - j

+

1) +2(j-

h)

=

U1 -

h

+

1)

+

(j - j1) > i1 - j1

+

1. (b) Leti 1 < j. Then as in case a,

m/q1:? 2(j - j1) U1 - j1

+

1)

+

(j - i1)

+

(j - 1 - j1)

LEMMA 4.3.5. Let

k;? 2, 0 ~ S ~ Yn, 0 ~ jh < ih, ih;? 1, mh

=

(1/2)(p - l)ps+ih Lh,

2mh ~ n, (Lh, p)

=

1(1 ~ h ~ k).

If n;? 2m1

+

2m2

+ · · · +

2mk, then

flL

1 T(s0(s, jh, mh)) E Z.

Proof. It suffices to consider the cases k = 2, 3. Fork = 2, by Lemma 4.2.4,

vp(T(s0_{(s, j1, m1)))}

+

_Vp(T(s0_(s,_h,_m2)))

=

tjl

+

tj2 - /i-h L1 - U1 - j1) - pi2-h L2 - U2 -

h).

By Lemma 4.1.12,

tj₁

+

_{tj2 ;? [n/(p - l)ps+h]}

+

[n/(p - I)ps+h] - 2

D

and since n;? 2m1

+

2m2, tj1

+

tj2 ;? [m1/qd

+

[m2/q2]

+

[mif q2]

+

[m2/q2] - 2,

where qh

=

(1/2)(p - l)ps+jhLh, (h

=

I, 2), m1/q1

=

pii-j1L1 and m2/q2

=

pii-h L2 and, thus,

Vp(T(s°(s,

h,

m1)))

+

Vp(T(s 0(s,

h,

m2)))

:? ([ :: ] -U1 -

h

+

1))

+ ([ :; ]-

U2 -

h

+

1)),

mifq2

=

pi1_{-hL 1;? pii-h;? i1 -}

h

+

_{1 and, hence,}_{[mifq2];? i1 -}

h

+

_{1 and,}

similarly, [mif qd;? i2 - j1

+

1.

For the case k

=

3, again by Lemma 4.2.4,

3

L

Vp(T(s0(s, jh, mh)))

(20)

46

and by exactly the same argument as for the case k

=

2,

3

L

Vp(T(s 0(s, }h, mh))) k=I

I. DIBAG

PROPOSITION 4.3.6. Let k ~ 2, 0 ::( s ::( rn, 0 ::( }h ::( rn - s, 1 ::( mh ::( [n/2] and sh E

s:,;fh,

(1 ::( h ::( k). /f n ~ 2m1

+

2m2

+ · · · +

2mk then n:=I T(sh) E Z.

Proof If sh is not a distinguished sequence Vl ::( h ::( k, the result follows from Lemma 4.3.1. Assume without loss of generality that sh is a distinguished sequence for 1 ::( h ::( l and sh is not a distinguished sequence for l

+

1 ::( h ::( k. If l ~ 2 then it follows from Lemma 4.3.5 that n~=I T(sh) E Zand from Lemma 4.3.1 that

n!=t+I T(sh) E Z. If l

=

1 then it follows from Lemma 4.3.4 that T(s1)T(s2) E Z and from Lemma 4.3.5 that n!=3 T(sh) E Z. D

COROLLARY 4.3.7. Let k ~ 2, 0 ::( s ::( rn, 0 ::( }h ::( rn - s, 1 ::( mh ::( [n/2] (1 ::( h ::( k). lfn ~ 2m1

+

2m2

+ · · · +

2mk then n:=I

b:,;{h

E Z.

Proof It follows from Definition 4.2.1 that n:=1

b;,{h

=

L

s-'·h

L

s'·iz · · ·

SJE ml SzE m2

Lskes'·ik ( n:=1 T (sh)) and n:=1 T (sh) E

z

by Proposition 4.3.6. D

mk

COROLLARY 4.3.8. Let k ~ 2, 0 ::( s ::( rn, }h, mh, ah E

z+

(1 ::( h ::( k) and expand s [n/2] p,(1/l{+jh (w))"'hp'jh

=

1

+

L

c;,ihwm. m=I c~ih

=

I:

a8+···+a/:,=ah a?+ zaq +·+ma!, =m

k

-n

_h=l

a8 +··+a/:,h =ah a~+zaq+-·+mha/:.h =mh

(21)

aJ+-+a,l11=cr1

a/+2al+---+m1a,l11 =m1

a~+···+a~1k=a1

af +2a~+-·-+mkaitk =mk X

Since k ~ 2 and n ~ 2a1m1

+ · · · +

2akmk ~ 2a}m1

+ · · · +

_{2a~ 1m1}

+ · · · +

2a}mk

+ · · · +

2a!kmk. It follows from Corollary 4.3.7 that

D

+ . p ps+j

COROLLARY4.3.9. Leta E Z , O~s~rn, O~J ~rn-sandexpandp!R(if!IR

(w))ap'i

=

1

+

_L~~11_{c~/wm. Thenc~j =ab~\modZ).}

Proof 1

+

L~~~l c~j wm = (1

+

L~~~l b~j wm)a. Thus c~j = ab~j

+

d!/,

where d:,;j=

I:

----·---(bS,J)al ... (bS,J

a'

.

)am-I ' ' ' I m-1 • ao.a1. ···am-I· n~2m ao+a1 +···+am-I =a a1 +2a2+·-+(m-l)am-l =m

2(a1

+

2a2

+ · · · +

(m - l)am-1) 2a1

+

4a2

+ · · · +

2(m - l)am-1

and a 1

+

a2

+ · · · +

am-I ~ 2 and it follows from Corollary 4.3.7 that

(b~'j)a 1 · · · (b~~1)am-l E Zand thus d:,;j E Z. D

COROLLARY 4.3.10. Let ajEZ (O~s~rn,O~j~rn - s) and expand

[l

r_.-s pP(,l,Ps+J (w))aip'J

=

1

+

'°'[n/ZJ as wm Then as

=

'°'~"-s a~bs,j (mod Z)

J=O JR 'f'JR_ L...m=l m · m L...1=0 J m ·

s+J as 1} [n/2) s ·

Proof Let p,(1/1: (w)) JP

=

1

+

Lm=l c,;/ wm. Then by definition,

as= _m

mo+m 1 +···+m,n -s=m

Cs,OCs,l · · · C s.,,,-s.

mo mi mrn-s

If mi > 0 for more than once in the above product then c~i ·. · c~~n-s E Z by Corollary 4.3.8. Hence the only possibly non-integral terms in the above sum cor-respond to mi = m for some O ~ i ~ rn - s and m j = 0, j -:/- i; e.g. the terms,

(22)

48 I. DIBAG

COROLLARY 4.3.11. Let a

1

E Z, (0 ~ s ~ r n, 0

:s;

j

:s;

r11 - s) and expand

1

+

~l11 /ZJ as wm Jr+ bs,J E Z VO~ 1· ~ r - s then

L.,m=I m · :J m --::: --::: 11

4.4. THE BIJECTION BETWEEN Mn,s AND <1>~.s

DEFINITION 4.4.1. Let O

:s;

s

:s;

rn. Then 1

:s;

m

:s;

[n /2] is called a singular s-exponent if and only if there exists O

:s;

j

:s;

r11 - s such that b~J

</.

Z (equivalently,

s,j 0) 8m > .

It follows from Corollary 4.3.2 that if m is a singulars-exponent then it can be writ-ten in the form, m = (l/2)(p -1) ps+i L', (i ;?; }, (L',, p) = 1). If mis not a singular

s-exponent then b~1 E Z, VO

::s;

j

:s;

r n - s. We let Mn,s denote the set of singular s-exponents. Then Mn= Mn,O

2

M11,l

2 · · · 2

Mn,s

2

Mn,s+I

2 · · ·.

LEMMA 4.4.2. Let O

:s;

s

:s;

rn, i;?; 1 and m

=

(l/2)(p - l)ps+i L', E Mn,s,

1

:s;

m

:s;

[n/2], (L',, p)

=

1. Then tf

=

L', - 1.

Proof.

(i) Assume tf

:s;

L', - 2. Then by Corollary 4.1.8,

n

:s;

ps+i (p _ l)tf + ps+I (pi _ 1)

:s;

ps+i (p _ 1) (L', _ 2) +

+

ps+I (pi _ l)

=

2m _ ps+1(/ _ 2pi-l

+

1)

=

2m - p5₊1_[pi-1_{(p - 1) -} _{(pi-I -} _1)]_<2m contradiction.

(ii) Assume tf ;?; L',. Then for O

:s;

j < i, by definition,

c~1

=

pi-i L',

+

i - j - tj

:s;

pi-i tf

+

i - j - tj < 0

by Corollary 4.1. 7. This contradicts the fact that m E Mn,s. Hence tf

=

L', - 1.

D

LEMMA 4.4.3. Let O

:s;

s

:s;

rn, i;?; 1 and m

=

(l/2)(p - l)ps+i L', E Mn,s,

1

:s;

m

:s;

[n/2], (L',, p)

=

1. Then there exists a unique O

,:s;

k < i such that tt

=

pv L', E <l>~.s (v;?; 1), i

=

k

+

v and tj

= pi-j

L', - 1, (k

+

1

:s;

j

:s;

i) and

cs,J

=

{ V

'

j

=

k,

m i-}+1, k+l:s;J:s;i.

Proof. Assume tt

</.

<I>~ s, VO

:s;

k < i. Then

n

:s;

ps+i(p - l)tf

+

ps(p - l)(pi - 1)

=

ps+i(p _ l)(L', _ 1) + ps(p _ l)(pi _ 1)

=

2m - p5 _{(p -} ₁₎_<2m contradiction.

by Corollary 4.1.8 by Proposition 4.4.4

Thus, there exists O

:s;

k < i such that tt E <1>~.s. Let O

:s;

k < i be the greatest

integer such thattt E <l>~.s· By Observation 4.1.6, tt+i

:s;

i-k-l (L',-1)+ i-k-I _ 1

(23)

to distinguish: Case 1. t/

=

pt{+ 1

+

p - 1, Vk < l < i. By Observation 4.1.6,

ts _{k+l -}_ _pi-k-l A 1 d th ts _ i-k A C 2 ts ,,,- s

+

2 ;:

Ll - an us k - p Ll. ase . ₁~ pt₁+₁ p - ior some

k < l < i. Then by Observation 4.1.6, t/ ~ ptt+1

+

p - 2 ~ p(/-l-1 6. - 1)

+

p - 2

=

pi-I 6. - 2. By Observation 4.1.6,

t%+1

~ pl-k-ltt

+

pl-k-1 - 1 ~ /-k-l(pi-l 1::,. - 2)

+

/-k-1 - 1 i-k-J A /-k-J 1 p Ll-p - ,

and since t{

=

0 (mod p), t{

=

pt{+i

+

p ~ pi-k 6.-/-k ~ pi-k 6.-p since l > k.

By Corollary 4.1.8,

n ~ ps+k(p _ l)t{

+

ps+l(pk _ 1) ~ ps+k(p _ l)(pi-k 6. _ p)

+

ps+l(pk _ 1)

=

ps+i(p _ 1)6. _ ps+l(pk(p _ 2)

+

1)

2m - ps+1(pk(p - 2)

+

1) < 2m.

Thus, the second case leads to a contradiction. By the first case, t{

=

pi-k 6.. That

tj

=

pi-J 6. - 1, (k

+

1 ~ j ~ i) follows from Definition 4.1.9.

and for k

+

1 ~ j ~ i, s;/

=

pi-i 6.

+

i - j - tj

=

pi-i 6.

+

i - j -

(i-

i 6. - 1)

=

i-J+l. D

PROPOSITION 4.4.4. There exists a bijection, a:

<t>~.s

i-+ Mn,s given by a (tD

=

(l/2)(p -

l)ps+kt%-Proof Lett{

E <t>~.s·

Express t{ = pv 6., (v ~ 1, (6., p) = 1) and define

a(tD

=

(l/2)(p - l)ps+k+v 6.

=

(l/2)(p - l)ps+kt{ E Mn,s· By Lemma 4.4.3,

a is invertible. D

COROLLARY 4.4.5.

<t>~

=

<t>~.o

2 <t>~.1 2 · · · 2 <t>~.s 2 <t>~.s+1 2 · · · ·

Proof It follows from Proposition 4.4.4 and Definition 4.4.1. D

LEMMA 4.4.6. Let

O~s~.rn, S V A ,+.Q

tk

=

p Ll E '+'n,s•

(24)

50 I. DIBAG Then for j ~ k, k-j _ 1 ( k-j _ 1) i-j

A+

p $'.:'. t~ $'.:'. i-j

A+

_P_P _ _ _ _ p Ll 1 "1"P Ll 1

p-

p-For j < k, either the left or the right inequality is strict. Proof By induction on j starting with j

=

k and tf

inequalities, ptj+ 1

+

1 ~ tj ~ p(tj+ 1

+

1).

p v 6 and using the

D

COROLLARY 4.4.7. Let O ~ s ~ rn, 0 ~ k ~ rn - s, tf

=

pv 6 E <l>~.s' (v ~ 1,

(6, p)

=

1), m

=

a(tD

=

(1/2)(p-l)ps+k+v 6. Then {s:;/}j :(k-1 is a strictly in-creasing sequence bounded above by v. s:;;k-I = _{v if and only iftL 1}= pv+I 6

+

1.

Proof Letj~k.s:;;j =pi-j6+i-j-tj,s:;;j-I =pi-H 16+i-j+I-tj_ 1-1

(1) s:;;j - s:;;j-I

=

(tj_

_{1 - tj) - ((p - l)pi-j}6

+

1). By Lemma 4.4.6,

(2) tj_I ~ i-j+I 6

+

(pk-j+I - 1)/(p - 1); and

(3) tj ~ pi-j 6

+

(p(pk-j - 1)/(p - 1); thus, (4) t}-1 - tj ~ (p - l)pi-j 6

+

1.

Hence, s:;;j - s:;;j-I ~ 0. If we put j

=

kin the above and note that s:;;k

=

v

we obtain that s:;;k-I ~ v. Let j < k. Then by Lemma 4.4.6, either inequality 2 or

inequality 3 is strict and thus inequality 4 is strict, i.e. s:;;j - s:;;j-I > 0. tt_ 1

=

pv+16+8, (1~8~p).Then

8~k-I

=

pk+v-(k-1) 6

+

(k

+

v) _ (k _ 1) _ (pv+I 6

+

8)

pv+i 6

+

V

+

I _ pv+I 6 _ 8

v

+ (

1 - 8)

=

v if and only if 8

=

1.

OBSERVATION 4.4.8. Let for O ~ s ~ rn, 0 ~ h < k ~ rn - s, tfi, tt E <l>~.s be

consecutive elements such that Tt

n

Tt

=

{tf} and let tt

=

pv 6 (v ~ 1, (6, p)

=

1) and m

=

(1 /2) (p - 1) ps+k+v 6

=

a (tD E Mn,s. Then s!-I ,s

=

v if and only if p = 2 _{and tt_ 1}= 2tf

+

I.

Proof Let tfi

=

pv' 61 _(v'~ 1, ₍₆_1,_p)

=

_{1). Then}_{tf_}

1

=

p6' - 1 and tf

=

6

1

-1

=

pv 6. Hence, 6'

=

pv 6

+

1 and tk-I

=

p(pv 6

+

1)-1

=

pv+I 6

+

p-1.

s:;;k-1

=

-tt-1+k+v-(k-l)+pk+v-(k-I)6

=

-pv+I6-p+l+v+l+pv+I6

=

v - (p - 2)

=

v if and only if p

=

2. If p

=

2, tk-I

=

2v+I 6

+

I. D

COROLLARY 4.4.9. Let O ~ s ~ rn, 0 ~ k ~ rn - s, tt

=

pv 6 (v ~ 1, (6, p)

=

1), p odd, m

=

a(tD E Mn,s· Then, either, (i) tf_1 > pv+I 6

+

1 in which case the

sequence {s:;;j}j::;;:k is strictly increasing, or, (ii) tt_ 1

=

pv+I6

+

1 in which case

s:;/-

1 ₌_V_{and iftfi}_E_<l>~.s_{is the preceding element to tt in}_<l>~.s_{then Tt}

n

_Tt₌_0.

Proof It is an immediate consequence of Corollary 4.4. 7 and Observation 4.4.8. D

(25)

4.5. T(s0 )

LEMMA 4.5.1. Let O::;;

s::;;

rn, 0::;; j::;; i, i ~ 1 and m

=

(1/2)(p - l)ps+i 6 E

Mn,s, 1 ::;; m::;; [n/2], (ll, p)

=

1. Then T(s0_(s,_{j, m))}

=

_(-1)µ;;/_{(1/ 6.)(1/}_pc:;'./) (mod

Qp),

where i.e. . { pi-j 6. - 1 µ~J

=

1 Proof By definition,

if

either p =fa 2, or, m is not the least element of Mn,s, or, j =fa i - 1

if

p

=

2, m is the least element of Mn,s and j

=

i - 1.

0

(cp11)(1)/-j1:,.

T(s)

=

. .

-

,

p•-J 6. p

o

p1J (

p1J

) (

p1J

)

is 1

Ts _{( )}= - -_pi-j,6. -1 -1 ··· 1 l

-pi-jf'-.-1 pi-jf'-.-2 (p )pPi-j1:,."

(4.3) Let cr(tt)

=

m, where cr is the bijection of Proposition 4.4.4. We consider the following two cases. Case (i): k

+

1 ::;; j ::;; i and Case (ii): j ::;; k.

In Case (i), by Lemma 4.4.3, tj

=

pi-j 6. - 1. We divide into two sub-cases.

(i)-a: k + 1::;; j < i, i.e. i - j ~ 1 and the terms (pi-j 6. - l) (1::;; l::;; pi-j 6. - 1)

appearing in the denominators of terms on the RHS are prime to p. It follows from

Equation (4.3) that T(s0 )

=

u

+

v, where vis the term obtained by multiplying all

the ( -1) 's together and u is the sum of all products which contain at least a term

p1J

/(pi-j 6. -1), (1::;; l::;; pi-j 6. - 1). Thus,

V -

(-1)'•-1,_1:

Gt-1.,,-;-,; -

(-I)'H,-1:

G)'"'

On the other hand, vp(u) ~ 2tj-(i-j 6.+i-j) = 2(pi-j 6.-1)-(pi-j 6.+i-j) =

pi-j 6. - (i - j

+

2) ~ 0 and, hence, u E

Qp

except for p

=

2, 6

=

1, i - j

=

1.

For this exceptional case, tj = 1, e!:/ = 2.

o (2)

(1)

21

(1)

2 (l)c:;,/

T(s)

=

-

= -

=

-2 2 2 2

Since 6 = 1 and µ~j = 1, this agrees with the statement of the lemma. (i)-b:

j

=

i. Then tj

=

6. - 1 and it follows from Equation (4.3) that T(s0 )

=

u

+

v,

Determination of the J-groups of complex projective and lens spaces