TRACTABLE FORMULAS FOR v1-PERIODIC HOMOTOPY

               GROUPS OF SU(n) WHEN n  p2- p + 1



                   DONALD M. DAVIS AND HUAJIAN YANG



                        1.Statement of results


  Let p be a fixed odd prime. The (p-local) v1-periodic homotopy groups, v-11ss*
 **(X),

of a space X were defined in [8]. Roughly speaking, they tell the portion of th*
 *e p-local

homotopy groups of X detected by K-theory.  For a spherically resolved space X,

each group v-11ssi(X) is a direct summand of some actual homotopy group ssL(X).

We will focus on the case where X is one of the special unitary groups SU(n), w*
 *hich

are spherically resolved by the fibrations


(1.1)                  SU(n - 1) ! SU(n) ! S2n-1:


  Let (m) = p(m) denote the exponent of p in m. Let


(1.2)             ep(k; n) = min{p(j!S(k; j)) : n  j  k};


where S(k; j) is the Stirling number of the second kind, which satisfies

                                    Xj       ij
(1.3)                   j!S(k; j) =   (-1)j-i jiik:
                                    i=1

The following result was proved in [7].


Theorem 1.4.  If k  n, then v-11ss2k(SU(n))  Z=pep(k;n), while v-11ss2k-1(SU(n))

has order pep(k;n), but is not necessarily cyclic.


Periodicity in v-11ss*(SU(n)) would allow one to determine v-11ss2k(SU(n)) for *
 *smaller

or negative values of k from this, if one wished.

  Theorem 1.4 at first glance appears to be all that one might want to know abo*
 *ut

v-11ss*(SU(n)). However, it has two drawbacks. One is that it only gives the or*
 *ders
__________
  1991 Mathematics Subject Classification. 55T15.
  Key words and phrases. v1-periodic homotopy groups, unitary groups, Stirling *
 *numbers, unstable
Novikov spectral sequence.


                                   1



2                         D. DAVIS AND H. YANG


of the odd groups, and not their actual structure, and the other has to do with*
 * the

intractability of the formula (1.2).

  The numbers ep(k; n) which occur in Theorem 1.4 are in some sense given expli*
 *citly

in (1.2), and some of them can be computed by a computer, but it seems to be ve*
 *ry

difficult to obtain useful general formulas for them from (1.2). Indeed, despit*
 *e efforts

in [11], [9], [6], and [7], the only general results obtained from (1.2) seem t*
 *o be a

(sharp) lower bound for ep(k; n) when n  p, and the inequality max {ep(k; n) : *
 *k 2

Z}  n - 1. These were proved in [6], using Fermat's Little Theorem. Since SU(n)

localized at p splits as a product of spheres when n  p, the first result can a*
 *lso be

obtained easily from results for spheres.  The second result is more useful, si*
 *nce it

implies (1.7).

  The first main result of this paper is a tractable formula for v-11ss2k(SU(n)*
 *), pro-

vided n  p2- p + 1.



Theorem 1.5.  Suppose p is odd, k = N + (p - 1)m with 1  N < p, and


                N + 1 + (p - 1)i  n < N + 1 + (p - 1)(i + 1)


with 0  i  p - 1.  Define m^ by 0  m^ < p and m  m^ mod p.  Then

v-11ss2k(SU(n))  Z=pe, where e is equal to

   8
   >>>i + 1                                ifi < N and i < ^m
   >>>
   >><i                                    ifN  i and (i < ^mor ^m= 0)


   >>>min(Ni+ (p - 1)m^; i + (m - ^m) + 1) ifi < N and ^m i
   >>>minN + (p - 1)m^+ 1;
   >>:                        iNj         j
          i + (m - ^m+ (-1)m^^mm^pm^(p-1)) ifN  i and 1  ^m i


The smallest n for which this theorem fails to give complete information is n =

N + 1 + (p - 1)i with N = 1 and i = p.

  The proof of this result makes no use of formula (1.2); it is an independent *
 *cal-

culation of v-11ss*(SU(n)). These separate calculations of the same homotopy gr*
 *oup

give a topological proof of a result in number theory, a tractable evaluation of


                      min {p(j!S(k; j)) : n  j  k}


when n  p2- p + 1, as given in Theorem 1.5. For example, the following corollary

is easily obtained from Theorem 1.5 and (1.2).



              FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS             3


Corollary 1.6. Suppose p is an odd prime, 0  ^m i < N < p, and i + p(`) + 2

N + (p - 1)m^. Then


            p(S(N + (p - 1)m^+ (p - 1)p`; N + (p - 1)i)) = p(p`):


We view this merely as an amusing offshoot of our work.

  An application of Theorem 1.5 to homotopy theory is a slightly improved lower

bound for the p-exponent of SU(n). Recall that the p-exponent of a space X, den*
 *oted

expp(X), is defined to be the largest e such that some homotopy group of X cont*
 *ains

an element of order pe.  As noted at the outset, if X is spherically resolved, *
 *its p-

exponent will be at least as large as that of the order of any element of v-11s*
 *s*(X). It

was shown in [7] that


(1.7)                      expp(SU(n))  n - 1:


This can be improved by 1 for certain values of n, as given in the following co*
 *rollary.


Corollary 1.8. If p is odd, then expp(SU(n))  n if, for some i < p,


                         i(p - 1) + 2  n  ip + 1:


Proof.We use Theorem 1.5 to determine max {ep(k; n) : k 2 Z}. In the notation of

Theorem 1.5, this maximum will occur when n = N + 1 + (p - 1)i and ^m= i. This

maximum will equal n - 1 if i < N, and n if i  N. The expression in the corolla*
 *ry

is obtained as N + 1 + (p - 1)i for 1  N  i.  ||



  The other main result is an explicit determination of the groups v2k-1(SU(n))*
 * when

n  p2- p + 1. Recall that Theorem 1.4 only gave their order.


Theorem 1.9.  Use the notation of Theorem 1.5, and let t = min{N; (m) + 1}.

Then
                       8
                       >>>Z=pe         ifi  N
                       >><
                        Z=p  Z=pe-1    ifi > N and ^m6= 0
    v-11ss2k-1(SU(n))  >     t    i-t
                       >>>Z=p  Z=p     ifi > N and ^m= 0 and t  i - N
                       >:  N     i-N
                        Z=p   Z=p      ifi > N and ^m= 0 and t  i - N


  A description of the conditions under which these groups are cyclic can be gi*
 *ven

without resorting to all the special notation of Theorem 1.5. The following res*
 *ult is

easily obtained from Theorem 1.9.



4                         D. DAVIS AND H. YANG

                    __                 __           __
Corollary 1.10. Let k be defined by 1  k  p - 1 and k  k mod p - 1.  Then
                                    __
v-11ss2k-1(SU(n)) is cyclic if n < (k + 1)p, and is the direct sum of two cycli*
 *c sum-

mands if
                          __           2
                         (k + 1)p  n  p - p + 1:



  The proofs of both of our main theorems will involve a delicate analysis of t*
 *he

unstable Novikov spectral sequence (UNSS) based on BP .  We let Es;t2(X) denote

the E2-term of this spectral sequence. The first part of Theorem 1.4 was an imm*
 *ediate

consequence of the following result, the first part of which was proved in [3],*
 * and the

second part in [7].



Theorem 1.11.     (1)If k  n, then E1;2k+12(SU(n))  Z=pep(k;n).

   (2)If k  n, then v-11ss2k(SU(n))  E1;2k+12(SU(n)).



  Theorem 1.5 is proved by computing E1;2k+12(SU(n)) by increasing induction on*
 * n.

By contrast, Theorem 1.11(1) was proved by computing E1;2k+12(SU(n)) by downward

induction on n, starting with E1;2k+12(SU(k))  Z=k!.  The methods of calculation

of the UNSS used in proving Theorems 1.5 and 1.9 extend those of [5]; we think

that the methods introduced here of calculating this spectral sequence for mult*
 *icell

complexes should be useful for other computations.

  In Section 2, we present the requisite background on the UNSS. In Section 3, *
 *we

outline the proof of Theorem 1.5, with details relegated to Section 4. In Secti*
 *on 5,

we prove Theorem 1.9. This paper overlaps substantially with the second author's

thesis, [12].



                      2.Background in the UNSS


  Let BP be the Brown-Peterson spectrum corresponding to the prime p. Then


                      BP* = ss*(BP )  Z(p)[v1; v2; : :]:;


where viare the Hazewinkel generators of BP*. Let  = BP*(BP )  BP*[t1; t2; : :]*
 *:,

where tiare Quillen's generators. We have |vi| = |ti| = 2(pi-1). Let c : BP*(BP*
 * ) !

BP*(BP ) be the conjugation, and define hi= c(ti). Then  = BP*[h1; h2; : :]:: L*
 *et

j = jR : BP* ! BP*(BP ) be the right unit.  We write hivj interchangeably with

j(vj)hi; this is the right action of BP* on .



              FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS             5


  Let M be a -comodule with coaction map  M : M!  BP*M. The stable cobar

complex C(T (M)) is defined as follows:


                 Cs(T (M)) =  BP*  . .B.P* BP*M;


with s copies of .  We use the cobar notation, and write fl0[fl1| : :|:fls]m fo*
 *r fl0

fl1 : : :fls m. The differential d  is given by
                               X   0h  00         i
      d(fl0[fl1| . .|.fls]m) =   fl0 fl0| . .|.flsm

           Xs         h            0   00         i      X     h               *
 * 0i00
(2.1)     +   (-1)jfl0 fl1| . .|.flj|flj|m.+.|.fls(-1)s+1   fl0 fl1| . .|.fls|f*
 *lm
           j=1
               P   0   00                        P   0  00
where  (fli) =   flj fljfor 0  j  s and  M (m) =   fl  m :

  The unstable cobar complex {C*(U(M)); d} is a subcomplex of {C*(T (M)); d},

consisting of terms satisfying an unstable condition, introduced in the followi*
 *ng def-

inition.


Definition 2.2.[4, p. 243] If M is a nonnegatively graded free left A-module, t*
 *hen

U(M) is defined to be the BP*-span of


              {hI m : 2(i1+ i2+ i3+ . .).< |m|}   BP*M;


where I = (i1; i2; : :):is a sequence containing only finitely many nonzero ij'*
 *s, and

hI = hi11hi22... .


  Define Us(M) = U(Us-1(M)), and Cs(U(M)) = Us(M). If M is a -comodule,

then U(M)  T (M) =   M as a sub--comodule, and so the differential d of the

stable cobar complex {Cs(T (M)); d} induces a differential on {Cs(U(M))} which

defines a complex {Cs(U(M)); d}. We will usually replace it by the chain-equiva*
 *lent

reduced complex obtained by replacing U(M) by ker(U(M)-!fflM).  This has the

effect of only looking at terms which have positive grading in each position.  *
 *The

homology groups of this unstable cobar complex are denoted by Exts;tU(M).

  It was proved in [4] that, if X is a simply-connected CW -space, there is a s*
 *pectral

sequence {Es;tr(X); dr} which converges to the homotopy groups of X localized a*
 *t p,

and if the integral cohomology H*(X) is a free algebra, then


                        Es;t2(X) = Exts;tU(P (BP*X));


where P (BP*X) denotes the sub--comodule of BP*X consisting of the primitives

under the coproduct. This is the UNSS for the space X.



6                         D. DAVIS AND H. YANG


  The following basic formulas were used or proved in [5].



Lemma 2.3.     (1)v1 = ph1+ j(v1) and  (h1) = h1 1 + 1  h1;
                                                          P p     p+1-ii i
   (2)v2 = ph2 + (1 - pp-1)hp1v1 + j(v2) - (p + 1)vp1h1 +   i=2aiv1   p h1, whe*
 *re

      ai2 Z;

   (3)d(v1) = j(vn1) - vn1and



           d(vahbvc) = (j(va) - va)  hbvc- va (hb)vc- vahb (j(vc) - vc)



      where  (hb) =  (hb) - hb 1 - 1  hb.



The first part of this lemma will be used very frequently in the context of rep*
 *lacing

ph1 by v1- j(v1).
      P F*       P F           P F
  Let     hi = c(   c(hi)) = c(   ti), where x +F y is the formal group sum
                                                 P      i+1         P      i+1
defined by x +F y = exp(logx + logy) with logx =   i0mix   , expx =   i0bix   ,

and exp(logx) = log(exp x) = x. Here {mi} and {bi} are two different polynomial

generators sets for H*BP with |mi| = |bi| = 2(pi- 1). Then the following lemma *
 *of

Bendersky ([3]) is useful.



Lemma 2.4.  The primitives P (BP*(SU(n)) form a free BP*-module generated by

elements {x3; x5; x7; :::; x2n-1} with coaction given by

                                X  X  F*   j
                      (x2k+1) =   (     hi)k-j x2j+1:
                                 j  i


The subscript k - j refers to the component in grading q(k - j).  Here we have

introduced the notation q = 2(p - 1), which will be used frequently throughout *
 *the

paper.

  We will need the following explicit computation.



Proposition 2.5. Mod terms of degree greater than 3q,

                  X F*                          p-1
                       hi= 1 + h1- h1v1+ h1v21- ___2h21v1:
                   i

                                  P F*
The only terms hj1which appear in     hiare 1 + h1.



              FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS             7


Proof.From


            x =   exp(logx)


              =   b0logx + b1(logx)2+ b2(logx)3+ . . .
                    X      i     X      i       X      i
              =   b0   mixp + b1(   mixp )2+ b2(   mixp )3+ . .;.
                    i0           i0             i0


we deduceithatjthe first nonzero bi's are b0 = 1, bp-1 = -m1, b2p-2= pm21, and

b3p-3=  p2- p(2p - 1). Then, mod terms of degree greater than 3q,



   X F*         X  F
       hi =   c(    ti)
    i          i i
                X      X     pj    j
          =   c    bi(    mjtk )i+1
                i0    j;k0
               i
          =   c 1 + m1+ t1- m1(1 + m1+ t1)p+ pm21(1 + m1+ t1)2p-1
                i j              j
              +( p2- p(2p - 1))m31
               i                             ij
          =   c 1 + m1+ t1- m1(1 + pm1+ pt1+  p2(m21+ t21) + p(p - 1)m1t1)
                                             i j                j
              +pm21(1 + (2p - 1)(m1+ t1)) + ( p2- p(2p - 1))m31)
               i                        ij     j
          =   c 1 + t1- pm1t1+ p2m21t1-  p2m1t21:


The desired result is now obtained using v1 = pm1 and c(v1) = j(v1).

  The second statement follows since the only powers of t1appearing in the expa*
 *nsion
   P    P    pj i+1
of   bi(  mjtk )   are 1 + t1.   ||



  This coaction formula in SU(n) will be extremely important, as it determines *
 *the

boundary homomorphism in the exact sequence associated to the fibration (1.1).

Indeed, there is an exact sequence


    ! Es;t2(SU(n - 1)) ! Es;t2(SU(n)) ! Es;t2(S2n-1)-@!Es+1;t2(SU(n - 1)) !;


with


(2.6)                      @(y) = y   (x2n-1):


This boundary formula is true since   gives the coboundary for the unstable cob*
 *ar

complex of SU(n). The term 1  x2n-1will not appear in our reduced complex.



8                         D. DAVIS AND H. YANG


  We prefer to work with the v1-periodic UNSS of [1]. This has the advantage th*
 *at

for X = SU(n) or S2n+1,
                          8
                          <v-1sst-s(X) ifs = 1 or 2, and t odd
(2.7)          Es;t2(X) = : 1
                           0           otherwise.


We can still work with unstable cobar names for classes; we just view the perio*
 *dic

spectral sequence as a direct summand of the actual UNSS, after multiplication *
 *by

sufficiently many v1's. So from now on, E2 refers to the periodic spectral sequ*
 *ence.

  We will make frequent use of the following result for the spheres. The 1-line*
 * part

was proved in [4], and the 2-line part in [2]. We introduce here terminology x *
 * y

mod S2n-1to mean that x - y desuspends to (or is defined on) S2n-1. For elements

of Es2(S2n+1), we frequently abbreviate y2n+1as y.


Theorem 2.8.     (1)The only nonzero groups Es;t2(S2n+1) are


                            Es;2n+1+qm2(S2n+1)  Z=pe


      with s = 1 or 2 and e = min(n; (m) + 1).

   (2)The generator of E1;2n+1+qm2(S2n+1) is ffm=e= d(vm1)=pe and satisfies


                          ffm=e -vm-e1he1 mod S2e-1:


   (3)Suppose n  (m) + 1. Then for 1  j  n, there is an element of order pj

      in E2;2n+1+qm2(S2n+1) which equals vm-j-11h1 hj1mod S2j-1.

   (4)If (m) + 1  n, then d(vm-n-11hn+11)2n+1has order p in E2;2n+1+qm2(S2n+1),

      and, for 0  j  (m), pj times a generator of E2;2n+1+qm2(S2n+1) equals

      vm-n+j-11h1 hn-j1mod S2(n-j)-1.

   (5)The homomorphism 2 : E2;2n-1+qm2(S2n-1) ! E2;2n+1+qm2(S2n+1) is injective

      if n  (m) + 1 and is multiplication by p otherwise.


  The formulas for leading terms of classes on the 2-line in parts 3 and 4 diff*
 *er

from those in [2], which have ve1on the right of the first h1.  We write them as

we have because that is the way in which they will occur in our applications.  *
 *To

see that h1  ve1hj1 ve1h1  hj1mod S2j-1, we use Lemma 2.3(1) to write the dif-
           P e-1ieji i+1 e-i j                                  i-1 i+1 e-i j-1
ference as   i=1 ip h1  v1  h1.  The ith term is a multiple of p   h1  v1  h1  *
 * =
P ii-1j j 2 e-j j-1                       2j-1
    j  v1h1v1  h1  , which is defined on S    . The following corollary encapsu*
 *lates
a common feature of parts 3 and 4 of the above theorem. We use the term "leading

term" to mean the term which desuspends least far.



              FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS             9


Corollary 2.9. If an element of E22(S2n+1) has order pf and leading term hhj2n+*
 *1,

then

                        j + (|E22(S2n+1)|) = f + n:


  We will also need the following result about the 2-line for spheres, proved i*
 *n [2].


Lemma 2.10.   Let Es2(M) = Exts(BP*; BP*=p) denote the E2-term of the stable

Novikov spectral sequence for the mod p Moore spectrum M. There is a homomor-

phism

                     H0: Es;t2(S2n+1) ! Es-1;t-2pn-12(M)

satisfying H0(x) = 0 if and only if x double desuspends, and if x = z  hn12n+1w*
 *ith

z2pn-1defined, then H0(x) = z.


  We will need the following fact about E2(M), proved in [5, 2.11].


Lemma 2.11.   vk1h1 6= 0 2 E12(M) and v1hk(p-1)+11= vk(p-1)+11h1 in E12(M).


  The following result will be very useful.


Lemma 2.12.     (1)hn+p-11 h12n+1 -vp-11h1 hn12n+1mod S2n-1.

   (2)hn+p-11 h12n+1double desuspends iff n = 1.


Proof.We use [5, 2.13], which says that


                 (hp1 hK(p-1)+n1+ hK(p-1)+p+n-11 h1)vK12n+1


is the sum of a boundary and a class which double desuspends, where n  1 and

K  0. As noted in [5, 2.14], this gives immediately that hp1h13 double desuspen*
 *ds,

since 2 is a unit. It also gives


                hn+p-11 h12n+1 -hp1 hn12n+1  mod S2n-1:


Thus, if n > 1, H0(hn+p-11 h12n+1) = -hp1by Lemma 2.10. By Lemma 2.11, this

equals -vp-11h1 6= 0, so that hn+p-11 h12n+1does not double desuspend. However,


        H0(hn+p-11 h12n+1+ vp-11h1 hn12n+1) = -vp-11h1+ vp-11h1 = 0;


and so the sum double desuspends.   ||



  The following lemma will also be used many times. It is the one place where t*
 *he

number of v1's on the left is important. We begin a policy of writing h for h1,*
 * and v

for v1.



10                        D. DAVIS AND H. YANG


Lemma 2.13.   For j  0,


     d(v`hn+1vj)  -(` + n + 1)v`h  hnvj+ jv`+1hn  vj-1h  mod  S2n-1-qj:


Proof.We evaluate the desired boundary using Lemma 2.3, expanding


                          d(va) = (v - ph)a - va;


to obtain

           X` ij                      Xn i  j
               ` `-i     i  n+1 j         n+1 ` i  n+1-i j
               iv   (-ph)  h   v   -       i v h  h     v
           i=1                        i=1
                                      Xj ij
                                   -      jiv`hn+1 vj-i(-ph)i:
                                      i=1

For each term in each sum, we study whether it satisfies the unstable condition*
 * on

S2n-1-qj(number of h's  1_2degree of stuff to the right of it) for both the par*
 *t on

the left side of the  and the part on the right side of the . In the first sum,*
 * hion

the left of the  satisfies the unstable condition on any sphere since pi with i*
 *t can

be used (via ph = v - jv) to make the exponent of h small. Terms with i > 1 can

use p2 to bring the hn+1 on the right side of  down to hn-1; then hn-1vj is def*
 *ined

on S2n-1-qj. The (i = 1)-term is, mod S2n-1-qj,


       -`v`-1h  (v - jv)hnvj  -`v`-1(v - ph)h  hnvj  -`v`h  hnvj:


  All terms in the second sum with i > 1 are defined on S2n-1-qj.  In the third

sum, terms with i > 1 can use p2 to bring hn+1 down to hn-1, so that it satisfi*
 *es

the unstable condition. Remaining p's can be used to bring hi down to h2, so th*
 *at

it satisfies the unstable condition. The term with i = 1 gives the second term *
 *in the

lemma since phn+1 = vhn - hnv.   ||



The following corollary will be useful.



Corollary 2.14. If k + n 6 0  mod p, and z = (vkhn+p-1 h + pw0)2n+1is a cycle

with w02n+1satisfying the unstable condition 2.2, then


                    z = d(__1__k+n+pvk+p-1hn+12n+1+ w2n+1)


with w2n+1unstable.



              FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS            11


Proof.Lemmas 2.12 and 2.13 imply that the leading term is correct.  The group

E2n+1+q(k+n+p)2(S2n+1) has order p, and by Lemma 2.12 and Theorem 2.8(4), both *
 *of

these leading terms yield nonzero classes. Since the other terms pw0 are p time*
 *s an

unstable class, they cannot cancel the leading term, and the tail terms on the *
 *second

representation must be an unstable boundary.   ||



  Our final preliminary is to introduce the spaces which are the factors in the*
 * de-

composition of [10] of the p-localization of SU(n) and its quotients as a produ*
 *ct of

p - 1 spaces.


Definition 2.15.If 1  N  p - 1, let Xij(N) denote the direct factor space of the

p-localization of the space SU(N + i(p - 1) + 1)=SU(N + j(p - 1)) which is buil*
 *t up

from fibrations involving p-local spheres S2N+1+kqfor j  k  i.


  The UNSS of Xij(N) has E2-term Exts;tU(P (BP*(Xij(N)))), where P (BP*(Xij(N)))

has BP*-basis {x2N+jq+1; x2N+(j+1)q+1; : :;:x2N+iq+1}, with coaction induced fr*
 *om

Lemma 2.4.  Because of the sparseness results for spheres given in (2.7) and Th*
 *e-

orem 2.8, the following is immediate.


Proposition 2.16.   (1) For s = 1 or 2, v-11ss2k+1-s(Xij(N))  Es;2k+12(Xij(N)),

      and is 0 unless k  N mod p - 1.

   (2)In the notation of Theorems 1.5 and 1.9,


                     v-11ss2k+1-s(SU(n))  v-11ss2k+1-s(Xi0(N)):


  Thus our efforts in the remainder of the paper will be to show that for s = 1*
 * or 2,

Es;2N+1+qm2(Xi0(N)) is as stated in Theorems 1.5 and 1.9. The fibrations whose *
 *exact

sequence in E2(-) will be studied are of the form


                       Xjk(N) ! Xik(N) ! Xij+1(N)


for k  j < i.



                  3. The cellular spectral sequence


  In this section, we sketch the proof of Theorem 1.5 by showing (modulo details

postponed until Section 4) that E1;2N+1+qm2(Xi0(N))  Z=pe, where e is as in that

theorem. Here Xi0(N) is the space introduced at the end of the previous section*
 *, and

all notation of Theorem 1.5 is in effect; in particular, N < p and i < p.



12                        D. DAVIS AND H. YANG


  Throughout the remainder of the paper, we consider N and m to be fixed, so th*
 *at

they may be omitted from some notation. The expression Es2(-) will always mean

Es;2N+1+qm2(-), and we let Xij= Xij(N). Note that Xjj= S2N+1+qj. Our goal is to

determine Es2(Xi0) for s = 1 and 2.

  We can organize this computation of Es2(Xi0) as a (cellular) spectral sequenc*
 *e with

Es;j1= Es2(Xjj) for 1  s  2 and 0  j  i, and dr : E1;jr! E2;j-rrinduced by pull*
 *ing

the class in E12(Xjj) back to E12(Xjj-r+1) and then applying -@!E22(Xj-rj-r). O*
 *ur desired

Es2(Xi0) is filtered with subquotients Es;j1, 0  j  i. For s = 1, we know E12(X*
 *i0) is

cyclic by Theorem 1.11, and so we just need to compute the sum

                                        Xi
(3.1)                    (|E12(Xi0)|) =    (|E1;j1|):
                                        j=0

Note that for 1  s  2
                        8
                        <Z=p                   ifj 6= ^m
                 Es;j1= :
                         Z=pmin(N+(p-1)j;(m-j)+1)ifj = ^m:


Thus every differential involves at least one group which is Z=p or 0, and so i*
 *t suffices

to tell which differentials are nonzero.

  The precise form of this spectral sequence falls into three different pattern*
 *s, deter-

mined by (1) ^m= 0, (2) N < ^m, and (3) N  ^m> 0. We begin by calculating the

cellular spectral sequence when ^m= 0, this being the easiest of the three case*
 *s. It is

easiest to describe the spectral sequence when i = p - 1, the maximal value tha*
 *t we

are considering, and then to obtain the spectral sequence for smaller values of*
 * i by

restriction.


Theorem 3.2.  If ^m= 0, then in the cellular spectral sequence for Xp-10,
                         8
                         >>>Z=pmin(N;(m)+1)s = 1; j = 0
                         >><
                          Z=p            s = 1; 1  j < N - (m)
           Es;j1= Es;j1= >
                         >>>Z=p          s = 1; N < j  p - 1
                         >:
                          Z=p            s = 2; 1  j  p - 1:

                                         dj  2;0                   2;0
If max (1; N - (m))  j  N, then Z=p  E1;jj-!Ej  is nonzero. Hence E1  = 0,

and E1;j1= 0 if max (1; N - (m))  j  N.


  If i < p - 1, then E1;j1(Xi0) will equal the group described in Theorem 3.2 i*
 *f j  i,

and will equal 0 if j > i. The same holds for E2;j1(Xi0) if j > 0 or i  N; if i*
 * < N,



              FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS            13


            Table 1. Spectral sequence when p = 11, N = 6, ^m= 0


                          |        j
                          ||                            |

                __________|0__1_2__3__4_5__6_7__8__9_10_|_
                 (m)     1 ||21  1 1  1  /1/1 1  1 1  1  ||

                         2 ||31  1 1  /1 /1/1 1  1 1  1  ||

                         3 ||41  1 /1 /1 /1/1 1  1 1  1  ||

                         4 ||51  /1/1 /1 /1/1 1  1 1  1  ||

                ________5_||6_/1_/1/1_/1_/1/1_1__1_1__1__||



then E2;01(Xi0) = Z=pmin(N-i;(m)+1). The reader can easily verify that the sum *
 *(3.1)

when applied to 3.2 yields the desired result

                              8
                              <i                   ifN  i
               (|E12(Xi0)|) = :
                               min(N; i + (m) + 1) ifi < N


for the case ^m= 0.

  The proof of Theorem 3.2 will be given in Section 4. In Table 1, we illustrat*
 *e the

spectral sequence in the case p = 11, N = 6. For each value of j from 0 to p - *
 *1, we

list


                     (|E1;j1|) = (|E1;13+20m2(S13+20j)|):


A slash (=) through a number indicates that it supports a differential. In this*
 * case

(m^= 0), all differentials hit into E2;0.



  Next we describe the cellular spectral sequence when N < ^m.



Proposition 3.3. Let  = (m - ^m). If N < ^m, then in the spectral sequence for

Xi0,

            8
            >>>Z=p                if0  j < N
            >>>
            >>>0                  ifj = N
            ><
             Z=p                  ifN < j < ^m
     E1;j1= >     min(m^(p-2)+N+2;+1)
            >>>Z=p                ifj = ^m
            >>>
            >>>Z=p                if^m< j  ^m(p - 1) + N + 1 -
            :0                    ifj > ^mand j > ^m(p - 1) + N + 1 - ;



provided j  i.



14                        D. DAVIS AND H. YANG


  It is then an easy exercise to verify that the sum (3.1) has the desired value
             8
             >><i + 1                           ifi < N


             >>:i                               ifN  i < ^m
              min(m^(p - 1) + N + 1; i + (m - ^m))if^m i:

Thus Theorem 1.5 in the case N < ^mwill be proved once we have proved Propositi*
 *on

3.3.

  In Table 2, we illustrate the spectral sequence in a particular case, namely *
 *p = 11,

N = 2, and ^m= 7. Let  = (m - ^m) = (m - 7). For each value of  and each j

from 0 to 10, we list

                      (|E1;j1|) = (|E1;5+20m2(S5+20j)|):

Directly beneath this number, we list a number k if there is a nonzero differen*
 *tial

E1;j! E2;k. Thus for example if  = 68, there are nonzero differentials


 E1;22! E2;02;   E1;74! E2;34;   E1;76! E2;16;   E1;84! E2;44;   E1;94! E2;54; *
 *  E1;104! E2;64:


In this case, we have        8
                             >><Z=p69if1  r  4

                       E1;7r >Z=p68  if5  r  6
                             >:  67
                              Z=p    if7  r  1:



  Proposition 3.3 is an immediate consequence of the following result, which de*
 *scribes

the nonzero differentials in the spectral sequence when N < ^m. In Section 4 we*
 * will

prove this result and its analogues in the other cases. We will frequently omit*
 * writing

the subscript of the differential and the E-groups. It just equals the differen*
 *ce of the

second superscripts.


Theorem 3.4.  If N < ^m, the nonzero differentials in the spectral sequence con*
 *verg-

ing to E2(Xi0) are those described below on groups E1;jwith j  i.

   (1)dN 6= 0 : E1;NN! E2;0N.

   (2)For t > 1, there is a nonzero differential from xm^(p-2)+N+1+tto zt, where
                8
                < element of order pk in E1;m^ ifk  (|E1;m^|)
          xk =  :               1;m^+     1            1   1;m^
                  generator of E1              if = k - (|E1  |) > 0
                8
                >><generator of E2;t-11        if1 < t  N

          zt =  > generator of E2;t1           ifN < t < ^m
                >:                  t-m^+1    2;m^
                  element of order p      in E1if^m t



  FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS            15



Table 2. Spectral sequence when p = 11, N = 2, ^m= 7


          |        j
          ||                                 |

          |0__1_2__3__4_5__6____7____8_9__10_|
      63 ||1  1 1  1  1 1  1   + 1   1 1  1  ||
          |     0                            |
          |__________________________________|
     = 64 ||1 1 1  1  1 1  1   65    1 1  1  ||
          |     0                         1  |
          |__________________________________|
        65 ||11 1  1  1 1  1   66    1 1  1  ||
          |     0                      1  3  |
          |__________________________________|
        66 ||11 1  1  1 1  1   67    1 1  1  ||
          |     0                    1 3  4  |
          |__________________________________|
        67 ||11 1  1  1 1  1   68    1 1  1  ||
          |     0               1    3 4  5  |
          |__________________________________|
        68 ||11 1  1  1 1  1   69    1 1  1  ||
          |     0              1,3   4 5  6  |
          |__________________________________|
        69 ||11 1  1  1 1  1   70    1 1  1  ||
          |     0             1,3,4  5 6  7  |
          |__________________________________|
        70 ||11 1  1  1 1  1   71    1 1  1  ||
          |     0            1,3,4,5 6 7  7  |
          |__________________________________|
      71 ||1  1 1  1  1 1  1   72    1 1  1  ||
          |     0             1,3-6  7 7  7  |
   _______|__________________________________|



16                        D. DAVIS AND H. YANG


We take a little liberty with notation here; although the elements xk and zt are

defined as elements of E1, the differentials generally involve the projections *
 *of these

elements in Er with r > 1.

  Next we present the analogues of Proposition 3.3 and Theorem 3.4 when N  ^m>

0. The condition

                               i j
(3.5)            m  ^m- (-1)m^^mN^mpm^(p-1)mod pm^(p-1)+1



will be important here.



Proposition 3.6. If N  ^m > 0, then the nonzero groups E1;j1in the spectral se-

quence converging to E12(Xi0) are the following groups which also satisfy j  i.*
 * Here

 = (m - ^m).

     8
     >>>Z=p              j < ^m
     >>>
     >>>Z=pmin(+1;N+m^(p-2))j = ^m
     >>>
     >>>Z=p                ^m(p - 1) and ^m< j < N
     ><
      Z=p                ^m(p - 1) <   ^m(p - 2) + N - 2

     >>>                    andm^ < j < ^m(p - 1) + N -
     >>>
     >>>Z=p              j = N + 1 and not (3:5)
     >>>
     >>>Z=p              j = N and (3:5)
     >:
      Z=p                 < ^m(p - 1) and N < j  ^m(p - 1) + N + 1 -



  As in the previous two cases, one can easily show that the sum (3.1) when app*
 *lied

to the groups of 3.6 yields the desired result, Theorem 1.5, when N  ^m> 0. Als*
 *o as

in the previous case, Proposition 3.6 is easily derived from a listing of the d*
 *ifferentials

in the cellular spectral sequence, which we now describe. The proof of the foll*
 *owing

result will be given in Section 4.



Theorem 3.7.  If N  m^ > 0, the nonzero differentials in the spectral sequence

converging to E2(Xi0) are those described below on groups E1;jwith j  i.


   (1)If (m - ^m) < ^m(p - 1), then dN 6= 0 : E1;NN! E2;0N.

   (2)If (3:5) is satisfied, then dN+1 6= 0 : E1;N+1N+1! E2;0N+1.

   (3)If t  1 or if t = 0 and (m - ^m)  ^m(p - 1) and (3:5) is not satisfied, t*
 *hen

      there is a nonzero differential from xm^(p-2)+N+1+tto zt, where



              FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS            17


            Table 3. Spectral sequence when p = 11, N = 7, ^m= 3


                              |     j
                              ||                               |

         _____________________|0_1__2___3___4__5__6_7__8_9__10_|
                   28        ||1  1 1   + 1  1 1  1  1 1  1  1  ||
                              |                      0         |
                              |________________________________|_
                  = 29        ||1 1 1   30   1 1  1  1 1  1  1  ||
                              |                      0       1  |
                              |_________________________________|
          m  3 + 6p30 mod p31 ||1 1 1   31   1 1  1  1 1  1  1  ||
                              |                        0  1  2  |
                              |_________________________________|
            = 30; m 6 3 + 6p30 ||11 1   31   1 1  1  1 1  1  1  ||
                              |                      0    1  2  |
                              |_________________________________|
                  = 31        ||1 1 1   32   1 1  1  1 1  1  1  ||
                              |                   0  1    2  3  |
                              |_________________________________|
                  = 32        ||1 1 1   33   1 1  1  1 1  1  1  ||
                              |                0  1  2    3  3  |
                              |_________________________________|
                  = 33        ||1 1 1   34   1 1  1  1 1  1  1  ||
                              |              0 1  2  3    3  3  |
                              |_________________________________|
                  = 34        ||1 1 1   35   1 1  1  1 1  1  1  ||
                              |         0    1 2  3  3    3  3  |
                              |_________________________________|
                  = 35        ||1 1 1   36   1 1  1  1 1  1  1  ||
                              |         0,1  2 3  3  3    3  3  |
                              |_________________________________|
                   36        ||1  1 1   37   1 1  1  1 1  1  1  ||
                              |        0,1,2 3 3  3  3    3  3  |
         _____________________|________________________________ |
         8
         >><elt of order pk in E1;m^1ifk  (|E1;m^1|)

  xk =   >gen of E1;m^+1          if = k - (|E1;m^1|) > 0 and ^m+   N
         >:       1;m^++1                     1;m^
          gen of E1               if = k - (|E1  |) > 0 and ^m+  > N
         8
         <gen ofE2;t              ift < ^m
  zt =   :       1                2;m^
          elt of order pt-m^+1in E1ift  ^m


  We close this section by illustrating 3.6 and 3.7 when p = 11, N = 7, and ^m=*
 * 3 in

Table 3. The conventions are as in Table 2. In particular, if in column j, a nu*
 *mber

k sits directly beneath a number e, then E1;j1 Z=pe and d 6= 0 : E1;j! E2;k.



 4.Determination of differentials in the cellular spectral sequence


  In this section, we determine the differentials in the cellular spectral sequ*
 *ence,

proving Theorems 3.2, 3.4, and 3.7, and thereby implying Theorem 1.5. We contin*
 *ue



18                        D. DAVIS AND H. YANG


with the notational conventions of Section 3. In particular, N and m are fixed,*
 * and

Es2(-) means Es;2N+1+qm2(-).

  We will also take a number of liberties with notation, which we now explain. *
 *First,

we omit v1-powers on the left of most terms. All of our terms have degree 2N +1*
 *+qm,

and so the number of v1's can be determined. This number of v1's does not affect

most formulas significantly, although it is important in applications of Lemma *
 *2.13.

Second, we frequently omit units in Z(p)which appear as coefficients in express*
 *ions.

There will be some cases, giving rise to (3.5), where a comparison of two terms*
 * will

be essential, and then we will of course keep track of coefficients, but usuall*
 *y we only

need to know whether or not they are zero mod p. Third, we write x  y  mod L to

mean that x - y desuspends to (or is defined on) lower spheres. "Lower than wha*
 *t"

should be clear from the context. Sometimes we will omit writing "mod L." Final*
 *ly,

we will often write yj for x2N+1+qjin BP*(Xik) and for 2N+1+qjin S2N+1+qj= Xjj.

  We begin with some lemmas, the first of which is an elementary fact involving

binomial coefficients.



Lemma 4.1.  If 0 < a < p, and 0  b; c < p, then

                               !!   8
                          ap + b    <1  ifb < c
                                  =
                            c       :0  ifb  c:


  The following lemma is a basic technique that will be used often.  We deal he*
 *re

and throughout this section with projection maps Xja! Xjj.



Lemma 4.2.  Suppose that (1) 0  a  j < p, (2) either ^m< a or ^m> j, and (3) if

a = 0, then j < N. Thinking of j and N as being fixed, let

                                8
                                <1  if`  N < j
(4.3)                     ffl`= :
                                 0  otherwise:


Then there is an element z 2 E12(Xja) satisfying


                            Xj
(4.4)                    z     hj+1-`-ffl`y` mod  L
                            `=a


which projects to a generator of E12(Xjj)  Z=p.  Hence dr = 0 : E1;jr! E2;j-rrf*
 *or

r  j - a.



              FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS            19


Proof.The proof is by downward induction on a, and is true for a = j by parts 1

and 2 of Theorem 2.8. Assume that it is true for a certain a satisfying a > N +*
 * 1 if

N < j, and also satisfying a - 1 > ^m. Let za denote the sum (4.4), which will *
 *have

all ffl`= 0. We will show below that


                          @ : E12(Xja) ! E22(Xa-1a-1)


satisfies

                       Xj i          j
              @(za)        N+(p-1)(a-1)`-a+1hj+1-` h`-a+1ya-1
                       `=a

(4.5)                  h  hj-a+1ya-1 d(hj-a+2ya-1)  mod L:


Thus za- hj-a+2ya-1is a cycle in E12(Xja-1), which extends the induction in thi*
 *s case

(a > N + 1). (Remember that we are not usually worrying about unit coefficients*
 *.)

  Now we prove the first  in (4.5).  By (2.6), Lemma 2.4, and Proposition 2.5,

@(hj+1-`y`) = hj+1-` Cya-1, where C is the component of
                                      i  j
                  (1 + h - hv + hv2 -  p-12h2v)N+(a-1)(p-1)


in degree q(` - a + 1). We will see that terms of degree greater than 3q in the*
 * sum of

2.5 would not play an important role here. The term that appears in the sum of *
 *(4.5)

is obtained from choosing (`-a+1) h's. If any other terms of 2.5 are chosen, th*
 *e term

will desuspend farther because of the v's on the right. For terms in the sum of*
 * 2.5

of degree greater than 3q, only pure powers of h1 could yield terms which desus*
 *pend

as far as the one already considered, and such terms were shown in Proposition *
 *2.5

not to exist. Note that a term such as h2 occurring in 2.5 desuspends no farthe*
 *r than

h1, but it uses up many more degrees. If Ly` is a term that desuspends lower th*
 *an

hj+1-`y`, then L is effectively htfor some t < j + 1 - `, and so @(Ly`) will de*
 *suspend

farther than hj+1-` h`-a+1.

  The second  of (4.5) is true because the unstable condition easily implies th*
 *at

other terms desuspend farther than the (` = j)-term. Also we need that the bino*
 *mial

coefficient is a unit. This follows from Lemma 4.1 and the hypothesis that we d*
 *o not

have a - 1  N < j. The third  is true by Lemma 2.13, in which the coefficient of

the first term is

            1_
(4.6)       q((2N + 1 + qm) - (2N + 1 + q(a - 1))) = m - (a - 1);

which is not a multiple of p, since a - 1 > ^m.



20                        D. DAVIS AND H. YANG

                                                      i          j
  The proof is similar when a - 1  N < j, except that  N+(p-1)(a-1)j-a+1is now p

times a unit, by Lemma 4.1. Thus the last terms of (4.5) become ph  hj-a+1ya-1

h  hj-aya-1 d(hj-a+1ya-1), where we have used ph = v - j(v) to "cancel" the ph,

and then argued as before.

  The condition that j + 1  N if a = 0 is required in order that all terms sati*
 *sfy the

unstable condition, Definition 2.2. The term hj+1-aya is closest to failing. It*
 * requires

j + 1 - a  N + a(p - 1), which is satisfied if a > 0 or j + 1  N.

  The statement about dr's being 0 follows from the definition of the spectral *
 *se-

quence.  If an element in E12(Xjj) pulls back to E12(Xja), then it survives to *
 *E1;jj-a.

||



  Lemma 4.2 must be modified as follows when a = ^m.



Lemma 4.7.  Define  = (m - ^m) and ffl` as in (4.3). Suppose ^m< j and


(4.8)                    j + 1 +  - fflm^ N + ^mp:


Then, under the projection map,

                                     Xj
                 hj+1-m^-ffl^m+ym^+      hj+1-`-ffl`y`2 E12(Xj^m)
                                    `=m^+1


maps to a generator of E2(Xjj)  Z=p. Hence dr = 0 : E1;jr! E1;j-rrfor r  j - ^m.



  The statement of this lemma initiates another abuse of notation which we will

allow ourselves. The class here is really only a cycle mod L. We mean here only*
 * the

sort of thing that was stated explicitly in Lemma 4.2.


Proof.The generator of E12(Xjj) pulls back to z 2 E12(Xj^m+1) as in Lemma 4.2, *
 *with

                     i       j
               @(z)   N+(p-1)m^j-m^h  hj-m^ym^ h  hj-m^-ffl^mym^:


When we try to apply Lemma 2.13 to write this as a boundary, the first coeffici*
 *ent

is m - ^m= sp , where s is a unit is Z(p). We don't worry about units, but the p

is missing from our term. To accommodate this, we note that, with e = j - ^m- f*
 *flm^

and t + 1 + e = m - ^m,


                  vth  he  vt- h  v he  vt- h  p he+



              FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS            21


mod L. Here the first  follows from the remarks after Theorem 2.8, and the sec-

ond follows from expanding p h  = (v - j(v)) , noting that terms with j(v)'s wi*
 *ll

desuspend farther than other terms. We obtain


               @(z)  p h  hj-m^-ffl^m+ym^ d(hj+1-m^-ffl^m+ym^);


which satisfies the unstable condition by (4.8).  The argument is completed as *
 *in

Lemma 4.2.   ||



  A similar lemma tells how classes in E1;m^can be pulled back.


Lemma 4.9.  Let ffl` = 1 if `  N < ^m, and ffl` = 0 otherwise. Suppose 0  a  ^m

and that t - ffla  N + a. Then hm^(p-2)+tym^2 E12(Xm^^m) pulls back, mod L, to *
 *a cycle
P ^m  `(p-2)+t-ffl`1  m^                   1;m^  1;m^-r
  `=ah        y`2 E2(Xa ). Also, dr = 0 : Er  ! Er    for r  ^m- a.


Proof.We assume that the formula is true for a (with class za), and will deduce*
 * it

for a - 1. We assume at first that it is not true that a - 1  N < ^m, so that a*
 *ll ffl's

are 0. We claim

             X^mi         j
    @(za)        N+(a-1)(p-1)`-a+1h`(p-2)+t h`-a+1ya-1
             `=a

(4.10)       ha(p-2)+t hya-1 h  h(a-1)(p-2)+t-1ya-1 d(h(a-1)(p-2)+tya-1):


Then za-1  za - h(a-1)(p-2)+tya-1 is a cycle, extending the induction.  The cla*
 *im

about dr = 0 is immediate, as in the proof of Lemma 4.2.

  The first  in (4.10) follows by the argument used for the first  of (4.5). The

second  says that the term with ` = a desuspends least far of all terms, and has

coefficient a unit mod p. The coefficient is a unit since a - 1 6= N. The measu*
 *re of

the size of the smallest sphere on which hi hj is defined is


                     exc(hi hj) = min(j; i - (p - 1)j):


For the terms in the sum of (4.10), this is largest when ` = a. The third  foll*
 *ows

from Lemma 2.12, and the fourth from Lemma 2.13, where the first coefficient is,

similarly to (4.6), m - (a - 1) 6 0  mod p.

  This completes the proof of the lemma as longias it isjnot true that a-1  N <*
 * ^m.

If a - 1 = N, then the binomial coefficient  N+(a-1)(p-1)1is p times a unit, an*
 *d this p

can be used to cancel one h, as in the proof of Lemma 4.2; this is accommodated*
 * by



22                        D. DAVIS AND H. YANG


ffl = 1. If a - 1 < N, then the t in the exponent has already been decreased to*
 * t - 1,

and subsequent steps will leave it there.

  The condition t-ffla  N +a is necessary in order that all terms satisfy the u*
 *nstable

condition.   ||



  Now we are ready to establish the differentials claimed in Theorems 3.2, 3.4,*
 * and

3.7.  The reader is encouraged to refer to Table 1 when considering this proof *
 *of

Theorem 3.2.

  Proof of Theorem 3.2. We emphasize that this is the case ^m= 0. Let 1  j  p-1,

 = (m), and ffl` be as in (4.3). By Lemma 4.2, a generator of E12(Xjj) pulls ba*
 *ck
       P j   j+1-`-ffl                 1;j   2;r-j
to z1    `=1h       y``, and dr = 0 : Er  ! Er    for r < j.  By Lemma 4.7,

dj = 0 : E1;jj! E2;0jif j < N - . If max(1; N - )  j  N, then, as in the proof *
 *of

Lemma 4.7, @(z1)  hhjy0, and by Theorem 2.8, this is an element of order pN+1-j

in E2;01= E22(S2N+1). Thus E2;0N+1= 0, completing the proof.   ||



  The reader is encouraged to refer to Table 2 in the proof of Theorem 3.4 which

follows.

  Proof of Theorem 3.4. If j < N, then dr = 0 on E1;jrfor all r by Lemma 4.2. A*
 *lso

by Lemma 4.2, we have E1;jj= E1;j1if N  j < ^m. We will show that


(4.11)                     dN 6= 0 : E1;NN! E2;0N:


Then E2;0N+1= 0 and hence dj is 0 on E1;jjwhen j > N, since there is nothing fo*
 *r it to

hit. To prove (4.11), we use Lemma 4.2 to pull the generator of E1;N1back to

                            NX
                        z1    hN+1-`y`2 E12(XN1);
                            `=1

which satisfies @(z1)  h  hN y0 by part of (4.5). This is nonzero by Lemmas 2.10

and 2.11.

  Next we consider the differentials from E1;m^.  We first show that elements x*
 * =

xm^(p-2)+N+1+tof order pm^(p-2)+N+1+twith N < t < ^msupport a nonzero differen-

tial to E2;t.  By Theorem 2.8(2), x is represented, mod L, by hm^(p-2)+N+1+tym^*
 * 2

E12(S2N+1+qm^). By Lemma 4.9, x can be extended, mod L, to

                          ^mX
                   xt+1=     h`(p-2)+N+1+ty`2 E12(Xm^t+1);
                         `=t+1



              FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS            23


which implies that differentials from x into E2;jwith j  t + 1 are 0. Then


(4.12) @(xt+1)  (N + (p - 1)t)h(t+1)(p-2)+N+1+t hyt6 0 2 E22(S2N+1+tq):


The first  here is obtained similarly to the first two steps of (4.10), while t*
 *he last

step is from Lemma 2.12.

  We have now completed the proof of the differential from xm^(p-2)+N+1+tto zt

asserted in Theorem 3.4 when t > N and ^m(p - 2) + N + 1 + t  (|E1;m^1|). If in*
 *stead

we have t  N, but still ^m(p - 2) + N + 1 + t  (|E1;m^1|), then in Lemma 4.9 ff*
 *l`= 1

for `  N, and so x extends to

                  X^m                    XN
              x0      h`(p-2)+N+1+ty`+        h`(p-2)+N+ty`:
                 `=N+1                 `=max(t;1)


If t  1, this shows that dr(x) = 0 for r < ^m, and hence x is a permanent cycle,

since d2;0^m= 0. If t > 1, then, by part of (4.10), @(x0)  ht(p-2)+N+t hyt-1, w*
 *hich is

nonzero by Lemma 2.12.

  This completes the analysis of the differentials of Theorem 3.4 on E1;jrwith *
 *j  ^m.

Now we will establish the differential on x = xm^(p-2)+N+1+twhen


(4.13)                ^m(p - 2) + N + 1 + t > (|E1;m^1|):


Let e = (|E1;m^1|). Then x  hym^(p-1)+N+1+t-e, and can be extended by Lemma 4.2

to


(4.14)        x0 hym^(p-1)+N+1+t-e+ . .+.hm^(p-2)+N+1+t-eym^+1;


with


(4.15)                 @(x0)  h  hm^(p-2)+N+1+t-eym^:


By Corollary 2.9, this equals the element of order pt-m^+1on S2N+1+qm^, establi*
 *shing

the claimed differential to ztwhen t  ^m(so that the order is greater than 1).

  Now we assume t < ^m. Let  = (m - ^m). Recall that


                  e = (|E1;m^1|) = min( + 1; N + ^m(p - 1)):


Combining t < ^mwith (4.13) yields e < N + ^m(p - 1), and hence e =  + 1. By

Lemma 4.7, the class x0of (4.14) extends to


       x00 hym^(p-1)+N+t-+ . .+.hm^(p-2)+N+t-ym^+1+ hm^(p-2)+N+t+1ym^:



24                        D. DAVIS AND H. YANG


By Lemma 4.9, x00can be extended to x000satisfying
            8
            <x00+ Pm^-1 h`(p-2)+N+t+1y`                  ift > N
       x000 :     P`=t+1^m-1              PN
             x00+  `=N+1 h`(p-2)+N+t+1y`+  `=th`(p-2)+N+ty`ift  N:


Then                        8
                            <h(t+1)(p-1)+N hyt ift > N
                   @(x000)  :
                             ht(p-1)+N hyt-1   ift  N:

These are nonzero by Lemma 2.12, establishing the final differentials of Theore*
 *m 3.4,

namely those from the second type of xk to the first two types of zt.   ||



  The proof of Theorem 3.7 proceeds similarly, except that in a certain case we

must keep track of unit coefficients because two terms are trying to have cance*
 *lling

differentials. It is recommended that the reader consult with Table 3 while stu*
 *dying

this proof.

Proof of Theorem 3.7. We divide into cases determined by the value of j, where *
 *the

differential emanates from E1;j.

  Case 1: j  ^m. If j < ^m, then all differentials on E1;jare 0 by Lemma 4.2. F*
 *or

j = ^m, we wish to show the differential asserted in 3.7 from xm^(p-2)+N+1+tto *
 *the

generator of E2;t1. Here we must have t < ^min order that E1;m^1has an element *
 *of the

asserted order. By Lemma 4.9 and (4.10), this xm^(p-2)+N+1+tpulls back to


              x0 hm^(p-2)+N+1+tym^+ . .+.h(t+1)(p-2)+N+1+tyt+1;


which has @(x0)  hN+(t+1)(p-1) hyt6 0, by Lemma 2.12.

  Case 2:  < ^m(p - 1) and ^m< j  N. Here and throughout the remainder of

this proof,  = (m - ^m). The claim here is that E1;j1consists of permanent cycl*
 *es if

j < N, while dN 6= 0 : E1;NN! E2;0N. To establish this, we begin by using Lemma*
 * 4.7

to pull the generator of E1;j1back to


(4.16)            x0  hyj+ . .+.hj-m^ym^+1+ hj-m^++1ym^:


Let k be the integer satisfying k(p - 1)   < (k + 1)(p - 1). Note that k satisf*
 *ies

0  k < ^m. Next we show that x0 pulls back to a cycle x0in E12(Xj1) of the form

                     ^m-1X                       ^m-k-1X
(4.17)       x0 x0+      hj-m^++1-(m^-`)(p-2)y`+      hj+1-`y`:
                    `=m^-k                        `=1

(Actually, as will be discussed in the third bullet below, it is possible that *
 *one of the

terms in this sum is incorrect, but this turns out to be inconsequential.)



              FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS            25


  To obtain (4.17), we note that if x0 has been pulled back to x002 E12(Xj`+1) *
 *with

` + 1  ^m- k, with terms like those in (4.16) and the first sum of (4.17), then*
 * @(x00)

can be evaluated by applying (4.5) to the terms of (4.16), and (4.10) to the te*
 *rms of

(4.17). The leading terms will be hhj-`y`and hhj-m^+-(m^-`)(p-2)y`, respectivel*
 *y.

Now there are three possibilities.


    o If `  ^m- k and it is not the case that ` = ^m- k and  = k(p - 1), then

      j - ^m+  - (m^- `)(p - 2) > j - `, and so the leading term of @(x00) will*
 * be

      the second of those listed above. Thus, mod L, the extension of x00to E12*
 *(Xj`)

      is obtained by extending the first sum of (4.17).

    o If ` = ^m- k - 1, then the leading term of @(x00) will be the first of th*
 *ose listed

      above.  Thus, mod L, the extension of x00to E12(Xj`) contains the top term

      in the last sum of (4.17). From this point on, subsequent extensions will*
 * be

      determined as they were in (4.5), by the very top term, hyj.

    o If  = k(p-1) and ` = ^m-k, these two leading terms will be equal. Actuall*
 *y,

      they may have different units as coefficients. If, when the units are tak*
 *en into

      account, these terms do not sum to 0, then the extension of x00to E12(Xj`*
 *) is

      obtained by extending the first sum of (4.17), just as it was in the firs*
 *t case.

      If, when the units are taken into account, these terms sum to 0 mod p, th*
 *en

      the powers of p in this coefficient can be used to cancel some of the h's*
 * in the

      ym^-k-term, rendering the expression (4.17) incorrect in this term. Howev*
 *er, at

      the next step, i.e., in going to ym^-k-1, and in subsequent steps, the ve*
 *ry top

      term will provide the leading term, just as it did in the second bullet a*
 *bove,

      and so this ambiguity in the ym^-k-term will be inconsequential.


  If x0is as in (4.17), then the leading term in @(x0) will be h  hjy0 2 E12(S2*
 *N+1).

This is a consequence of the assumption that  < ^m(p - 1), using the same sort *
 *of

excess comparisons that occurred in the three possibilities above. This @(x0) i*
 *s 0 if

j < N, since h  hj2N+1 desuspends. Here and elsewhere, we use freely the fact t*
 *hat

when E22(S2N+1)  Z=p, then elements in it which desuspend are 0. If j = N, then

h  hN 2N+1 6= 0 by Lemmas 2.10 and 2.11. This completes the proof of this case.

  Case 3:  = ^m(p - 1) and j < N. Here E1;j1consists of permanent cycles by the

argument just completed. Indeed, in this case, the first sum of (4.17) extends *
 *down

to ` = 0 with all terms satisfying the unstable condition.

  Case 4:  = ^m(p - 1) and j = N. This is the delicate case, requiring that we



26                        D. DAVIS AND H. YANG


keep track of units. We will show that, under these hypotheses, dN : E1;NN! E2;*
 *0Nis

zero if and only if (3.5) is satisfied.

  As in (4.17), a generator of E1;N1pulls back to



(4.18)     hyN + . .+.hN-m^ym^+1+ hN+m^(p-2)+1ym^+ . .+.hN+p-1y1;



but now both the first and last terms hit a nonzero element of E22(X00) when @ *
 *is

applied. We will show that if the coefficient of hyN is chosen to be 1, and if *
 *m - ^m=

sp , with s 6 0  mod p, then, for 0  k < ^m, the coefficient of hN+(m^-k)(p-2)+*
 *1ym^-k

in (4.18) is, mod p,

                                              !
                                  1  N - ^m+ k
(4.19)                  uk = (-1)k_            :
                                  s      k



Then E12(XN1)-@!E22(X00) satisfies, mod L,

                                               i  j
(4.20)   @(hyN + um^-1hN+p-1y1)   (h  hN + um^-1N1 hN+p-1  h)y0


                                  (1 - um^-1N)h  hN 2N+1:



We have used Lemma 2.12(2) at the last step.

  When @ is applied to (4.18), modified so as to properly incorporate unit coef*
 *ficients

on all terms, the intermediate terms desuspend farther than the end terms, and *
 *the

desired dN is determined by (4.20). Incorporating (4.19), we find that the imag*
 *e of

this dN is

                                      i   j
                        1 + (-1)m^1_sN N-1^m-12 Z=p;



and this is 0 if and only if (3.5), as claimed.

  It remains to prove (4.19), which we do by induction on k.  For k = 0, we note

that, similarly to the proof of Lemma 4.7, for any numbers c`,


           N-m^X             i       j
   @(hyN +     c`h`yN+1-`)    N+m^(p-1)N-m^h  hN-m^ym^
           `=2

                             1_s(m - ^m)h  hN-m^+ ym^ -1_sd(hN-m^++1 ym^);



and so adding 1_shN-m^++1 ym^extends the cycle, with u0 = 1_s.



              FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS            27


  Assume now that uk satisfies (4.19). Let z = hyN + . .+.ukhN+(m^-k)(p-2)+1ym^*
 *-k.

Then, mod L, we have

                   i            j
        @(z)     uk N+(m^-k-1)(p-1)1hN+(m^-k)(p-2)+1 hym^-k-1


                 -uk(N - ^m+ k + 1)h  hN+(m^-k-1)(p-2)ym^-k-1


                 ___1____m-m^+k+1uk(N - ^m+ k + 1)d(hN+(m^-k-1)(p-2)+1ym^-k-1)
                       i       j
                 (-1)k1_sN-m^+k+1k+1d(hN+(m^-k-1)(p-2)+1ym^-k-1):

                    i       j
Thus adding (-1)k+11_sN-m^+k+1k+1hN+(m^-k-1)(p-2)+1ym^-k-1to z yields a cycle m*
 *od L

in E12(XN^m-k-1) which projects to z. The coefficient here is just what we have*
 * claimed

is uk+1, and so this extends the induction.

  Case 5:  > ^m(p - 1) and ^m< j  N. These are the second type of elements

xk in Theorem 3.7. As before, let e = (|E1;m^1|) = min( + 1; N + ^m(p - 1)). Th*
 *en,

in the notation of Theorem 3.7, our j becomes ^m(p - 1) + N + 1 + t - e. We show

in the next paragraph that if t  ^m, then the differential from E1;jhits the el*
 *ement

of order pt-m^+1in E2;m^1. Of course, this hit element should really be viewed *
 *as the

element of order p in E2;m^j-m^, elements of order less than pt-m^+1already hav*
 *ing been

hit by shorter differentials.

  By the first sentence of the proof of Lemma 4.7, a generator of E12(Xjj) pull*
 *s back

to an element x 2 E12(Xj^m+1) satisfying @(x)  h  hj-m^ym^.  This latter element

has order pt-m^+1in E22(S2N+1+qm^) by Corollary 2.9.  Indeed, the equation of t*
 *hat

corollary becomes


       (m^(p - 1) + N + 1 + t - e - ^m) + e = (t - ^m+ 1) + (N + ^m(p - 1)):



  If, on the other hand, t < ^m, then by Lemma 4.7 the generator of E1;jpulls b*
 *ack

to a cycle
                 Xj
             x0      hj+1-`y`+ hm^(p-2)+N+t-e++2ym^2 E12(Xj^m):
                `=m^+1

We can simplify this by noting that the hypotheses imply e =  + 1. (If not, then

e = ^m(p - 1) + N, so j = t + 1, contradicting j > ^m> t.) Now by Lemma 4.9 we

can pull x0back to

                          ^m-1X
                  x00 x0+     h`(p-2)+N+t+1y`2 E12(Xjt+1):
                          `=t+1



28                        D. DAVIS AND H. YANG


As in (4.10), @(x00)  h(t+1)(p-1)+N hyt, which is nonzero by Lemma 2.12.  We

observe that @(hyj) cannot interfere here, since this yields a term which desus*
 *pends

unless j = N and t = 0, and this is impossible since, if t = 0,


              j = ^m(p - 1) + N + 1 - e = ^m(p - 1) + N -  < N:


  Case 6: j > N. In the notation of the theorem, we have


      j =  m^+  + 1 = ^m+ k - e + 1


(4.21)  =  m^+ (m^(p - 2) + N + 1 + t) - e + 1 = ^m(p - 1) + N + 2 + t - e:


The generator of E12(Xjj) can be pulled back as in Lemma 4.2 to a class x 2 E12*
 *(Xj^m+1)

satisfying @(x)  h  hj-m^-1ym^, similarly to part of (4.5) except that ffl = 1.*
 * Using

Corollary 2.9 and (4.21), we obtain that the order of hhj-m^-12N+1+qm^is pt-m^+*
 *1, as

claimed, provided t  ^m. Note that if j = N +1, then the class which this diffe*
 *rential

would like to hit has already been killed by a differential from E1;N.

  If t < ^m, then by Lemma 4.7 a generator of E12(Xjj) pulls back to


              x  hyj+ . .+.hj-m^-1ym^+1+ hj-m^+ym^2 E12(Xj^m):


By Lemma 4.9, x can be extended farther, to

                       ^m-1X
                x0 x +     h`(p-2)+N+2+t-e+y`2 E12(Xjt+1);
                       `=t+1


with @(x0)  h(t+1)(p-1)+N hyt 6= 0 by Lemma 2.12.  We must worry here about

possible cancellation from @(hyj)  ch  hj-tyt. This will desuspend, and hence be

0, if t > 0. Thus it remains to consider the possible cancellation when t = 0.

  If  > ^m(p - 1), then t must be greater than 0. This can be deduced using j >*
 * N,

(4.21), and e = min( + 1; N + ^m(p - 1)).

  If   ^m(p - 1) and (3.5) is not satisfied, then it was proved in Cases 2 and 4

that E2;0N+1= 0, and so we need not worry about the case t = 0 here.  Finally, *
 *we

must establish that dN+1 6= 0 : E1;N+1N+1! E2;0N+1if (3.5) is satisfied. To pro*
 *ve this by

the methods employed so far in this section is more delicate than we care to pr*
 *esent.

Instead, we deduce it using differentials already determined, together with res*
 *ults for

cyclicity of certain E2-groups which will be established in the next section.

  So, we are assuming (3.5). In particular,  = ^m(p - 1) and e =  + 1. Earlier *
 *in

Case 6, we verified that under this hypothesis dN+1 6= 0 : E1;N+2N+1! E2;1N+1. *
 *This says



              FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS            29


that the composite ae O @ 6= 0 in (4.22), and that | coker(i2)| = p.

                                                           ae 2  1
(4.22)     E12(XN2)-i1!E12(XN+12)-i2!E12(XN+22)-@!E22(X10)-! E2(X1)


By Proposition 5.2, ae is a surjection from Z=p2 to Z=p.  Thus @ is surjective,*
 * and

hence @ O i2 6= 0, hitting E22(X00)  E22(X10). On the other hand, the 0-differe*
 *ntial

from E1;NNto E2;0Nestablished in Case 4 implies that the composite @ O i2O i1 i*
 *n (4.22)

is 0. Thus @ O i2 is nonzero on an element which projects nontrivially to E12(X*
 *N+1N+1),

and this establishes dN+1 6= 0 : E1;N+1N+1! E2;0N+1, as desired.   ||



                          5.Group structure


  In this section, we prove Theorem 1.9; we determine the group structure on the

2-line of the UNSS of SU(n) for n  p2 - p + 1.  We also prove a few subsidiary

results about group structure of 1-line groups E12(Xba).

  The conventions of the previous sections are still in force. Thus p, N, and m*
 * are

fixed, with 1  N < p.  The second superscript of all E2-groups is 2N + 1 + qm

(unless specifically stated to the contrary), and ^m is the least nonnegative r*
 *esidue

of m mod p. Also, Xbais a direct factor of a complex Stiefel manifold (quotient*
 * of

SU(n)'s) built from spheres 2N +1+qj for a  j  b. Also, we continue the practice

of omitting v1's on the left and units in Z(p)whenever they are unimportant.

  Although Theorem 1.11 implies that E12(Xb0) is cyclic, it will not always be *
 *the

case that E12(Xba) is cyclic. The following result presents one case in which E*
 *12(Xba)

will be cyclic.


Proposition 5.1. The group E12(Xba) is cyclic if there are no numbers j congrue*
 *nt

to N mod p satisfying a  j < b.


Proof.The proof is by induction on b. The result is true when b = a by Theorem

2.8(1).  We assume it is true for Xba, and will deduce it for Xb+1a, provided b*
 * 6 N

mod  p. We consider the commutative diagram of exact sequences below.
                         j*  1  b+1   ae* 1  b+1    @   2  b
           0 !  E12(Xba)-!? E2(Xa? ) -!  E2(Xb+1)? -!  E2(Xa)
                  ??           ?             ?
                  y f*         ?yg*          ?y=
                        j1*  1  b+1       1  b+1
           0 !  E12(Xbb)-!  E2(Xb  ) -!  E2(Xb+1)


If j* is surjective, then E12(Xb+1a) is cyclic by the induction hypothesis, and*
 * so we are

done. Thus we may assume that there is an element fi such that ae*(fi) = ff, wh*
 *ere



30                        D. DAVIS AND H. YANG


ff has order p. Since b 6 N  mod p, the attaching map in Xb+1bis nontrivial, an*
 *d so

[5, 2.23] implies that there is a generator fl of E12(Xbb) such that j1*(fl) = *
 *p . g*(fi).

Exactness implies that there is y such that j*(y) = pfi.  Then j1*(f*(y) - fl) *
 *= 0,

so that f*(y) = fl, and hence y is a generator of the cyclic group E12(Xba). Th*
 *us the

extension is cyclic.   ||



  Next we prove a result that implies the first case of Theorem 1.9.


Proposition 5.2. The group E22(Xba) is cyclic if either b  N or N < a  b < p.


Proof.The proof is by induction on b and is true for b = a by 2.8(1). We assume*
 * the

result for Xba, and consider the exact sequence

                                          ae* 2  b+1  @
                   E22(Xba) ! E22(Xb+1a) -!  E2(Xb+1)-! 0;


with b 6= N. If


(5.3)                   E22(Xb+1b+1)  Z=pN+(b+1)(p-1);


then by Theorems 3.4 and 3.7, E22(Xba) = 0, and so we are done in this case, si*
 *nce ae*

will be iso. The easiest way to believe this claim that E22(Xba) = 0 is to look*
 * at the

last row of Tables 2 and 3. In the first one, it is the assertion that E1;7kill*
 *s E2;jfor

3  j  6, and in the second one, that E1;3kills E2;jfor 0  j  2.

  Thus we may assume that (5.3) is not true, and so d(hN+(b+1)(p-1)+1)yb+1has o*
 *rder

p in E22(Xb+1b+1), by Theorem 2.8(4). Since @ annihilates this class, there is *
 *w 2 E21(Xba)

such that z = d(hN+(b+1)(p-1)+1)yb+1- w is a cycle in E22(Xb+1a). We wish to sh*
 *ow

that pz is the image of a generator of E22(Xba).

  We have


(5.4)                    d(hty) = d(ht)y + htd(y);


where d(ht) is as in Lemma 2.3(3). Note here that the minus sign which is attac*
 *hed

to  (fl1) in (2.1) has been incorporated into d(ht) (which is really d(vsht)) i*
 *n Lemmas

2.3(3) and 2.13. Now, mod L,


       pz  d(hN+(b+1)(p-1)yb+1) - (N + b(p - 1))hN+(b+1)(p-1) hyb- pw:


Here we have used ph = v - j(v) and (5.4). The first term is a boundary, and so*
 * is

ignored. Note that the exponent of h had to be brought down to N + (b + 1)(p - *
 *1)

in order that it be placed in front of yb+1. Since b 6= N, the coefficient of t*
 *he second



              FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS            31


term is a unit, and so will be ignored. Since w was defined on Xba, pw desuspen*
 *ds

below SN+b(p-1), and so may be incorporated into L. Thus, by Lemma 2.12(2), pz

generates E22(Xbb) mod L, and so pz is the image of a generator of E22(Xba), as*
 * desired.

||



  Next we prove the second part of Theorem 1.9, which we restate as follows.


Theorem 5.5.  Suppose i > N and ^m> 0. If e is as in Theorem 1.5, then E22(Xi0)

Z=pe-1 Z=p.


Proof.In the exact sequence

                                    OE 2  i  ae 2  i
(5.6)         E12(XiN+1)-@!E22(XN0)-! E2(X0)-! E2(XN+1) ! 0;


the groups E22(XN0) and E22(XiN+1)  Z=pa are cyclic by Proposition 5.2.  We will

show that if x 2 E22(Xi0) is such that ae(x) is a generator, then pax = pOE(g),*
 * with g

a generator of E22(XN0). The result is immediate from this, with x and OE(g) - *
 *pa-1x

generating the summands.

  Case 1:  N = ^m. The thing that distinguishes this case is that pg lives on t*
 *he

top cell of XN0, i.e., E22(XN0) ! E22(XNN) sends pg nontrivially. Thus the exte*
 *nsion

question can be studied in the exact sequence


                 0 ! E22(XNN) ! E22(XiN) ! E22(XiN+1) ! 0:


We let x also equal the image of a generator of E22(Xi0) in E22(XiN). As in the*
 * proof

of 5.2, we can write


              pa-2x = d(hN+(N+2)(p-1)+1)yN+2 + w0yN+1 + w1yN ;


with each term unstable. Now we can use (5.4), (2.6), 2.4, and 2.5 to write


 pa-1x =  d(hN+(N+2)(p-1)yN+2) + pw0yN+1 + pw1yN
                         i                                     i  j     j
          -hN+(N+2)(p-1)  (N + (N + 1)(p - 1))hyN+1 - NphvyN +  Np2h2yN  :


We ignore the boundary term and apply Corollary 2.14 to the terms on yN+1. We

also write hv = vh - ph2 and ignore the p2w0yN term (with w0yN defined) which t*
 *his

yields. We obtain


pa-1x  =  -(N+1)p-1_m-N-1d(hN+(N+1)(p-1)+1+ w)yN+1 - NphN+(N+2)(p-1) (vh + uh2)*
 *yN


          +pw1yN ;



32                        D. DAVIS AND H. YANG


where wyN+1 is defined, and u is a unit. We multiply by another p and ignore te*
 *rms

which are p2 times an unstable class on yN . We obtain


    pax     u0d(hN+(N+1)(p-1)+ pw)yN+1
              i                                                        j
            u0 d((hN+(N+1)(p-1)+ pw)yN+1) - Np(hN+(N+1)(p-1)+ pw)  hyN


with u0a unit. Here we have applied (5.4) again. By Lemma 2.12, this is p times*
 * a

generator of E22(XNN).

  Case 2: N > ^mor N < i < ^mor (N < ^m i and i+(m-m^) < ^m(p-1)+2N).

In these cases, E2;j1= E2;j1for j  N - 1, and so in the short exact sequence


(5.7)            0 ! coker(@) ! E22(Xi0) ! E22(XiN+1) ! 0;


the generator g of coker(@) lives on yN , and pg lives on yN-1. Hence (5.7) map*
 *s to a

short exact sequence

                               OE02  i       2  i
                0 ! E22(XNN-1)-! E2(XN-1) ! E2(XN+1) ! 0;


and it suffices to show OE0(pg) = pax in this latter sequence.

  The analysis is quite similar to Case 1. We write


          pa-2x = d(hN+(N+2)(p-1)+1)yN+2 + w0yN+1 + w1yN + w2yN-1


with each term unstable. Then, ignoring terms which are p times an unstable cla*
 *ss

on yN-1, and letting u, u0, etc., denote units in Z(p), we obtain


    pa-1x     d(hN+(N+2)(p-1)yN+2) + pw0yN+1 + pw1yN
                             i                               i  j
              -hN+(N+2)(p-1)  ((N + 1)p - 1)hyN+1 - NphvyN +  Np2h2yN
                                                j
                             +(uhv2 + u0h2v)yN-1 :


Now we ignore boundaries and use 2.14, 2.3(1), and 2.3(2) to obtain


     pa-1x     -(N+1)p-1_m-N-1d(hN+(N+1)(p-1)+1+ w)yN+1


               +NphN+(N+1)(p-1) hyN + pw1yN + u00phN+(N+2)(p-1) h2yN


               -hN+(N+2)(p-1) (uhv2 + u0h2v)yN-1:


Now we multiply by p again, again omit p times unstable classes on yN-1, and in*
 *cor-

porate the two surrounding terms into w1, obtaining


                pax  u000d(hN+(N+1)(p-1)+ pw)yN+1 + p2w01yN :



              FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS            33


Now we apply (5.4) again, and omit writing the d(hy)-term, obtaining

                                       i

     pax     -u000(hN+(N+1)(p-1)+ pw)   NphyN - ((N - 1)p + 1)hvyN-1


                      i       j       j!
                     + (N-1)p+12h2yN-1   + p2w01yN :



We omit lots of terms which are p times an unstable class on yN-1.  We omit the

-u000-factor, which can be considered as a factor of the entire expression.  We*
 * also

combine together several terms that are p2 times an unstable class on yN . Fina*
 *lly,

we apply Lemma 2.3(1) to hv. This yields



        pax  Np(hN+(N+1)(p-1) h + pw001)yN - hN+(N+1)(p-1) vhyN-1:



Now we apply 2.14 to the first term and 2.3(2) to the second term, which effect*
 *ively

cancels hp-1v. This yields



           pax     _N__m-Npd(hN+N(p-1)+1+ w3)yN - hN+N(p-1) hyN-1


                   _N__m-Nd(hN+N(p-1)+ pw3)yN - hN+N(p-1) hyN-1;



with w3yN defined. We apply (5.4) once again, and omit writing the term of the *
 *form

d(hy), obtaining



  pax     -__N_m-N((N - 1)p + 1)(hN+N(p-1)+ pw3)  hyN-1 - hN+N(p-1) hyN-1


          -(__N_m-N+ 1)hN+N(p-1) hyN-1:



Since __N_m-N+ 1 = __m_m-N, this is nonzero by Lemma 2.12 since m 6 0  mod p by

assumption.

  Case 3: N < ^m i and i+(m-m^)  ^m(p-1)+2N. In this case, Theorem 3.7

(with Table 2 again recommended to provide insight) implies (1) E2;j1= 0 for j *
 *< N,

and hence E22(XN0)= im(@)  Z=p, and (2) E22(XiN+1) ! E22(Xi+1N+1) is multiplica*
 *tion

by p of groups each isomorphic to Z=pa. To see (2), note first that the generat*
 *or of

E22(Xi+1N+1) lives on yi+1, and second that E12(Xi+1i+1)-@!E22(XiN+1) hits the *
 *elements of

order p. If i = p - 1, then we need to extend Theorem 3.7 to include the case i*
 * = p.

This extension is straightforward; it is only Theorem 3.2 that must be modified*
 * when

i = p.



34                        D. DAVIS AND H. YANG


  Now the splitting follows from the following commutative diagram of exact se-

quences.

             0 ! E22(XN0)=?im(@) ! E22(Xi0)?! E22(XiN+1)?! 0
                      ??              ?            ?
                      y=              ?yOE         ?y.p

             0 ! E22(XN0)= im(@) !E22(Xi+10)! E22(Xi+1N+1) ! 0

Indeed, suppose to the contrary that E22(Xi0)  Z=pa+1 with generator G.  Then

the left square implies paOE(G) 6= 0, and so we must have E22(Xi+10)  Z=pa+1 wi*
 *th

generator OE(G).  Then the composition around the bottom of the right square is

surjective, but the composite around the top is not surjective.   ||



  Finally, we prove the third and fourth cases of Theorem 1.9, which we restate*
 * as

follows.  Again, we remind the reader that the conventions which were restated *
 *at

the beginning of this section continue to be in effect.


Theorem 5.8.  Suppose ^m= 0, i > N, and t = min(N; (m) + 1). Then
                                       8
                                       <Z=pt Z=pi-t    ift  i - N
         E22(Xi0)  E22(Xi1)  E12(Xi1)  :
                                        Z=pN  Z=pi-N   ift  i - N


Proof.The first isomorphism is immediate from E2;01= 0 in Theorem 3.2, and is

clearly depicted in Table 1. The second and third isomorphisms could be general*
 *ized

to interesting results regarding isomorphisms of 1- and 2-line groups, and eval*
 *uation of

noncyclic 1-line groups of complex Stiefel manifolds, but, in the interest of e*
 *xpediency,

we will just deal with the case at hand.

  The middle isomorphism of 5.8 is the composite

                            OE 2;2N+1+qm+qL i    2;2N+1+qm i
           E1;2N+1+qm2(Xi1)-! E2          (X1)  E2       (X1);


with (L) sufficiently large. The second isomorphism here is due to the periodic*
 *ity

of the groups E2;*2(Xi1), while OE is defined by


                         OE(x) = ffL=(N+m(p-1)) x;


where ffL=eis as in Theorem 2.8. We need (L)  N + m(p - 1) - 1. Note that OE(x)

is defined unstably since ffL=eis defined on S2e+1.

  We will show that OE is injective, from which its bijectivity will follow sin*
 *ce |E22(Xi1)| =

|E12(Xi1)|. If x 6= 0 2 E12(Xi1), choose the largest integer j such that the pr*
 *ojection

map ae sends x to a nonzero element ae*(x) of E12(Xij). Then there is a nonzero*
 * element



              FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS            35


x1 2 E12(Xjj) such that g*(x1) = ae*(x), where g : Xjj,! Xij. Let ff = ffL;N+m(*
 *p-1).

Then g*(ff  x1) = ae*(ff  x). By Lemma 5.10 below, ff  x1 6= 0. By Theorem 3.2,


(5.9)            g* : E2;2N+1+qm+qL2(Xjj) ! E2;2N+1+qm+qL2(Xij)


is injective, and hence ae*(ff  x) = g*(ff  x1) 6= 0. Thus ff  x 6= 0, establis*
 *hing the

desired injectivity. The injectivity of (5.9) is the crucial hypothesis in gene*
 *ralizing

this result.  The addition of qL to the second superscript doesn't matter here *
 *by

periodicity.  The injectivity in (5.9) is the statement that E2;j1= E2;j1for j *
 *> 0 in

Theorem 3.2, or that in Table 1 the only elements hit by differentials have j =*
 * 0.

  Now we prove the last isomorphism of (5.8). We use the following commutative

diagram of short exact sequences.

                 0 ! E12(XN0)?! E12(Xi0)!?E12(XiN+1)?! 0
                       ?? 0        ?           ?
                       yOE         ?yOE        ?y=

                0 ! E12(XN1)-!  E12(Xi1)! E12(XiN+1) ! 0

By Theorem 3.2, illustrated in Table 1, the diagram is

                    0 ! Z=pN?!   Z=pi? ! Z=pi-N?! 0
                         ?? t      ?         ?
                         y.p       ?yOE      ?y=

                    0 ! Z=pN -!   E    ! Z=pi-N ! 0:


The key observation here is that OE0= .pt, which we now explain. By Theorem 3.2,

there are no differentials in the cellular spectral sequence for XN1, but in th*
 *e spectral

sequence for XN0, there are differentials from E1;jto E2;0for N - t + 1  j  N.

Thus the generator of E12(Xi0) sits on yN-t, which is where pttimes the generat*
 *or of

E12(Xi1) lives.

  Now it is a matter of simple algebra to determine the structure of the abelian

group E = E12(Xi1) in the diagram above. Let g denote a generator of E12(Xi0), *
 *and

let h denote a generator of E12(XN1). Then pi-NOE(g) = pt(h). If t = N, this im*
 *plies

pi-NOE(g) = 0, and so E  Z=pN  Z=pi-N with generators (h) and OE(g). If t < N

and t  i - N, then E  Z=pt Z=pi-twith generators (h) - pi-N-tOE(g) and

OE(g). If t < N and t  i - N, then E  Z=pN  Z=pi-N with generators (h) and

OE(g) - pt-i+N(h).   ||



  We close with a lemma which was used in the preceding proof. Previous convent*
 *ions

for N and ^mapply.



36                        D. DAVIS AND H. YANG


Lemma 5.10.   Suppose (L)  N + m(p - 1) - 1, ^m= 0, and 1  j < p. Then


        ffL;N+m(p-1) : E1;2N+1+qm2(S2N+1+qj) ! E2;2N+1+qm+qL2(S2N+1+qj)


is bijective.


Proof.Since both groups have order p, it suffices to show ffL=(N+m(p-1))ffm-j2N*
 *+1+qj

is nonzero in E2. Let s = L-N -m(p-1). By Theorem 2.8(2), this class is equival*
 *ent,

mod terms that desuspend, to


                      vshN+m(p-1) vm-j-1h2N+1+qj:


By Lemma 2.13, this is equivalent to
        i                                                        j
     _1_    s-1 N+m(p-1)+1m-j            s-1    N+m(p-1)m-j
     m-j d(v   h         v   )2N+1+qj+ Lv   h  h       v   2N+1+qj:


Since L is highly p-divisible, the second term is 0 in E2. The first term is si*
 *mplified

by using Lemma 2.12(2) to replace (hpv)m-j by (vph)m-j. Thus the desired term i*
 *s a

unit times d(vL-N+m-jp-1hN+j(p-1)+1)2N+1+qj, which is nonzero by Theorem 2.8(4).

||



                              References

 1.M. Bendersky, The v1-periodic unstable Novikov spectral sequence, Topology 3*
 *1 (1992) 47-64.
 2.________, Unstable towers in the odd primary homotopy groups of spheres, Tra*
 *ns Amer Math
   Soc 276 (1985) 529-542.
 3.________, Some calculations in the unstable Adams-Novikov spectral sequence,*
 * Publ RIMS 16
   (1980) 739-766.
 4.M. Bendersky, E. B. Curtis, and H. R. Miller, The unstable Adams spectral se*
 *quence for gener-
   alized homology, Topology 17 (1978) 229-248.
 5.M. Bendersky, D. M. Davis, and M. Mimura, v1-periodic homotopy groups of exc*
 *eptional Lie
   groups: torsion-free cases, Trans Amer Math Soc 333 (1992) 115-135.
 6.M. C. Crabb and K. Knapp, The Hurewicz map on stunted complex projective spa*
 *ces, Amer
   Jour Math 110 (1988) 783-809.
 7.D. M. Davis, v1-periodic homotopy groups of SU(n) at an odd prime, Proc Lond*
 *on Math Soc
   43 (1991) 529-544.
 8.D. M. Davis and M. Mahowald, Some remarks on v1-periodic homotopy groups, Lo*
 *ndon Math
   Soc Lecture Note Series 176 (1992) 55-72.
 9.A. T. Lundell, A divisibility property for Stirling numbers, Jour Number The*
 *ory 10 (1978)
   35-54.
10.M. Mimura and H. Toda, Cohomology operations and the homotopy of compact Lie*
 * groups-I,
   Topology 9 (1970) 317-336.
11.G. Walker, Estimates for the complex and quaternionic James numbers, Quar Jo*
 *ur Math Oxford
   32 (1981) 467-489.



              FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS            37


12.H. Yang, Tractable formulas for v1-periodic homotopy groups of SU(n) when n *
 * p2- p + 1,
   Lehigh Univ. thesis, 1994.


  Lehigh University, Bethlehem, PA 18015
  E-mail address: dmd1@lehigh.edu


  Lehigh University, Bethlehem, PA 18015
  E-mail address: hy02@lehigh.edu