3
           _   as an upper triangular matrix
                Jonathan Barker and Victor Snaith

                                Abstract

         In the 2-local stable homotopy category the group of left-bu-module
      automorphisms of bu^bo which induce the identity on mod 2 homology
      is isomorphic to the group of infinite upper triangular matrices with
      entries in the 2-adic integers. We identify the conjugacy class of the
      matrix corresponding to 1 ^ _3, where _3 is the Adams operation.
1     Introduction


Let bu and bo denote the stable homotopy spectra representing 2-adically com-

pleted unitary and orthogonal connective K-theory respectively.  The main

result of [7] is the existence of an isomorphism of groups

                              ~=     0
                     : U1 Z2 -!  Aut left-bu-mod(bu ^ bo).


Here U1 Z2 is the group of upper triangular matrices with coefficients in the

2-adic integers and Aut0left-bu-mod(bu^bo) denotes the group of left bu-module

automorphisms of bu ^ bo in the stable homotopy category of 2-local spectra,

which induce the identity on mod 2 singular homology.  The details of this

isomorphism are recapitulated in x2.

    This isomorphism is defined up to inner automorphisms of U1 Z2. Given

an important automorphism in Aut 0left-bu-mod(bu ^ bo) one is led to ask what

is its conjugacy class in U1 Z2. By far the most important such automorphism

is 1 ^ _3, where _3 : bo -! bo denotes the Adams operation.

    The following is our main result, which proved by combining the discussion

of x3.5 with Theorem 4.2.


Theorem 1.1.

    Under the isomorphism   the automorphism 1 ^ _3 corresponds to an

                                    1

element in the conjugacy class of the matrix
                        0                      1
                           1  1  0   0  0  . . .
                        B                      C
                        B                      C
                        B                      C
                        B  0  9  1   0  0  . . .C
                        B                      C
                        B                      C
                        B         2            C
                        B  0  0  9   1  0  . . .C.
                        B                      C
                        B                      C
                        B             3        C
                        B  0  0  0   9  1  . . .C
                        B                      C
                        @                      A
                           .. .  .   .   .  .
                           .  .. ..  ..  .. ..


    By techniques which are described in ([1] pp.338-360) and reiterated in x3,

Theorem 1.1 reduces to calculating the effect on 1 ^ _3 on ss*(bu ^ bo) modulo

torsion.  The difficulty arises because, in order to identify  -1(1 ^ _3) one

must compute the map on homotopy modulo torsion in terms of an unknown

2-adic basis defined in terms of the Mahowald splitting of bu ^ bo (see x2 and

x3). On the other hand a very convenient 2-adic basis is defined in [3] and the

crucial fact is that 1 ^ _3 acts on the second basis by the matrix of Theorem

1.1. This fact was pointed out to one of us (VPS) by Francis Clarke in 2001

and led to the confident prediction appearing as a footnote in ([7] p.1273).

Verifying the prediction has proved a little more difficult than first imagined!

    Once one has Theorem 1.1 a number of homotopy problems become merely

a matter of matrix algebra.  In x5 we give an example concerning the maps

1^(_3-1)(_3-9) . .(._3-9n-1) where we prove a vanishing result (Theorem

5.4) which is closely related to the main theorem of [5], as explained in Remark

5.5.  In subsequent papers we shall give further applications to homotopy

theory and to algebraic K-theory.
2     2-adic homotopy of bu ^ bo


2.1. Let bu and bo denote the stable homotopy spectra representing 2-adically

completed unitary and orthogonal connective K-theory respectively. We shall

begin by recalling the 2-local homotopy decomposition of bu ^ bo which is one

of a number of similar results which were discovered by Mark Mahowald in

the 1970's. These results may be proved in several ways [1], [4] and [5]. For

notational reasons we shall refer to the proof, a mild modification of ([1]

pp.190-196), which appears in ([7] x2).

    Consider the second loopspace of the 3-sphere,  2S3. There exists a model

for  2S3 which is filtered by finite complexes ([2] , [6])

                                                 [
                 S1 = F1   F2   F3   . . . 2S3 =    Fk

                                                 k 1


                                    2

and there is a stable homotopy equivalence, an example of the so-called Snaith

splitting, of the form

                            2S3 ' _k 1Fk=Fk-1.

There is a 2-local homotopy equivalence of left-bu-module spectra (see [7]

Theorem 2.3(ii)) of the form


                   ^L: _k 0bu ^ (F4k=F4k-1) -'! bu ^ bo.



The important fact about this homotopy equivalence is that its induced map

on mod 2 homology is a specific isomorphism which is described in ([7] x2.2)

    From this decomposition we obtain left-bu-module spectrum maps of the

form

                 'k,l: bu ^ (F4k=F4k-1) -! bu ^ (F4l=F4l-1)

where 'k,k= 1, 'k,l= 0 if l > k and, as explained in ([7] x3.1), 'k,lis defined

up to multiplication by a 2-adic unit when k > l.

    Consider the ring of left bu-module endomorphisms of degree zero in the

stable  homotopy  category  of  spectra  [1],  which  we  shall  denote  by

End left-bu-mod(bu ^ bo).  The group of units in this ring will be denoted

by Aut left-bu-mod(bu ^ bo), the group of homotopy classes of left bu-module

homotopy equvialences and let Aut 0left-bu-mod(bu ^ bo) denote the subgroup

of left bu-module homotopy equivalences which induce the identity map on

H*(bu ^ bo; Z=2).

    Let U1 Z2 denote the group of infinite, invertible upper triangular matrices

with entries in the 2-adic integers. That is, X = (Xi,j) 2 U1 Z2 if Xi,j2 Z2

for each pair of integers 0   i, j and Xi,j= 0 if j > i and Xi,iis a 2-adic

unit. This upper trianglular group is not equal to the direct limit lim!nUnZ2

of the finite upper triangular groups. The main result of [7] is the existence

of an isomorphism of groups

                              ~=     0
                     : U1 Z2 -!  Aut left-bu-mod(bu ^ bo).


By the Mahowald decomposition of bu ^ bo the existence of _ is equivalent to

an isomorphism of the form

                       ~=     0
              : U1 Z2 -!  Aut left-bu-mod(_k 0bu ^ (F4k=F4k-1)).


If we choose 'k,lto satisfy 'k,l= 'l+1,l'l+2,l+1. .'.k,k-1for all k - l   2 the*
 *n,

for X 2 U1 Z2, we define ([7] x3.2)

             X
    (X-1 ) =    Xl,k'k,l: bu ^ (_k 0F4k=F4k-1) -! bu ^ (_k 0F4k=F4k-1).

             l k


The ambiguity in the definition of the 'k,l's implies that   is defined up to

conjugation by a diagonal matrix in U1 Z2.



                                    3

2.2. Bases for ss*(bu^bo)_Z2Torsion

    Let Gs,tdenote the 2-adic homotopy group modulo torsion


                            sss(bu ^ F4t=F4t-1)   Z2
                      Gs,t= _______________________
                                    Torsion

so
                    sss(bu ^ F4t=F4t-1)   Z2  ss*(bu ^ bo)   Z2
          G*,*=  s,t_______________________~= ________________.
                            Torsion               Torsion

From [1] or [7]            8

                           <  Z2  ifs even, s   4t,

                     Gs,t~=
                           :  0   otherwise



and if "Gs,tdenotes sss(bu ^ F4t=F4t-1)   Z2 then "Gs,t~=Gs,t  Ws,twhere Ws,t

is a finite, elementary abelian 2-group.

    In [3] a Z2-basis is given for G*,*consisting of elements lying in the subr*
 *ing

Z2[u=2, v2=4] of Q2[u=2, v2=4]. One starts with the elements
                           _            !
                             v2 - 9i-1u2
                 c4k =  ki=1 ___________ ,    k = 1, 2, . . .
                              9k - 9i-1


and "rationalises" them, after the manner of ([1] p.358), to obtain elements of

Z2[u=2, v2=4]. In order to describe this basis we shall require a few well-known

preparatory results about 2-adic valuations.


Proposition 2.3.
                             n        n+3
    For any integer n   0, 92  - 1 = 2   (2s + 1) for some s 2 Z.


    Proof

    We prove this by induction on n, starting with 9 - 1 = 23. Assuming the

result is true for n, we have

                 (n+1)         2n       2n
               92    - 1   = (9   - 1)(9   + 1)


                                n       2n
                           = (92  - 1)(9   - 1 + 2)



                           = 2n+3(2s + 1)(2n+3(2s + 1) + 2)



                           = 2n+4(2s + 1) (2n+2(2s + 1) + 1)
                                          ________-z_______"
                                                 odd

as required.


Proposition 2.4.

    For any integer l   0, 9l- 1 = 2 2(l)+3(2s + 1) for some s 2 Z, where  2(l)

denotes the 2-adic valuation of l.



                                    4

    Proof

    Write l = 2e1 + 2e2 + . .+.2ek with 0   e1 < e2 < . . .< ek so that

 2(l) = e1. Then, by Proposition 2.3,

                    e1+2e2+...+22k
         9l- 1  = 92            - 1



                = ((2s1 + 1)2e1+3+ 1)) . .(.(2sk + 1)2ek+3+ 1)) - 1



                  (2s1 + 1)2e1+3 (modulo 2e1+4)



                = 2e1+3(2t + 1)


as required.

Proposition 2.5.           Q

    For any integer l   1,   li=1(9l- 9i-1) = 2 2(l!)+3l(2s + 1) for some s 2 Z.


    Proof

    By Proposition 2.4 we have
             Q l     l   i-1    Q l    l-i+1     i-1
               i=1 (9 - 9   ) =   i=1(9     - 1)9

                                Q l
                              =   i=1 2 2(l-i+1)+3(2ti+ 1)9i-1



                              = (2t + 1)2 2(l!)+3l,


as required.

Proposition 2.6. For any integer l   0, 2 2(l!)+3l= 24l-ff(l)where ff(l) is

equal to the number of 1's in the dyadic expansion of l.


    Proof

    Write l = 2e1+2e2+. .+.2ek with 0   e1 < e2 < . .<.ek so that ff(l) = k.

    9l- 1 = 2 2(l)+3(2s + 1) for some s 2 Z, where  2(l) denotes the 2-adic

valuation of l. Then

             2(l!) =      2ff1-1 +  2ff2-1  +  . . .+   2ffk-1

                      +   2ff1-2 +  2ff2-2  +  . . .+   2ffk-2
                          ..                            .
                          .                             ..

                      +   1      +  2ff2-ff1+  . . .+   2ffk-ff1

                                 +  1       +  . . .+   2ffk-ff2

                                                    +   1


because the first row counts the multiples of 2 less than or equal to l, the

second row counts the multiples of 4, the third row counts multiples of 8 and

so on. Adding by columns we obtain


               2(l!) = 2ff1- 1 + 2ff2- 1 + . .+.2ffk- 1 = l - k



                                    5

which implies that 23l+ 2(l!)= 23l+l-ff(l)= 24l-ff(l), as required.


2.7. Bases continued
                                      2-9i-1u2
    Consider the elements c4k =  ki=1v_______9k-9i-1, introduced in x2.2, for a*
 * par-

ticular k = 1, 2, . ...For completeness write c0 = 1 so that c4k 2 Q2[u=2, v2=4*
 *].

Since the degree of the numerator of c4k is 2k, Proposition 2.6 implies that


                                                    v2 - 9i-1u2
              f4k = 24k-ff(k)-2kc4k = 22k-ff(k) ki=1___________
                                                     9k - 9i-1


lies in Z2[u=2, v2=4] but 24k-ff(k)-2k-1c4k =2Z2[u=2, v2=4]. Similarly (u=2)f4k*
 * =

24k-ff(k)-2k-1uc4k 2 Z2[u=2, v2=4] but 24k-ff(k)-2k-2uc4k =2Z2[u=2, v2=4] and so

on. This process is the "rationalisation yoga" referred to in x2.2. One forms

ujc4k and then multiplies by the smallest positive power of 2 to obtain an

element of Z2[u=2, v2=4].

    By Proposition 2.6, starting with f4l= 24l-ff(l)-2lc4lthis process produces

the following set of elements of Z2[u=2, v2=4]


      f4l, (u=2)f4l, (u=2)2f4l, . . ., (u=2)2l-ff(l)f4l,



               u(u=2)2l-ff(l)f4l, u2(u=2)2l-ff(l)f4l, u3(u=2)2l-ff(l)f4l, . ...


    As explained in ([1] p.352 et seq), the Hurewicz homorphism defines an

injection of graded groups of the form


                     ss*(bu_^_bo)___Z2_          2
                                     -! Q2[u=2, v =4]
                         Torsion


which, by the main theorem of [3], induces an isomorphism between ss*(bu^bo)_Z2*
 *Torsion

and  the  free  graded  Z2-module  whose  basis  consists  of  the  elements  of

Z2[u=2, v2=4] listed above for l = 0, 1, 2, 3, . ...

    From this list we shall be particularly interested in the elements whose

degree is a multiple of 4. Therefore denote by g4m,4l2 Z2[u=2, v2=4] for l   m

the element produced from f4lin degree 4m. Hence, for m   l, g4m,4lis given

by the formula
                    8
                    >< u2m-4l+ff(l)u2l-ff(l)f4l_if4l - ff(l)   2m,
                                   22l-ff(l)
            g4m,4l=
                    >:  u2(m-l)f4l_
                         22(m-l)            if 4l - ff(l) > 2m.

Lemma 2.8.

    In the notation of x2.2, let   denote the projection


                 ss*(bu ^ bo)   Z2
               : ________________~= G*,*-! G*,k=  m  Gm,k.
                     Torsion

Then  (g4k,4i) = 0 for all i < k.



                                    6

    Proof

    Since Gm,k is torsion free it suffices to show that  (g4k,4i) vanishes in

G*,k  Q2. When i < k, by definition


                     ss4i(bu ^ bo)   Z2      ss4k(bu ^ bo)   Z2
       g4k,4i2 u2k-2i_________________  Q2   _________________  Q2.
                          Torsion                 Torsion


However   projects onto  s sss(bu^F4k=F4k-1)_Z2Torsionand commutes with multi-

plication by u so the result follows from the fact that the homotopy of

bu ^ F4k=F4k-1 is trivial in degrees less than 4k (see [7] x3).


2.9. Recall from x2.2 that G4k,k~=Z2 for k = 0, 1, 2, 3, . .s.o we may choose

a generator z4k for this group as a module over the 2-adic integers (with the

convention that z0 = f0 = 1).  Let "z4kbe any choice of an element in the

2-adic homotopy group "G4k,k~=G4k,k  W4k,kwhose first coordinate is z4k.


Lemma 2.10.

    Let B denote the exterior subalgebra of Z=2 Steenrod algebra generated by

Sq1 and Sq0,1. In the collapsed Adams spectral sequence (see [1] or [7])


                 Es,t2~=Exts,tB(H*(F4k=F4k-1; Z=2), Z=2)



                          =) sst-s(bu ^ (F4k=F4k-1))   Z2



the homotopy class "z4kis represented either in E0,4k2or E1.4k+12.


    Proof

    Recall from x2.2 that ss4k(bu ^ (F4k=F4k-1))   Z2 = G"4k,k~=Z2   W4k,k.

The following behaviour of the filtration coming from the spectral sequence

is well-known, being explained in [1]. The group "G4k,khas a filtration


                   . . .F i  . .F.2  F 1  F 0= "G4k,k



with F i=F i+1~=Ei,4k+i2and 2F i  F i+1. Also 2 . W4k,k= 0, every non-trivial

element of W4k,kbeing represented in E0,4k2. Furthermore for i = 1, 2, 3, . . .

we have 2F i= F i+1and F 1~=Z2.

    Now suppose that "z4kis represented in Ej,4k+j2for j    2 then "z4k2

F j. From the multiplicative structure of the spectral sequence there exists a

generator ^z4kof F 1such that 2j^z4kgenerates F j+1and therefore 2jflz^4k= 2"z4k

for some 2-adic integer fl. Hence 2(2j-1flz^4k-z"4k) = 0 and so 2j-1flz^4k-z"4k2

W4k,kwhich implies the contradiction that the generator z4k is divisible by 2

in G4k,k.


Theorem 2.11.
                                    7

    In the notation of x2.7 and x2.9


                                                ss4k(bu ^ bo)   Z2
               z4k =  ki=02fi(k,i)~4k,4ig4k,4i2 _________________
                                                     Torsion


with ~s,t2 Z2, ~4k,4k2 Z*2and

                      8
                      <  4(k - i) - ff(k) + ff(i)if4i - ff(i) > 2k,

           fi(k, i) =
                      :  2k - ff(k)             if4i - ff(i)   2k.



    Proof

    From [3], as explained in x2.7, a Z2-module basis for G4k,*is given by

{g4k,4l}0 l k. Hence there is a relation of the form


           z4k = ~4k,4kg4k,4k+ "~4k,4(k-1)g4k,4(k-1)+ . .+."~4k,0g4k,0



where  "~4k,4iand  ~4k,4k are  2-adic  integers.    Applying  the  projection

  : G4k,*-! G4k,kwe see that z4k =  (z4k) = ~4k,4k (g4k,4k), by Lemma 2.8.

Hence, if ~4k,4kis not a 2-adic unit, then z4k would be divisible by 2 in G4k,k

and this is impossible since z4k is a generator, by definition.

    Multiplying the relation


                z4k = ~4k,4k (g4k,4k) = ~4k,4k (f4k) 2 G4k,k.


by (u=2)2k-ff(k)we obtain (u=2)2k-ff(k)z4k = ~4k,4k ((u=2)2k-ff(k)f4k), which

lies in G8k-2ff(k),k, by the discussion of x2.7. Therefore, in G8k-2ff(k),k  Q2*
 * we

have the relation

                                             k-1X

        (u=2)2k-ff(k)z4k = (u=2)2k-ff(k)f4k+    "~4k,4i(u=2)2k-ff(k)g4k,4i.

                                             i=0


Since the left hand side of the equation lies in G8k-2ff(k),k, the Q2 coefficie*
 *nts

must all be 2-adic integers once we re-write the right hand side in terms of

the basis of x2.7.

    For i = 0, 1, . .,.k - 1

                         8  u2k-ff(k)+2k-4i+ff(i)+2i-ff(i)
                         <  _____________________22k-ff(k)+2i-ff(i)f4iif4i - ff*
 *(i)   2k,

    (u=2)2k-ff(k)g4k,4i=
                         :  u2k-ff(k)+2k-2i_
                            22k-ff(k)+2k-2if4i       if4i - ff(i) > 2k

                        8    u4k-2i-ff(k)
                        <  ____________22k+2i-ff(k)-ff(i)f4iif4i - ff(i)   2k,

                      =
                        :  u4k-2i-ff(k)_
                           24k-2i-ff(k)f4i if 4i - ff(i) > 2k.


                                    8

Now we shall write (u=2)2k-ff(k)g4k,4ias a power of 2 times a generator derived

from f4i in x2.7 (since we did not define any generators called g4k+2,4ithe

generator in question will be g8k-2ff(k),4ionly when ff(k) is even).

    Assume that 4i - ff(i)   2k so that 2i - ff(i)   4k - 2i - ff(k) and


          __u4k-2i-ff(k)_       1     4k-4i-ff(k)+ff(i) 2i-ff(i)
                        f4i= ________u             (u=2)      f4i
          22k+2i-ff(k)-ff(i) 22k-ff(k)


which implies that "~4k,4iis divisible by 22k-ff(k)in the 2-adic integers, as

required.

    Finally assume that 4i-ff(i) > 2k. We have 2i-ff(i)   4k-2i-ff(k) also.

To see this observe that ff(i) + ff(k - i) - ff(k)   0 because, by Proposition
                                                                   k
2.6, this equals the 2-adic valuation of the binomial coefficient (i). Therefore


                 ff(k) - ff(i)   ff(k - i)   k - i < 4(k - i).


Then, as before,


         u4k-2i-ff(k)_        1        4k-4i-ff(k)+ff(i) 2i-ff(i)
                   f4i= ______________u             (u=2)      f4i
         24k-2i-ff(k)   24k-4i-ff(k)+ff(i)


which implies that "~4k,4iis divisible by 24k-4i-ff(k)+ff(i)in the 2-adic integ*
 *ers,

as required.


Theorem 2.12.

    (i)  In the collapsed Adams spectral sequence and the notation of Lemma

2.10 "z4kmay be chosen to be represented in E0,4k2.

    (ii)  In fact, "z4kmay be taken to be the smash product of the unit j of the

bu-spectrum with the inclusion of the bottom cell jk into F4k=F4k-1


                                j^jk
                       S0 ^ S4k -!  bu ^ F4k=F4k-1.


    Proof

    For part (i), suppose that "z4kis represented in E1,4k+12. By Lemma 2.10

we must show that this leads to a contradiction. From [7] we know that on the

s = 1 line the non-trivial groups are precisely E1,4k+12, E1,4k+32, . .,.E1,8k+*
 *2-2ff(k)2

which are all of order two. From the multiplicative structure of the spectral

sequence, if a homotopy class w is represented Ej,4k+2j-12and Ej,4k+2j+12is

non-zero then there is a homotopy class w0 represented in Ej,4k+2j+12such

that 2w0 = uw.  Applied to "z4kthis implies that the homotopy element

u2k-ff(k)+1"z4kis divisible by 22k-ff(k)+1.  Hence u2k-ff(k)+1z4k is divisible *
 *by

22k-ff(k)+1in G*,*, which contradicts the proof of Theorem 2.11.

    For part (ii) consider the Adams spectral sequence


   Es,t2= Exts,tB(H*(F4k=F4k-1; Z=2), Z=2) =) sst-s(bu ^ F4k=F4k-1)   Z2.



                                    9

We have an isomorphism


                              Ht(F4k=F4k-1; Z=2)
  E0,t2= Hom(____________________________________________________, Z=2).
             Sq1Ht-1(F4k=F4k-1; Z=2) + Sq0,1Ht-3(F4k=F4k-1; Z=2)


The discussion of the homology groups H*(F4k=F4k-1; Z=2) given in ([1] p.341;

see also x3.1) shows that E0,4k2~=Z=2 generated by the Hurewicz image of

j ^ jk. Therefore the generator of E0,4k2represents j ^ jk. Since there is only

one non-zero element in E0,4k2it must also represent "z4k, by part (i), which

completes the proof.
3     The Matrix


3.1. Consider the left-bu-module spectrum map of x2.1


                 'k,l: bu ^ (F4k=F4k-1) -! bu ^ (F4l=F4l-1)


when l > k. This map is determined up to homotopy by its restriction, via

the unit of bu, to (F4k=F4k-1). By S-duality this restriction is equivalent to a

map of the form


                  S0 -! D(F4k=F4k-1) ^ bu ^ (F4l=F4l-1),


which DX denotes the S-dual of X. Maps of this form are studied by means

of the (collapsed) Adams spectral sequence (see [7] x3.1), where B is as in

Lemma 2.10,


      Es,t2= Exts,tB(H*(D(F4k=F4k-1); Z=2)   H*(F4l=F4l-1; Z=2), Z=2)



                =) sst-s(D(F4k=F4k-1) ^ (F4l=F4l-1) ^ bu)   Z2.


    Recall from ([1] p.332) that  a is the (invertible) B-module given by

Z=2 in degree a,  -a  = Hom( a, Z=2) and I is the augmentation ideal,

I = ker(ffl : B - ! Z=2).  Hence, if b > 0, I-b = Hom(Ib, Z=2), where Ib is

the b-fold tensor product of I. These duality identifications may be verified

using the criteria of ([1] p.334 Theorem 16.3) for identifying  aIb.

    In ([1] p.341) it is shown that the B-module given by


              H-* (D(F4k=F4k-1); Z=2) ~=H*(F4k=F4k-1; Z=2)

                              r-1+1 2r-1-1                     r
is  stably  equivalent  to   2     I       when  0  <  4k  =  2 .   Therefore
                                                  r-1+1)1-2r-1
H*(D(F4k=F4k-1); Z=2) is stably equivalent to  -(2     I       when 0 <

4k = 2r. If k is not a power of two we may write 4k = 2r1+ 2r2+ . .+.2rt

with 2   r1 < r2 < . .<.rt. In this case


              H*(F4k=F4k-1; Z=2) ~= rtj=r1H*(F2j=F2j-1; Z=2)



                                    10

which is stably equivalent to  2k+ff(k)I2k-ff(k), where ff(k) equals the num-

ber of 1's in the dyadic expansion of k, as in Proposition 2.6.  Similarly,

H*(D(F4k=F4k-1); Z=2) is stably equivalent to  -2k-ff(k)Iff(k)-2k. From this,

for all s > 0, one easily deduces a canonical isomorphism ([7]p.1267) of the

form
           Es,t2 ~=Exts,tB( 2l-2k+ff(l)-ff(k)I2l-2k-ff(l)+ff(k), Z=2)



                 ~=Exts+2l-2k-ff(l)+ff(k),t-2l+2k-ff(l)+ff(k)B(Z=2, Z=2).


Also there is an algebra isomorphism of the form Ext*,*B(Z=2, Z=2) ~=Z=2[a, b]

where a 2 Ext1,1B, b 2 Ext1,3B. As explained in ([7] p.1270) ik,lis represented

in


 E4(k-l)+ff(l)-ff(k),4(k-l)+ff(l)-ff(k)2~=Ext2k-2l,6k-6l)B(Z=2, Z=2) ~=Z=2 = <b*
 *2k-2l>.


Proposition 3.2.

    For l < k, in the notation of x2, the homomorphism


                          ('k,l)* : G4k,k-! G4k,l


satisfies ('k,l)*(z4k) = ~4k,4l22k-2l-ff(k)+ff(l)u2k-2lz4lfor some 2-adic unit *
 *~4k,4l.


    Proof

    Let "z4k2 "G4k,kbe as in x2.9 so that, proved in a similar manner to Lemma

2.10, 2"z4kis represented in E1,4k+12in the spectral sequence


                 Es,t2= Exts,tB(H*(F4k=F4k-1; Z=2), Z=2)



                          =) sst-s(bu ^ (F4k=F4k-1))   Z2.


where, from x3.1, we have
     E1,4k+12~=Ext1+2k-ff(k),4k+1-2k-ff(k)B(Z=2, Z=2) ~=Z=2 = <a2k+1-ff(k)>.


The  multiplicative  pairing  between  these  spectral  sequences  shows  that

('k,l)*(2"z4k) 2 "G4k,lis represented in the spectral sequence


                 Es,t2= Exts,tB(H*(F4l=F4l-1; Z=2), Z=2)



                          =) sst-s(bu ^ (F4l=F4l-1))   Z2.



by the generator of E1+4k-4l-ff(k)+ff(l),1+8k-4l-ff(k)+ff(l)2because a2k+1-ff(k*
 *)b2k-2l

is the generator of


  E1+4k-4l-ff(k)+ff(l),1+8k-4l-ff(k)+ff(l)2~=Ext1+4k-2l-ff(k),1+8k-6l-ff(k)B(Z=*
 *2, Z=2).



                                    11

    Since multiplication by a and b in the spectral sequence corresponds to

multiplication by 2 and u respectively on homotopy groups we have the fol-

lowing table of representatives in ss*(bu ^ (F4l=F4l-1))   Z2.


            _homotopy_element___||representative||dimension___

            ________2z4l________||_a2l-ff(l)+1_||_____4l______

            _____(u=2)(2z4l)_____||_a2l-ff(l)b____||4l_+_2____

            ____(u=2)2(2z4l)____||a2l-ff(l)-1b2___||4l_+_4____
                      ..               .              .
            __________._________||_____..______||_____..______

            __(u=2)2l-ff(l)(2z4l)__||ab2l-ff(l)_||8l_-_2ff(l)_

            __u(u=2)2l-ff(l)(2z4l)_b||2l-ff(l)+1||8l_-_2ff(l)_+ 2

            _u2(u=2)2l-ff(l)(2z4l)_||b2l-ff(l)+28||l_-_2ff(l)_+ 4
                      ..               .              .
            __________._________||_____..______||_____..______


    Therefore there are two cases for ('k,l)*(2"z4k). If 2k-2l   2l-ff(l)+1 then

b2k-2lrepresents u2k-2l-(2l-ff(l))(u=2)2l-ff(l)"z4l= u2k-4l+ff(l))(u=2)2l-ff(l)*
 *"z4land,

up   to   multiplication   by   2-adic   units,   ('k,l)*(2"z4k)   is   equal  *
 * to

21+2k-ff(k)u2k-4l+ff(l))(u=2)2l-ff(l)"z4l,  as  required.   On  the  other  han*
 *d,  if

2k - 2l   2l - ff(l) then a2l-ff(l)+1-(2k-2l)b2k-2l= a4l-2k-ff(l)+1b2k-2lrepres*
 *ents

(u=2)2k-2l(2"z4l) which shows that, up to 2-adic units, ('k,l)*(2"z4k) is equal*
 * to

21+2k-ff(k)-(4l-2k-ff(l)+1)(u=2)2k-2l(2"z4l) = 24k-ff(k)-4l+ff(l)(u=2)2k-2l(2"z*
 *4l), as re-

quired.


Proposition 3.3.

    Let _3 : bo -! bo denote the Adams operation, as usual.  Then, in the

notation of x2.7,

                           8
                           >> 9kg4k,4k+ 9k-12 2(k)+3g4k,4k-4ifk   3,
                           >>
                           >>
                           >>  2           3
                           <  9 g8,8+ 9 . 2 g8,4           ifk = 2,

       (1 ^ _3)*(g4k,4k) =
                           >>
                           >> 9g4,4+ 2g4,0                 ifk = 1,
                           >>
                           >>
                           :
                              g0,0                         ifk = 0.

                                    12

    Proof

    The map (1^_3)* fixes u, multiplies v by 9 and is multiplicative. Therefore

                         2  i-1 2
 (1 ^ _3)*(c4k) =  ki=19v_-9___u_9k-9i-1


                       (9v2-9ku2+9ku2-u2) k  (v2-9i-2u2)
                = 9k-1 ___________________i=2______ k    k  i-1
                                                    i=1(9 -9   )

                       (9v2-9ku2) k  (v2-9i-2u2)     (9ku2-u2) k  (v2-9i-2u2)
                = 9k-1 ___________i=2_______ k+ 9kk-1i-1_______i=2_______kki-1
                                             i=1(9 -9   )  i=1(9 -9   )

                     (v2-9k-1u2) k  (v2-9i-2u2)     (9ku2-u2) k  (v2-9i-2u2)
                = 9k ____________i=2_______ k+ 9kk-1i-1_______i=2_______kki-1
                                            i=1(9 -9   )  i=1(9 -9   )

                                       u2 k-1(v2-9i-1u2)
                = 9kc4k+ 9k-1(9k - 1) ____i=1_________(9k-1) k-1  k  i-1
                                                             i=1(9 -9   )


                = 9kc4k+ 9k-1u2c4k-4.


    Hence, for k   1, we have


 (1 ^ _3)*(f4k) = 22k-ff(k)(1 ^ _3)*(c4k)



                = 22k-ff(k)9kc4k+ 9k-1u222k-ff(k)-2k+2+ff(k-1)+2k-2-ff(k-1)c4k-4



                = 9kf4k+ 9k-1u222-ff(k)+ff(k-1)f4k-4



                = 9kf4k+ 9k-1u22 2(k)+1f4k-4,


which yields the result, by the formulae of x2.7.


Proposition 3.4.

    When k > l

                  8   l         l-1
                  >> 9 g4k,4l+ 9   g4k,4l-4        if4l - ff(l)   2k,
                  >>
                  >>
                  >>  l         l-1 4l-ff(l)-2k
                  <  9 g4k,4l+ 9   2        g4k,4l-4if4l - ff(l) -  2(l) - 3

(1^_3)*(g4k,4l) =                                           2k < 4l - ff(l),
                  >>
                  >>
                  >>  l         l-1 3+  (k)
                  >> 9 g4k,4l+ 9   2   2  g4k,4l-4 if2k < 4l - ff(l) -  2(l) - 3
                  :
                                                          < 4l - ff(l)


                                    13

    Proof

    Suppose that 4l - ff(l)   2k then, by Proposition 3.3 (proof),


               (1 ^ _3)*(g4k,4l)


                                      u2l-ff(l)f
               = (1 ^ _3)*(u2k-4l+ff(l)_______224ll-ff(l))


                            u2l-ff(l)(9lf +9l-1u22 2(l)+1f   )
               = u2k-4l+ff(l)____________4l___________22l-4l-4ff(l)


                                          u2l-ff(l)u22 2(l)+1f
               = 9lg4k,4l+ 9l-1u2k-4l+ff(l)________________22l4l-4-ff(l)


                                          u2l+2-ff(l)2 2(l)+1f
               = 9lg4k,4l+ 9l-1u2k-4l+ff(l)________________22l4l-4-ff(l).


Then, since  2(l) = 1 + ff(l - 1) - ff(l),


4(l - 1) - ff(l - 1) = 4l - ff(l) + ff(l) - ff(l - 1) - 4 = 4l - ff(l) - 3 -  2*
 *(l) < 2k


so that                                      2l-2-ff(l-1)

                 g4k,4l-4 = u2k-4l+4+ff(l-1)u________f4l-4_22l-2-ff(l-1)


                                         u2l+2-ff(l)f
                          = u2k-4l+ff(l)_____________4l-4_22l-2-ff(l)+ff(l)-ff(*
 *l-1)


                                       u2l+2-ff(l)f
                          = u2k-4l+ff(l)___________4l-422l-ff(l)- 2(l)-1


so that, for 0 < l < k suppose that 4l - ff(l)   2k,


                  (1 ^ _3)*(g4k,4l) = 9lg4k,4l+ 9l-1g4k,4l-4.


    Similarly, for 0 < l < k if 4l - ff(l) > 2k then, by Proposition 3.3 (proof*
 *),


                    (1 ^ _3)*(g4k,4l)


                                  2(k-l)f4l
                    = (1 ^ _3)*( u______22(k-l))


                        2(k-l)(9lf4l+9l-1u22 2(k)+1f4k-4)
                    =  u________________________22(k-l)


                                     u2k-2l+22 2(k)+1f
                    = 9lg4k,4l+ 9l-1 _________________4k-422(k-l).


This situation splits into two cases given by

    (i)  4l - ff(l) -  2(l) - 3   2k < 4l - ff(l) or

    (ii)  2k < 4l - ff(l) -  2(l) - 3 < 4l - ff(l).


                                    14

    In case (i) 4l - 4 - ff(l - 1) = 4l - ff(l) -  2(l) - 3   2k and so again we

have                                         2l-2-ff(l-1)

                 g4k,4l-4 = u2k-4l+4+ff(l-1)u________f4l-4_22l-2-ff(l-1)


                              2k-2l+2f4l-4
                          = _u__________22l-1- 2(l)-ff(l)


                             2k-2l+221+ 2(l)f4l-4
                          = u_______________22k-2l+4l-ff(l)-2k

so that

              (1 ^ _3)*(g4k,4l) = 9lg4k,4l+ 9l-124l-ff(l)-2kg4k,4l-4.

    In case (ii)
                                     u2k-2l+2f
                         g4k,4l-4 =  _________4k-4_22(k-l+2)

so that

               (1 ^ _3)*(g4k,4l) = 9lg4k,4l+ 9l-123+ 2(k)g4k,4l-4.



3.5. In the notation of x2.1, suppose that A 2 U1 Z2 satisfies

                 (A-1) = [1 ^ _3] 2 Aut 0left-bu-mod(bu ^ bo).

    Therefore, by definition of   and the formula of Theorem 2.11
           P                            3
             l kAl,k('k,l)*(z4k)= (1 ^ _ )*(z4k)


                               =  ki=02fi(k,i)~4k,4i(1 ^ _3)*(g4k,4i).

On the other hand
      P
         l kAl,k('k,l)*(z4k)

                 P
      = Ak,kz4k+   l<kAl,k~4k,4l22k-2l-ff(k)+ff(l)u2k-2lz4l



      = Ak,k ki=02fi(k,i)~4k,4ig4k,4i

              P
            +   l<k  li=0Al,k~4k,4l22k-2l-ff(k)+ff(l)u2k-2l2fi(l,i)~4l,4ig4l,4i.

    In order to determine the Ak,l's it will suffice to express u2k-2lg4l,4ias a

multiple of g4k,4iand then to equate coefficients in the above expressions. By

definition           8
                                           2i-ff(i)
                     >< u2k-2lu2l-4i+ff(i)u_____f4iif4i - ff(i)   2l,
                                         22i-ff(i)
       u2k-2lg4l,4i=
                     >:        u2(l-i)f4i
                        u2k-2l _______22(l-i)     if4i - ff(i) > 2l.

                    8   2k-2i
                    >< u____f4i_if4i - ff(i)   2l,
                        22i-ff(i)
                  =
                    >: u2k-2if4i_
                        22l-2i  if4i - ff(i) > 2l


                                    15

while               8
                    >< u2k-4i+ff(i)u2i-ff(i)f4i_if4i - ff(i)   2k,
                                   22i-ff(i)
            g4k,4i=
                    >:  u2(k-i)f4i_
                         22(k-i)            if4i - ff(i) > 2k.

From these formulae we find that
                        8
                        >> g4k,4i         if4i - ff(i)   2l   2k,
                        >>
                        <

          u2k-2lg4l,4i=    24i-ff(i)-2lg4k,4iif2l < 4i - ff(i)   2k,
                        >>
                        >>
                        :  22k-2lg
                                  4k,4i   if2l < 2k < 4i - ff(i).


    Now let us calculate Al,k.

    When k = 0 we have z0 = (1 ^ _3)*(z0) = A0,0('0,0)*(z0) = A0,0z0 so that

A0,0= 1.

    When k = 1 we have

       P                            3
          l 1Al,1('1,l)*(z4)= (1 ^ _ )*(z4)


                           = ~4,4(1 ^ _3)*(g4,4) + 2~4,0(1 ^ _3)*(g4,0)



                           = ~4,4(9g4,4+ 2g4,0) + 2~4,0g4,0


and

        P
          l 1Al,k('1,l)*(z4)= A1,1z4 + A0,1~1,02g4,0


                           = A1,1(2~4,0g4,0+ ~4,4g4,4) + A0,1~1,02g4,0



which implies that A1,1= 9 and A0,1= ~-11,0(~4,4- 8~4,0) so that A0,12 Z*2.

    When k = 2 we have

 P                            3
    l 2Al,2('2,l)*(z8)= (1 ^ _ )*(z8)


                     = (1 ^ _3)*(~8,8g8,8+ 23~8,4g8,4+ 23~8,0g8,0)



                     = ~8,8(92g8,8+ 9 . 23g8,4) + 23~8,4(9g8,4+ g8,0) + 23~8,0g*
 *8,0


                                    16

and

  P
    l 2Al,2('2,l)*(z8)= A2,2z8 + A1,2('2,1)*(z8) + A0,2('2,0)*(z8)


                     = A2,2(~8,8g8,8+ 23~8,4g8,4+ 23~8,0g8,0)



                         +A1,2(~8,422u2z4) + A0,2(~8,023u4z0)



                     = A2,2(~8,8g8,8+ 23~8,4g8,4+ 23~8,0g8,0)



                         +A1,2~8,422(2~4,0g8,0+ ~4,4u2g4,4) + A0,2~8,023g8,0



                     = A2,2(~8,8g8,8+ 23~8,4g8,4+ 23~8,0g8,0)



                         +A1,2~8,422(2~4,0g8,0+ ~4,42g8,4) + A0,2~8,023g8,0.


Therefore we obtain


          ~8,8(92g8,8+ 9 . 23g8,4) + 23~8,4(9g8,4+ g8,0) + 23~8,0g8,0



          = A2,2(~8,8g8,8+ 23~8,4g8,4+ 23~8,0g8,0)



                  +A1,2~8,422(2~4,0g8,0+ ~4,42g8,4) + A0,2~8,023g8,0


which yields


                92 = A2,2,



                ~8,8. 9 + ~8,4(9 - 92) = A1,2~8,4~4,4,



                ~8,4+ ~8,0(1 - 92) = A1,2~8,4~4,0+ A0,2~8,0.


Hence A1,22 Z*2.

    Now assume that k   3 and consider the relation derived above


       ki=02fi(k,i)~4k,4i(1 ^ _3)*(g4k,4i)



      = Ak,k ki=02fi(k,i)~4k,4ig4k,4i

              P
            +   l<k  li=0Al,k~4k,4l22k-2l-ff(k)+ff(l)u2k-2l2fi(l,i)~4l,4ig4l,4i.


The coefficient of g4k,4kon the left side of this relation is equal to ~4k,4k9k*
 * and

on the right side it is Ak,k~4k,4kso that Ak,k= 9k for all k   3.  From the



                                    17

coefficient of g4k,4k-4we obtain the relation


     ~4k,4k9k-12 2(k)+3+ 23+ 2(k)~4k,4k-49k-1



     = 9k23+ 2(k)~4k,4k-4



           +Ak-1,k~4k,4k-422-ff(k)+ff(k-1)22~4k-4,4k-423+ 2(k)~4k,4k-49k-1



     = 9k23+ 2(k)~4k,4k-4



           +Ak-1,k~4k,4k-423+ 2(k)~4k-4,4k-4


which shows that Ak-1,k2 Z*2for all k   3. This means that we may conju-

gate A by the matrix D = diag(1, A1,2, A1,2A2,3, A1,2A2,3A3,4, . .).2 U1 Z2 to

obtain                    0                          1
                             1  1  c1,3 c1,4c1,5 . . .
                          B                          C
                          B                          C
                          B                          C
                          B  0  9   1   c2,4c2,5 . . .C
                          B                          C
                          B         2                C

              DAD-1 = C = BB 0  0  9    1   c3,5 . . .CC.
                          B                          C
                          B                          C
                          B              3           C
                          B  0  0   0   9    1   . . .C
                          B                          C
                          @                          A
                             .. .   .    .    .   .
                             .  ..  ..   ..   ..  ..


    In the next section we examine whether we can conjugate this matrix

further in U1 Z2 to obtain the matrix

                          0                      1
                             1  1  0   0  0  . . .
                          B                      C
                          B                      C
                          B                      C
                          B  0  9  1   0  0  . . .C
                          B                      C
                          B         2            C

                      B = BB 0  0  9   1  0  . . .CC.
                          B                      C
                          B                      C
                          B            3         C
                          B  0  0  0  9   1  . . .C
                          B                      C
                          @                      A
                             .. .  .   .   .  .
                             .  .. ..  ..  .. ..
                                    18

4     The Matrix Reloaded


4.1. Let B, C 2 U1 Z2 denote the upper triangular matrices which occurred

in x3.5
        0                      1        0                          1
           1  1  0   0  0  . . .           1  1  c1,3 c1,4c1,5 . . .
        B                      C        B                          C
        B                      C        B                          C
        B                      C        B                          C
        B  0  9  1   0  0  . . .C       B  0  9   1   c2,4c2,5 . . .C
        B                      C        B                          C
        B         2            C        B         2                C

    B = BB 0  0  9   1  0  . . .CC, C = BB 0  0  9    1   c3,5 . . .CC.
        B                      C        B                          C
        B                      C        B                          C
        B            3         C        B              3           C
        B  0  0  0  9   1  . . .C       B  0  0   0   9    1   . . .C
        B                      C        B                          C
        @                      A        @                          A
           .. .  .   .   .  .              .  .   .    .    .   .
           .  .. ..  ..  .. ..             .. ..  ..   ..   ..  ..


    The following result is the main result of this section.  Along with the

discussion of x3.5 it completes the proof of Theorem 1.1.


Theorem 4.2.  There exists an upper triangular matrix U 2 U1 Z2 such that

U-1 CU = B.


    Proof

    Let U have the form
      0                                                            1
         1        u1,2                   u1,3             u1,4 . . .
      B                                                            C
      B                                                            C
      B  0   1 + (9 - 1)u                u                u    . . .C
      B                  1,2              2,3              2,4     C
      B                                                            C
      B                                                            C
  U = BB 0         0         1 + (9 - 1)u1,2+ (92 - 9)u2,3u3,4 . . .CC.
      B                                                            C
      B                                                            C
      B  . . .     . . .                 . . .            . . .. . .C
      B                                                            C
      @                                                            A

         . . .     . . .                 . . .            . . . . . .


Then (UB)j,j= Uj,jBj,j= Cj,jUj,j= (CU)j,jand, in fact, (UB)j,j+1=

(CU)j,j+1for all j, too. For any 1 < s < j we have


   (UB)j-s,j= uj-s,j9j-1 + uj-s,j-1and



   (CU)j-s,j= 9j-s-1uj-s,j+ uj-s+1,j+ cj-s,j-s+2uj-s+2,j+ . .+.cj-s,juj,j.


In order to prove Theorem 4.2 it suffices to verify that we are able to solve

for the ui,jin the equations (UB)s,t= (CU)s,tfor all s   t inductively in

such in a manner such that, for every k, the first k-columns of the equality

UB = CU is achieved after a finite number of steps. Lemma 4.3 provides a

method which proceeds inductively on the columns of U.



                                    19

Lemma 4.3.  For j   3 and 1 < s < j, uj-s,j-1may be written as a linear

combination of uj-2,j-1, uj-3,j-1, . .,.uj-s+1,j-1and uj-1,j, uj-2,j, . .,.u1,j


    Proof

    We shall prove the result by induction on j. Consider the case j = 3, we

have the following equation:


    u3-s,392 + u3-s,2= 92-su3-s,3+ u4-s,3+ c3-s,5-su5-s,3+ . .+.c3-s,3u3,3


for 1 < s < 3 =) s = 2.  Hence substituting s = 2 gives



    u1,392 + u1,2= u1,3+ u2,3+ c1,3u3,3



    =) u1,2= (1 - 92)u1,3+ u2,3+ c1,3(1 + (9 - 1)u1,2+ (92 - 9)u2,3)



    =) (1 - (9 - 1)c1,3)u1,2= (1 - 92)u1,3+ (1 + (92 - 9))c1,3u2,3+ c1,3

and since (1 - (9 - 1)c1,3) is a 2-adic unit we can write u1,2as a Z2-linear

combination of u1,3and u2,3as required.



    We now need to show that if the lemma is true for j = 3, 4, . .,.k - 1 then

it is also true for j = k. This means we need to solve


       uk-s,k9k-1 + uk-s,k-1



       = 9k-s-1uk-s,k+ uk-s+1,k+ ck-s,k-s+2uk-s+2,k+ . .+.ck-s,kuk,k


for uk-s,k-1for 1 < s < k. This equation may be rewritten


   uk-s,k-1



   = (9k-s-1 - 9k-1)uk-s,k+ uk-s+1,k+ ck-s,k-s+2uk-s+2,k+ . .+.ck-s,kuk,k



   = (9k-s-1 - 9k-1)uk-s,k+ uk-s+1,k+ ck-s,k-s+2uk-s+2,k+ . . .



   . .+.ck-s,k(1 + (9 - 1)u1,2+ (92 - 9)u2,3+ . .+.(9k-1 - 9k-2)uk-1,k).


Now consider the case s = k - 1


      u1,k-1



      = (1 - 9k-1)u1,k+ u2,k+ c1,3u3,k+ . . .



      . .c.1,k(1 + (9 - 1)u1,2+ (92 - 9)u2,3+ . .+.(9k-1 - 9k-2)uk-1,k)
             ___________________________-z__________________________"
                                        B



                                    20

By repeated substitutions the bracket B may be rewritten as a linear combi-

nation of u1,k-1, u2,k-1, . .,.uk-2,k-1and uk-1,k. The important point to notice

about this linear combination is that the coefficient of u1,k-1will be an even

2-adic integer.  Hence, we can move this term to the left hand side of the

equation to obtain a 2-adic unit times u1,k-1equals a linear combination of

u1,k, u2,k, . .,.uk-1,kand u2,k-1, u3,k-1, . .,.uk-2,k-1as required.

    Now consider s = k - 2

     u2,k-1



     = (9 - 9k-1)u2,k+ u3,k+ c2,4u4,k+ . . .



     . .+.c2,k(1 + (9 - 1)u1,2+ (92 - 9)u2,3+ . .+.(9k-1 - 9k-2)uk-1,k).
              ___________________________-z__________________________"
                                         B0

    As   before   B0  can   be   written   as   a   linear   combination   of

u1,k-1, u2,k-1, . .,.uk-2,k-1, uk-1,kand from the case s = k - 1, u1,k-1may be

replaced by a linear combination of u1,k, . .,.uk-1,kand u2,k-1, . .,.uk-2,k-1.

Again the important observation is that the coefficient of u2,k-1is an even

2-adic integer, hence this term can be moved to the left hand side of the

equation to yield a 2-adic unit times u2,k-1equals a linear combination of

u1,k, u2,k, . .,.uk-1,kand u3,k-1, . .,.uk-2,k-1as required.

    Clearly this process may be repeated for s = k - 3, k - 4, . .,.2 to get a

2-adic unit times uk-s,k-1as a linear combination of u1,k, u2,k, . .,.uk-1,kand

uk-s+1,k-1, . .,.uk-2,k-1as required.
5     Applications


5.1. bu ^ bu

    Theorem 1.1 implies that, in the 2-local stable homotopy category there

exists an equivalence C0 2 Aut 0left-bu-mod(bu ^ bo) such that
                                   X           X
                  C0(1 ^ _3)C0-1 =     9k'k,k+     'k,k-1

                                   k 0         k 1

where 'k,lis as in x2.1, considered as left bu-endomorphism of bu ^ bo via the

equivalence ^Lof x2.1.

    In [7] use is made of an equivalence of the form bu  '  bo ^  -2CP 2,

first noticed by Reg Wood (as remarked in [1]) and independently by Don

Anderson (both unpublished).   This is easy to construct.   By definition

bu0( -2CP 2) ~= bu2(CP 2) ~= [CP 2, BU] and from the cofibration sequence

S0 - !  -2CP 2 -! S2 we see that bu0( -2CP 2) ~= Z   Z fitting into the

following exact sequence


             0 -! bu0(S2) -! bu0( -2CP 2) -! bu0(S0) -! 0.



                                    21

Choosing any stable homotopy class x :  -2CP 2 - ! bu restricting to the

generator of bu0(S0) yields an equivalence of the form

                                                  ~
                     bo ^ ( -2CP 2) c^x-!bu ^ bu -!  bu


in which c denoted complexification and ~ is the product.

    In the 2-local stable homootopy category there is a map


                           :  -2CP 2 -!  -2CP 2


which satisfies  *(z) = _3(z) for all z 2 bu0( -2CP 2). For example, take

to be 3-1 times the double desuspension of the restriction to the four-skeleton

of the CW complex CP 1 = BS1 of the map induced by z 7! z3 on S1, the

circle. With this definition there is a homotopy commutative diagram in the

2-local stable homotopy category

        bo ^  -2CP 2   _________________-     bo ^  -2CP 2
                              _3 ^

            |                                    |
            |                                    |
            |                                    |
            |                                    |
            |                                    |
            |                                    |
            |'                                  '|
            |                                    |
            |                                    |
            |                                    |
            |                                    |
            |                                    |
            |?                                   |?


                              _3
          bo           _________________-        bu



in which the vertical maps are equal given by the Anderson-Wood equivalence.

    Now suppose that we form the smash product with  -2CP 2 of the 2-local

left bu-module equivalence bu ^ bo ' _k 0bu ^ (F4k=F4k-1) to obtain a left

bu-module equivalence of the form


                 bu ^ bu ' _k 0bu ^ (F4k=F4k-1) ^  -2CP 2.


For l   k set


  ~k,l= 'k,l^   : bu ^ (F4k=F4k-1) ^  -2CP 2 -! bu ^ (F4l=F4l-1) ^  -2CP 2


then we obtain the following result.


Theorem 5.2.

    In the notation of x5.1, in the 2-local stable homotopy category, there exi*
 *sts

C0 2 Aut 0left-bu-mod(bu ^ bo) such that


                        1 ^ _3 : bu ^ bu -! bu ^ bu



                                    22

satisfies
                                       X           X
            (C0^ 1)(1 ^ _3)(C0^ 1)-1 =     9k~k,k+     ~k,k-1.

                                       k 0         k 1

5.3. Endleft-bu-mod(bu ^ bo)

    In this section we shall apply Theorem 1.1 to study the ring of left-bu-

module homomorphisms of bu ^ bo.  As usual we shall work in the 2-local

stable homotopy category.  Let U"1Z2 denote the ring of upper triangular,

infinite matrices with coefficients in the 2-adic integers. Therefore the group

U1 Z2 is a subgroup of the multiplicative group of units of "U1Z2. Choose a

left-bu-module homotopy equivalence of the form

                   ^L: _k 0bu ^ (F4k=F4k-1) -'! bu ^ bo,


as in x2.1. For any matrix A 2 "U1Z2 we may define a left-bu-module endo-

morphism of bu ^ bo, denoted by ~A , by the formula
                                  X
                       ~A = ^L. (      Al,k'k,l) . ^L-1.

                                 0 l k

Incidentally here and throughout this section we shall use the convention that

a composition of maps starts with the right-hand map, which is the opposite

convention used in the definition of the isomorphism   of x2.1 and [7]. When

A 2 U1 Z2 we have the relation ~A =  (A-1). For A, B 2 "U1Z2 we have
                     P                                 P
     ~A . ~B = (L^. (   0 l kAl,k'k,l) . ^L-1) . (L^. (   0 t sBt,s's,t) . ^L-1)


                    P
             = ^L. (   0 l t sAl,tBt,s's,l) . ^L-1


                    P
             = ^L. (   0 l s(AB)l,s's,l) . ^L-1



             = ~AB .

    By Theorem 1.1 there exists H 2 U1 Z2 such that


                             1 ^ _3 = ~HBH-1


for                       0                      1
                             1  1  0   0  0  . . .
                          B                      C
                          B                      C
                          B                      C
                          B  0  9  1   0  0  . . .C
                          B                      C
                          B         2            C

                      B = BB 0  0  9   1  0  . . .CC.
                          B                      C
                          B                      C
                          B            3         C
                          B  0  0  0  9   1  . . .C
                          B                      C
                          @                      A
                             .. .  .   .   .  .
                             .  .. ..  ..  .. ..



                                    23

Hence, for any integer u   1, we have 1 ^ (_3 - 9u-1) = ~HBuH-1  where

Bu = B - 9u-1 2 U"1Z2 and 9u-1 denotes 9u-1 times the identity matrix.

Following [5] write OEn : bo -! bo for the composition OEn = (_3 - 1)(_3 -

9) . .(._3 - 9n-1). Write Xn = B1B2 . .B.n2 "U1Z2.


Theorem 5.4. (i)  In the notation of x5.3 1 ^ OEn = ~HXnH-1  for n   1.

    (ii)  The first n-columns of Xn are trivial.

    (iii)  Let Cn = Cone(L^: _0 k n-1bu ^ (F4k=F4k-1) -'! bu ^ bo,, which is

a left-bu-module spectrum. Then in the 2-local stable homotopy category there

exists a commutative diagram of left-bu-module maps of the form
                                  1 ^ OEn
             bu ^ bo   _________________________-      bu ^ bo
                 Q Q                               j3
                     Q Q                        j j^
                   ssn  Q Q                 j j    OEn
                           Qs           j j

                                 Cn



where ssn is the cofibre of the restriction of ^L.  Also ^OEnis determined up to

homotopy by this diagram.

    (iv)  More precisely, for n   1 we have



                      (Xn)s,s+j= 0 ifj < 0 or j > n

and the other entries are given by the formula
                              X
              (Xn)s,s+t=                A(k1)A(k2) . .A.(kt)

                         1 k1<k2<...<kt n


where                        Q n
                     A(k1) =   j1=n-k1+1(9s-1 - 9j1-1),


                             Q n-k1-1
                     A(k2) =   j2=n-k2+1(9s - 9j2-1),


                             Q n-k2-1
                     A(k3) =   j3=n-k3+1(9s+1 - 9j3-1),


                          ..    .      .     .
                          .     ..     ..    ..


                             Q n-kt-1
                     A(kt) =   jt=1  (9s+t+1- 9jt-1).
                                    24

    Proof

    Part (i) follows immediately from the discussion of x5.3. Part (ii) follows

from part (iv), but it is simpler to prove it directly.  For part (ii) observe

that the Bi commute, being polynomials in the matrix B so that Xn  =

Xn-1Bn.  Since (Bn)s,tis zero except when t = s, s + 1 so that (Xn)i,j=

(Xn-1)i,j(Bn)j,j+ (Xn-1)i,j-1(Bn)j-1,j, which is zero by induction if j < n.

When j = n by induction we have (Xn)i,j= (Xn-1)i,n(Bn)n,nwhich is trivial

because (Bn)n,n = 9n-1 - 9n-1.  In view of the decomposition of bu ^ bo,

part (iii) amounts to showing that HXnH-1 corresponds to a left-bu-module

endomorphismm of _0 k bu ^ (F4k=F4k-1) which is trivial on each summand

bu ^ (F4k=F4k-1) with k   n - 1.  The (i, j)-th entry in this matrix is the

multiple of 'j-1,i-1: bu ^ (F4j-4=F4j-5) -! bu ^ (F4i-4=F4i-5) given by the

appropriate component of the map. The first n columns are zero if and only

if the map has no non-trivial components whose domain is bu ^ (F4j-4=F4j-5)

with j   n. Since H is upper triangular and invertible, the first n columns of

Xn vanish if and only if the same is true for HXnH-1 . Finally the formulae

of part (iv) result from the fact that Bj has 9m-1 - 9j-1 in the (m, m)-th

entry, 1 in the (m, m + 1)-th entry and zero elsewhere.


Remark 5.5. Theorem 5.4 is closely related to the main result of [5]. Follow-

ing [5] let bo(n)- ! bo denote the map of 2-local spectral which is universal f*
 *or

all maps X -! bo which are trivial with respect to all higher Z=2-cohomology

operations of order less than n. Cf with [5] Theorem B. Milgram shows that

OE2n factorises through a map of the form `2n : bo -!  8nbo(2n-ff(n))and that

OE2n+1 factorises through a map of the form `2n+1 : bo -!  8n+4bsp(2n-ff(n))

and then uses the `m 's to produce a left-bo-module splitting of bo ^ bo. Using

the homotopy equivalence bu ' bo ^  -2CP 2 mentioned in [7] one may pass

from the splitting of bu ^ bo to that of bo ^ bo (and back again). In the light

of this observation, the existence of the diagram of Theorem 5.4 should be

thought of as the upper triangular matrix version of the proof that the `n's

exist. The advantage of the matrix version is that Theorem 5.4(iv) gives us

every entry in the matrix Xn, not just the zeroes in the first n columns.
References


  [1] J.F. Adams: Stable Homotopy and Generalised Homology; University of

      Chicago Press (1974).


  [2] E.H. Brown and F.P. Peterson: On the stable decomposition of  2Sr+2;

      Trans. Amer. Math. Soc. 243 (1978) 287-298.


  [3] F. Clarke, M. D. Crossley and S. Whitehouse: Bases for cooperations

      in K-theory; K-Theory (3) 23 (2001) 237-250.



                                    25

[4] M. Mahowald: bo-Resolutions; Pac. J. Math. (2) 92 (1981) 365-383.


[5] R.J. Milgram:  The Steenrod algebra and its dual for connective K-

    theory; Reunion Sobre Teoria de Homotopia, Universidad de North-

    western Soc. Mat. Mex. (1974) 127-159.


[6] V.P. Snaith: A stable decomposition of  nSnX; J. London Math. Soc.

    2 (1974) 577-583.


[7] V.P. Snaith:  The upper triangular group and operations in algebraic

    K-theory; Topology 41 (2002) 1259-1275.
                                  26