ROOT INVARIANTS IN THE ADAMS SPECTRAL SEQUENCE



                           MARK BEHRENS1


       Abstract.Let E be a ring spectrum for which the E-Adams spectral se-
       quence converges. We define a variant of Mahowald's root invariant calle*
 *d the
       `filtered root invariant' which takes values in the E1term of the E-Adam*
 *s spec-
       tral sequence. The main theorems of this paper concern when these filter*
 *ed
       root invariants detect the actual root invariant, and explain a relation*
 *ship
       between filtered root invariants and differentials and compositions in t*
 *he E-
       Adams spectral sequence. These theorems are compared to some known com-
       putations of root invariants at the prime 2. We use the filtered root in*
 *variants
       to compute some low dimensional root invariants of v1-periodic elements *
 *at
       the prime 3. We also compute the root invariants of some infinite v1-per*
 *iodic
       families of elements at the prime 3.


                              Contents

  1.  Introduction                                                    1
  2.  Filtered Tate spectra                                             6
  3.  Definitions of various forms of the root invariant                      8
  4.  The Toda bracket associated to a complex                          10
  5.  Statement of results                                             14
  6.  Proofs of the main theorems                                      17
  7.  bo resolutions                                                   25
  8.  The algebraic Atiyah-Hirzebruch spectral sequence                   26
  9.  Procedure for low dimensional calculations of root invariants           34
  10.  BP -filtered root invariants of some Greek letter elements             36
  11.  Computation of R(fi1) at odd primes                              43
  12.  Low dimensional computations of root invariants at p = 3            45
  13.  Algebraic filtered root invariants                                  50
  14.  Modified filtered root invariants                                  53
  15.  Computation of some infinite families of root invariants at p = 3       *
 *58
  References                                                         62



                           1.Introduction

  In his work on metastable homotopy groups [22], Mahowald introduced an in-
variant that associates to every element ff in the stable stems a new element R*
 *(ff)
called the root invariant of ff. The construction has indeterminacy and so R(ff*
 *) is
____________
   2000 Mathematics Subject Classification. Primary 55Q45; Secondary 55Q51, 55T*
 *15.
   1The author is partially supported by the NSF.

                                  1




2                           MARK BEHRENS


in general only a coset. The main result of [25] indicates a deep relationship *
 *be-
tween elements of the stable homotopy groups of spheres which are root invarian*
 *ts
and their behavior in the EHP spectral sequence. Mahowald and Ravenel conjec-
ture in [24] that, loosely speaking, the root invariant of a vn-periodic elemen*
 *t is
vn+1-periodic.  Thus the root invariant is related simultaneously to unstable a*
 *nd
chromatic phenomena.
  The conjectural relationship between root invariants and the chromatic filtra*
 *tion
is based partly on computational evidence. For instance, at the prime 2, we have
[25], [18]                    8

                              >>>ff4t   i = 4t
                              <ff       i = 4t + 1
                      R(2i) = >  4t+1
                              >>:ff4t+1ff1i = 4t + 2
                               ff4t+1ff21i = 4t + 3

while at odd primes we have [35], [25]

                              R(pi) = ffi

demonstrating that the root invariant sends v0-periodic families to v1-periodic*
 * fam-
ilies. For p   5 it is also known [35], [25] that

                              fii2 R(ffi)

and
                            fip=22 R(ffp=2).

These computations led the authors of [25] to regard the root invariants as def*
 *ining
the nth Greek letter elements as the root invariants of the (n - 1)stGreek lett*
 *er
elements when the relevant Smith-Toda complexes do not exist.
  Other evidence of the root invariant raising chromatic filtration is seen in *
 *the
cohomology of the Steenrod algebra. Mahowald and Shick define a chromatic fil-
tration on ExtA(F2, F2) in [26], and Shick proves that an algebraic version of *
 *the
root invariant increases chromatic filtration in this context in [36].
  The Tate spectrum computations of [11], [10], [27], [16], [2], and [15] indic*
 *ate
that the Z=p-Tate spectrum of a vn-periodic cohomology theory is vn-torsion. The
root invariant is defined using the Tate spectrum of the sphere spectrum, so the
results of the papers listed above provide even more evidence that the root inv*
 *ariant
of a vn-periodic element should be vn-torsion.
  The purpose of this paper is to introduce a new variant of the root invariant
called the filtered root invariant. We apply the filtered root invariant toward*
 *s the
computation of some new root invariants at the prime 3 in low dimensions. We al*
 *so
compute the root invariants of some infinite v1-families as infinite v2-familie*
 *s at the
prime 3. We will also describe how the theory of filtered root invariants given*
 * in
this paper works out to give an alternative perspective on known computations of
root invariants at the prime 2.
  Let X be a finite p-local spectrum, and let E be a ring spectrum for which the
E-Adams spectral sequence (E-ASS) converges.  Given ff 2 sst(X), we define its

filtered root invariants R[k]E(ff) to be a sequence of cosets of dr-cycles in t*
 *he Ek,*1
term of the E-ASS converging to ss*(X), where r depends on k. These invariants
govern the passage between the E-root invariant RE (ff), or the algebraic root *
 *in-
variant Ralg(ff), and the elements in the E-ASS that detect R(ff). The passage *
 *of
each of these filtered root invariants to the next is governed by differentials*
 * and




            ROOT INVARIANTS IN THE ADAMS SPECTRAL SEQUENCE           3


compositions in the E-ASS. Our method is to use algebraic and E-cohomology
computations to determine the first of these filtered root invariants and then *
 *itera-
tively deduce the higher ones from differentials and compositions. Under favora*
 *ble
circumstances, the last of these filtered root invariants will detect the root *
 *invariant.
  In Section 2, we discuss the E-Adams resolution, and various filtered forms of
the Tate spectrum which we shall be using to define the the filtered root invar*
 *iants.
  In Section 3, we define the filtered root invariants. We also recall the defi*
 *nitions
of the root invariant, the E-root invariant, and the algebraic root invariant. *
 * We
will sometimes refer to the root invariant as the "homotopy root invariant" to
distinguish it from all of the other variants being used.
  Our main results are most conveniently stated using the language of Toda brac*
 *k-
ets introduced in the appendix of [37]. In Section 4, we define a variant of th*
 *e Toda
bracket, the K-Toda bracket, which is taken to be an attaching map in a fixed
finite CW-complex K.  Some properties of K-Toda brackets are introduced.  We
also define a version on the Er term of the E-ASS.
  In Section 5, we state the main results which relate the filtered root invari*
 *ants to
root invariants, Adams differentials, compositions, E-root invariants, and alge*
 *braic
root invariants. If R[k]E(ff) contains a permanent cycle x, then x detects an e*
 *lement

of R(ff) modulo Adams filtration k + 1. If R[k]E(ff) does not contain a permane*
 *nt
cycle, there is a formula relating its E-Adams differential to the next filtere*
 *d root
invariant. Thus if one knows a filtered root invariant, one may sometimes deduce
the next one from its E-Adams differential.  There is a similar result concerni*
 *ng
compositions.  The zeroth filtered root invariant R[0]E(ff)_is the E-root_invar*
 *iant._

The first filtered root invariant R[1]E(ff) is the E ^ E-root invariant (E is t*
 *he fiber
of the unit of E). If E = HFp, then the first non-trivial filtered root invaria*
 *nt is
the algebraic root invariant Ralg(ff).
  We present the proofs of the main theorems of Section 5 in Section 6. The pro*
 *ofs
are technical, and it is for this reason that they have been relegated to their*
 * own
section.
  In Section 7 we give examples of these theorems with E = bo, and show how the
bo resolution computes the root invariants of the elements 2k at the prime 2. T*
 *his
was the motivating example for this project.
  We will need to make some extensive computations in an algebraic Atiyah-
Hirzebruch spectral sequence (AAHSS). This is the spectral sequence obtained by
applying ExtBP*BP(BP*, BP*(-)) to the cellular filtration of projective space. *
 *We
introduce this spectral sequence in Section 8. We compute the dr differentials *
 *for
small r using formal group methods.
  Calculating homotopy root invariants from our filtered root invariants is a d*
 *eli-
cate business. In an effort to make our low dimensional calculations easier to *
 *follow
and less ad hoc, we spell out our methodology in the form of Procedure 9.1 which
we follow throughout our low dimensional calculations.  The description of this
procedure is the subject of Section 9.
  In Section 10, we compute some BP -filtered root invariants of the elements pi
as well as of the elements ffi=j. We find that at every odd prime p



                             ffi2 R[1]BP(pi)

                            fii=j2 R[2]BP(ffi=j)




4                           MARK BEHRENS


These filtered root invariants hold at the prime 2 modulo an indeterminacy which
is identified, but not computed, in this paper. The only exceptions are the cas*
 *es
i = j = 1 and i = j = 2 at the prime 2 (these cases correspond to the existence*
 * of the
Hopf invariant 1 elements   and oe). These filtered root invariants are compute*
 *d by
means of manipulation of formulas in BP*BP arising from p-typical formal groups.
  In Section 11 we compute the root invariants

                              R(fi1) = fip1

at primes p > 2. These root invariants were announced without proof in [25]. Th*
 *is
computation is accomplished by applying our theorems to the Toda differential in
the Adams-Novikov spectral sequence (ANSS).
  In Section 12, we apply the methods described to compute some root invariants
of the Greek letter elements ffi=jthat lie within the 100-stem at the prime 3. *
 *At
the prime 3, fii is known to be a permanent cycle for i   0, 1, 2, 5, 6 (mod  9*
 *) [4]
and is conjectured to exist for i   3 (mod  9). The element fi3 is a permanent *
 *cycle.
One might expect from the previous work for p   5 that fii2 R(ffi) when fiiexis*
 *ts.
Surprisingly, there is at least one instance where fii exists, yet is not conta*
 *ined in
R(ffi). Our low dimensional computations of R(ffi=j) at p = 3 are summarized in
Table 1.


          Table 1. Low dimensional root invariants of ffi=jat p = 3

                       ________________________
                       _Element_|Root_Invariant_
                           ff1   |    fi1
                           ff2   |   fi21ff1
                          ff3=2  |   -fi3=2
                           ff3   |    fi3
                           ff4   |     fi51
                           ff5   |    fi5
                          ff6=2  |    fi6=2
                       ____ff6___|____-fi6_____



  All of these root invariants are v2-periodic in the sense that they are detec*
 *ted
in ss*(L2S0) [34].  A similar phenomenon happens at the prime 2 with the root
invariants of 2i: they are not all given by the Greek letter elements ffi, but *
 *the
elements R(2i) are nevertheless v1-periodic [25].  The lesson we learn is that *
 *if
one believes that the homotopy Greek letter elements should be determined by
iterated root invariants (as suggested in [25]) then they will not always agree*
 * with
the algebraic Greek letter elements, even when the latter are permanent cycles.
  These results will be partly generalized to compute R(ffi) for i   0, 1, 5 (m*
 *od  9)
at the prime 3 in Section 15. The remainder of this paper is devoted to providi*
 *ng
the machinery necessary for this computation.
  In the ANSS the fi family lies in low Adams-Novikov filtration, but in the ASS
this family is in high filtration.  For the purposes of infinite chromatic fami*
 *lies,
it is often useful to take both spectral sequences into account simultaneously.*
 * In
Section 13 we explain how our framework can be applied to the Mahowald spectral
sequence to compute algebraic root invariants. As mentioned earlier, algebraic *
 *root
invariants are the first non-trivial HFp filtered root invariants.




            ROOT INVARIANTS IN THE ADAMS SPECTRAL SEQUENCE           5


  Infinite families of Greek letter elements are constructed as homotopy classes
through the use of Smith-Toda complexes. In their computation of R(ffi) for p  *
 * 5,
Mahowald and Ravenel introduce modified root invariants [25] which take values
in the homotopy groups of certain Smith-Toda complexes. In Section 14 we adapt
our results to modified root invariants.
  Our modified root invariant methods are applied in Section 15 to make some
new computations of the root invariants of some infinite v1-periodic families a*
 *t the
prime 3. Specifically, we are able to show that

                           (-1)i+1fii2 R(ffi)

for i   0, 1, 5 (mod  9).
  This paper represents the author's dissertation work.  The author would like
to extend his heartfelt gratitude to his adviser, J. Peter May, for his guidance
and encouragement, and to Mark Mahowald, for many enlightening conversations
regarding the contents of this paper.

Conventions. Throughout this paper we will be working in the stable homotopy
category localized at some prime p. We will always denote the quantity q = 2(p-*
 *1),
as usual. All ordinary homology will be taken with Fp coefficients. If p = 2, l*
 *et PNM
denote the stunted projective space with bottom cell in dimension N and top cell
in dimension M. Here M and N may be infinite or negative. See [25] for details.*
 * If
p is odd, then projective space is replaced by B p. The complex B p has a stable
cell in every positive dimension congruent to 0 or -1 mod q, and we will use the
notation PNM to indicate the stunted complex with cells in dimensions between N
and M.  When M = 1, or when N = -1 the superscript or subscript may be
omitted.
  Given a spectrum E, the Tate spectrum

                     (E ^ P )-1  =  holim(E ^ P-n )

will be denoted tE. To relate this notation to that in [15], we have

                            tE = t('*E)Z=p.

There is a unit S0 ! tE. For E = S0, this is the inclusion of the 0-cell. If X *
 *is a
finite complex, then the Segal conjecture for the group Z=p [3], [9], [32] (als*
 *o known
as Lin's Theorem [20], [21] at p = 2 and Gunawardena's Theorem [17] for p > 2)
implies that the map

(1.1)                 X = X ^ S0 ! X ^ tS0 = tX

is p-completion (the last equality requires X to be finite).
  Suppose A and B are two subsets of a set C. We shall write A \=B to indicate
that A \ B is nonempty. This is useful notation when dealing with operations wi*
 *th
indeterminacy.  If we are working over a ring R, we shall use the notation =.to
indicate that two quantities are equal modulo multiplication by a unit in Rx . *
 *We

shall similarly use the notation 2 for containment up to multiplication by a un*
 *it.
  We will denote the regular ideal (p, v1, v2, . .,.vn-1)   BP* by In.
  Finally, we will be using the following abbreviations for spectral sequences.

    ASS:  The classical Adams spectral sequence.
    ANSS:   The Adams-Novikov spectral sequence derived from BP .
    E-ASS:  The generalized Adams spectral sequence derived from a ring spec-
       trum E.




6                           MARK BEHRENS


    AHSS:   The Atiyah-Hirzebruch spectral sequence. We will be using the form
       that computes stable homotopy groups from homology.
    AAHSS:   The algebraic Atiyah-Hirzebruch spectral sequence, which uses the
       cellular filtration to compute Ext(X).
    MSS:  The Mahowald spectral sequence, which computes Ext(X) by applying
       Ext(-) to an Adams resolution of X.



                      2. Filtered Tate spectra

  Given a ring spectrum E, we will establish some notation for dealing with the*
 * E-
Adams resolution. We will then mix the Adams filtration with the skeletal filtr*
 *ation
in the Tate spectrum tS0. These filtered Tate spectra will carry the filtered r*
 *oot
invariants defined in Section 3.  Our treatment of the Adams resolution follows
closely that of Bruner in [8,_IV.3]._
  For E a ring spectrum, let E be the fiber of the unit, so there is a cofiber *
 *sequence

                             __   0 j
                             E ! S  -!E.
                                                          __(k)
For X a spectrum, let Wk(X) denote the k-fold smash power E   ^ X. We shall
also use the notation Wkl(X) to denote the cofiber


                    Wk+l+1(X) ! Wk(X) ! Wkl(X).


We may drop the X from the notation when X = S0. Note that with our definitions
Wkk-1(X) ' *. The E-Adams resolution of X now takes the form


      X ______W0(X) oo___W1(X)  oo___W2(X)  oo___W3(X)  oo___. . .
                |           |           |           |
                |           |           |           |
                fflffl|     fflffl|     fflffl|     fflffl|
             W00(X)      W11(X)      W22(X)      W33(X)


The notation Wkl(X) is used because the E-ASS for Wkl(X) is obtained from the
E-ASS for X by setting Es,t1= 0 for s < k and s > l, and adjusting the differ-
entials accordingly. If the resolution converges to the p-completion (p-localiz*
 *ation)
of X, then W1 (X) ' * in the p-complete (p-local) stable homotopy category, and
Ws1(X) ' Ws(X).
  We shall denote Er(X) for the Er term of the E-ASS for X.  An element
of Es,t1(X) = sst-s(Wss(X)) is a dr-cycle if and only if it lifts to an element*
 * of
sst-s(Wss+r).  Given an element ff 2 ssn(X) we shall let filtE (ff) denote its *
 *E-
Adams filtration.
  In what follows, the reader may find it helpful to assume that the spectra Ws
are CW spectra and the maps

                             Ws ! Ws-1

are the inclusions of subcomplexes. If this is the case, then in what follows t*
 *he ho-
motopy colimits may simply regarded as unions. As Bruner points out [8, IV.3.1],
this assumption represents no loss of generality, since any infinite tower may *
 *be
replaced with a tower of inclusions of CW-spectra through the use of CW ap-
proximation and mapping telescopes. The Tate spectrum  -1tS0 is bifiltered, as




            ROOT INVARIANTS IN THE ADAMS SPECTRAL SEQUENCE           7


depicted in the following diagram.

               ..             .              .
               .OO            ..OO           ..OO
                |             |              |
                |             |              |
                |             |              |
           W0(P N+1)Ooo___W1(PON+1)Ooo___W2(PON+1)Ooo___.O. .

                |             |              |
                |             |              |
                |             |              |
            W0(PON)Ooo_____W1(PON)Ooo_____W2(PON)Ooo____. . .

                |             |              |
                |             |              |
                |             |              |
           W0(P N-1)Ooo___W1(PON-1)Ooo___W2(PON-1)Ooo___.O. .

                |             |              |
                |             |              |
                |             |              |
               ..             .              .
               .              ..             ..

In the above diagram, Wk(P N) is the spectrum Wk(P N)-1  = holimM Wk(P-NM),
where the homotopy inverse limit is taken after smashing with Wk. We emphasize
that this is in general quite different from what one obtains if one smashes wi*
 *th Wk
after taking the homotopy inverse limit.
  Given increasing sequences of integers

                       I = {k1 < k2 < . .<.kl}

                       J = {N1 < N2 < . .<.Nl}

we define subsets S(I, J) of Z x Z by

                             [l
                   S(I, J) =   {(a, b) : a   ki, b   Ni}.
                             i=1

We give the set of all multi-indices (I, J) the structure of a poset by declari*
 *ng
(I, J)   (I0, J0) if and only if S(I, J)   S(I0, J0).

Definition 2.1 (Filtered Tate spectrum). Given sequences

                       I = {k1 < k2 < . .<.kl}

                       J = {N1 < N2 < . .<.Nl}

with ki  0, we define the filtered Tate spectrum (of the sphere) as the homotopy
colimit                           G

                        WI(P J) =   Wki(P Ni).
                                   i
We allow for the possibility of Nl = 1.  More generally, given another pair of
sequences (I0, J0)   (I, J), we define spectra

                 0  J          i        J0-1        J j
              WII(PJ0) = cofiberWI0+1(P    ) ! WI(P  )


where I0+1 (respectively J0-1) is the sequence obtained by increasing (decreasi*
 *ng)
every element of the sequence by 1.




8                           MARK BEHRENS
                  Figure 1. The filtered Tate spectrum


  Figure 1 displays a diagram of S(I, J) intended to help the reader visualize *
 *the
filtered Tate spectrum. The entire Tate spectrum is represented by the right ha*
 *lf-
plane.  The shaded region S(I, J) is the portion represented by the filtered Ta*
 *te
spectrum WI(P J).

        3.Definitions of various forms of the root invariant

  In this section we will recall the definitions of the Mahowald root invariant*
 *, the
E-root invariant, and the algebraic root invariant. We will then define the fil*
 *tered
root invariants.

Definition 3.1 (Root invariant ). Let X be a finite complex, and ff 2 sst(X). T*
 *he
root invariant (also called the Mahowald invariant) of ff is the coset of all d*
 *otted
arrows making the following diagram commute.

                          St _____//___________ -N+1 X

                           |ff|        ||
                           fflffl|     ||

                          X            |
                           |           |
                           |           |
                           fflffl|     fflffl|
                          tX ____//_ P-N ^ X

This coset is denoted R(ff).  Here the map X ! tX p-completion by the Segal
conjecture (1.1), and N is chosen to be minimal such that the composite St !
 P-N ^ X is non-trivial.

  The root invariant is difficult to compute because it involves knowing the ho*
 *mo-
topy groups of the finite complex X and of X ^ P-N . For this reason, Mahowald
and Ravenel [25] introduced a calculable approximation to the root invariant ca*
 *lled
the E-root invariant, for E a ring spectrum.  We find it useful to generalize to
arbitrary spectra and to elements of ss*(E).




            ROOT INVARIANTS IN THE ADAMS SPECTRAL SEQUENCE           9


Definition 3.2 (E-root invariant ). Let E be a spectrum. Let x 2 sst(E). Define
the E-root invariant of x to be the coset RE (ff) of dotted arrows making the
following diagram commute.

                          St _____//___________ -N+1 E

                          x||          ||
                           fflffl|     ||

                           E           |
                           |           |
                           |           |
                           fflffl|     fflffl|
                          tE _____// E ^ P-N

It is quite possible that the composite St ! tE is trivial.  If this is the cas*
 *e the
E-root invariant is said to be trivial. Otherwise, in the diagram above we choo*
 *se
N be minimal such that the composite St !  E ^ P-N  is non-trivial.
  If E is a ring spectrum, and ff 2 ss*(X) for X a finite spectrum, let x = h(f*
 *f) 2
E*(X) be the Hurewicz image of ff. We will then refer to the E ^ X-root invaria*
 *nt
of h(ff) simply as the E-root invariant of ff. Therefore, by abuse of notation,*
 * we
have
                         RE (ff) = RE^X (h(ff)).

  Finally, one may define root invariants in Ext. These are called algebraic ro*
 *ot
invariants.

Definition 3.3 (Algebraic root invariant ). Let ff be an element of Exts,t(H*X).
We have the following diagram of Ext groups which defines the algebraic root
invariant Ralg(ff).

                                Ralg(-)    s,t+N-1
                Exts,t(H*X)  ///o//o///o//oooooExt(H*X)

                    f||                      'N ||
                     fflffl|                    fflffl|
            Exts,t-1(H*P-1  ^ X)___N__//Exts,t-1(H*P-N ^ X)

Here f is induced by the inclusion of the -1-cell of P-1 ,  N is the projection*
 * onto
the -N-coskeleton, 'N is inclusion of the -N-cell, and N is minimal with respec*
 *t to
the property that  N O f(ff) is non-zero. Then the algebraic root invariant Ral*
 *g(ff)
is defined to be the coset of lifts fl 2 Exts,t+N-1(H*X) of the element  N O f(*
 *ff).

  We wish to extend these definitions to a sequence of filtered root invariants
that appear in the E-Adams resolution. Suppose that X is a finite complex and
ff 2 sst(X).  We want to lift ff over the smallest possible filtered Tate spect*
 *rum
(Definition 2.1). To this end, we shall describe a pair of sequences

                     I = {k1 < k2 < . .<.kl}

                     J = {-N1 < -N2 < . .<.-Nl}

associated to ff, which we define inductively. Let k1   0 be maximal such that *
 *the
composite
               St-1 ff-! -1X !  -1tX ! W0k1-1(P ^ X)-1

is trivial. Next, choose N1 to be maximal such that the composite

          St-1 ff-! -1X !  -1tX ! W0(k1-1,k1)(P(-N1+1,1)^ X)




10                          MARK BEHRENS


is trivial. Inductively, given

                        I0= (k1, k2, . .,.ki)

                       J0 = (-N1, -N2, . .,.-Ni)

let ki+1be maximal so that the composite
                                      0-1,ki+1-1)
          St-1 ff-! -1X !  -1tX ! W0(I        (P(J0+1,1)^ X)

is trivial. If there is no such maximal ki+1, we declare that ki+1= 1 and we are
finished. Otherwise, choose Ni+1to be maximal such that the composite
                                0-1,ki+1-1,ki+1)
    St-1 ff-! -1X !  -1tX ! W0(I           (P(J0+1,-Ni+1+1,1)^ X)

is trivial, and continue the inductive procedure. We shall refer to the pair (I*
 *, J) as
the E-bifiltration of ff.
  Observe that there is an exact sequence

           sst-1(WI(P J^ X)) ! sst(tX) ! sst-1(W I-1(PJ+1 ^ X)).

Our choice of (I, J) ensures that the image of ff in sst-1(W I-1(PJ+1^X)) is tr*
 *ivial.
Thus ff lifts to an element fff2 sst-1(WI(P J^ X)).

Definition 3.4 (Filtered root invariants ). Let X be a finite complex, let E a *
 *ring
spectrum such that the E-Adams resolution converges, and let ff be an element of
sst(X) of E-bifiltration (I, J). Given a lift fff2 sst-1(WI(P J^ X)), the kthfi*
 *ltered
root invariant is said to be trivial if k 6= ki for any ki2 I. Otherwise, if k *
 *= ki for
some i, we say that the image fi of fffunder the collapse map

                 sst-1(WI(P J^ X)) ! sst-1(Wkkii( -NiX))

is an element of the kth filtered root invariant of ff. The kth filtered root i*
 *nvariant
is the coset R[k]E(ff) of Ek,t+k+Ni-11(X) of all such fi as we vary the lift ff*
 *f.

Remark 3.5. Let ri denote the difference ki+1- ki. Then there is a factorization

    sst-1(WI(P J^ X)) ! sst-1(Wkki+1-1i( -NkiX)) ! sst-1(Wkkii( -NkiX))


Thus any such fi 2 R[k]E(ff) is actually a dr-cycle for r < ri.

  Figure 2 gives a companion visualization to Figure 1, by displaying the bifil*
 *tra-
tions of the filtered root invariants in the filtered Tate spectrum.


            4.The Toda bracket associated to a complex

  In this section we introduce a variant of the Toda bracket.  This treatment is
essentially a specialization of the treatment of Toda brackets given in the app*
 *endix
of [37]. Suppose K is a finite CW-spectrum with one bottom dimensional cell and
one top dimensional cell. Suspend K accordingly so that it is connective and n-
dimensional with one cell in dimension zero and one cell in dimension n. For so*
 *me
other spectrum X we shall define the K-Toda bracket to be an operator which, wh*
 *en
defined, takes an element of sst(X) to a coset of sst+n-1(X). We shall present *
 *a dual
definition, and show this dual definition is equivalent to our original definit*
 *ion. We
will also define a variant on the Er term of the E-ASS.
  In what follows, we let Kj be the j-skeleton of K, and let Kjibe the quotient
Kj=Ki-1. We shall omit the top index for the i-coskeleton Ki= K=Ki-1.




            ROOT INVARIANTS IN THE ADAMS SPECTRAL SEQUENCE           11
       Figure 2. The the bifiltration of the filtered root invariants in the
       filtered Tate spectrum.



Definition 4.1 (K-Toda bracket ). Let

                            f :  -1K1 ! S0

be the attaching map of the 1-coskeleton of K to the 0-cell, so that the cofibe*
 *r of f
is K. Let   : K1 ! Sn be the projection onto the top cell. Suppose ff is an ele*
 *ment
of sst(X). We have


                 sst(X) - *sst+n(X ^ K1) f*-!sst+n-1(X).

We say the K-Toda bracket

                          <K>(ff)   sst+n-1(X)

is defined if ff is in the image of  *. Then the K-Toda bracket is the collecti*
 *on of all
f*(fl) 2 sst+n-1(X) where fl 2 sst+n(X ^ K1) is any element satisfying  *(fl) =*
 * ff.
If <K>(ff) contains only one element, we will say that it is strictly defined.

Remark 4.2. If the (n - 1)-skeleton Kn-1 is coreducible, then the attaching map
f factors as a composite

                        f :  -1K1 -!Sn-1 ,-!S0.

Then the product , . ff is in <K>(ff).

  We remark on the relationship to Shipley's treatment of Toda brackets in tria*
 *n-
gulated categories given in the appendix of [37]. In Shipley's terminology, if *
 *K is
an m-filtered object in {f1, f2, . .,.fm-1 } with F1K ' S0 and Fm K=Fm-1 K ' Sn,
then we have the equality (modulo indeterminacy)

                      <K>(ff) \=<f1, f2, . .f.m-1, ff>.

In particular, if we take K to be the 3-filtered object

                          *   S0   Kn-1   K




12                          MARK BEHRENS


with attaching maps


                          f1 : -1Kn-11! S0

                          f2 :Sn-1 ! Kn-11


then we have the equality (modulo indeterminacy)


                          <K>(ff) \=<f1, f2, ff>.


The reason we don't have exact equalities is that with the Toda bracket, you can
get more indeterminacy by varying the m-filtered object, whereas the m-filtered
object is fixed in the case of the K-Toda bracket. However, the K-Toda bracket
can also get indeterminacy that is not seen by the Toda bracket if X 6= S0, bec*
 *ause
there are potentially more choices of lift fl if you smash with X.
  In some of our applications the following dual variant will be more natural to
work with. Our use of the word `dual' stems from the use the skeletal instead of
coskeletal filtration of K.


Definition 4.3 (Dual definition of the K-Toda bracket ). Let


                           g : Sn-1 ! Kn-1


be the attaching map of the n-cell of K to the (n - 1)-skeleton, so that the co*
 *fiber
of g is K. Let ' : S0 ! Kn-1 be the inclusion of the bottom cell. Suppose ff is*
 * an
element of sst(X). We have


                sst(X) g*-!sst+n-1(X ^ Kn-1) -'*sst+n-1(X)


We say the (dual) K-Toda bracket


                          <K>(ff)   sst+n-1(X)


is defined if g*(ff) is in the image of '*. Then the K-Toda bracket is the coll*
 *ection
of all fl 2 sst+n-1(X) where '*(fl) = g*(ff).


  We reconcile our use of the same notation in both definitions with the follow*
 *ing
lemma.
Lemma 4.4.  The K-Toda bracket and the dual K-Toda bracket are equal.


Proof.Let ff, f, g, ', and   be the maps given in our two definitions of the K-*
 *Toda
bracket. For the purposes of this proof, we shall refer to the dual K-Toda brac*
 *ket
as <K>d(ff). The map of cofiber sequences makes the following diagram commute.


                                          f
                    S0______//K_____//K1______//S1

                   ' ||      ||||     ||        '||
                     fflffl| ||       fflffl|   fflffl|
                  Kn-1  ____//_K____//Sn_g__// Kn-1


Taking the last square in the above triangle, extending it to a map of cofiber
sequences the other way, and smashing with X, we get the following commutative




            ROOT INVARIANTS IN THE ADAMS SPECTRAL SEQUENCE           13


diagram whose columns are exact.

                 sst+n(X ^ Kn-11)______sst+n(X ^ Kn-11)

                      '0*||                  |f0*|
                        fflffl|   f*         fflffl|
                  sst+n(X ^ K1)_________//_sst+n-1(X)

                      * ||                   |'*|
                        fflffl|              fflffl|
                     sst(X)____g*____//_sst+n-1(X ^ Kn-1)

                     g0*||                   ||0*
                        fflffl|              fflffl|
                sst+n-1(X ^ Kn-11)____sst+n-1(X ^ Kn-11)

Here '0is the inclusion and  0 is the projection, and f0 and g0are determined f*
 *rom
the diagram. The bracket <K>(ff) is computed by taking the preimage under  * and
then applying f*. The bracket <K>d(ff) is computed applying g* and then taking
the preimage over '*. We see that for any lift fl 2 sst+n(X ^ K1) of ff, f*(fl)*
 * is a lift
of g*(ff). It follows that we have <K>(ff)   <K>d(ff).
  For the reverse containment, suppose that fi 2 sst+n-1(X) is a lift of g*(ff)*
 *. Then
fi is an arbitrary element of <K>d(ff). Let fl 2 sst+n(X ^ K1) be any lift if f*
 *f. Such
a lift exists since
                     g0*(ff) =  0*g*(ff) =  0*'*(fi) = 0.

We must add a correction term to fl since it is not necessarily the case that f*
 **(fl) =
fi.  Let ffi be the difference f*(fl) - fi.  Then '*(ffi) = 0, so there is a li*
 *ft to effi2
sst+n(X ^ Kn-11). Then fl0= fl - '0*(effi) is another lift of ff, and f*(fl) = *
 *fi. We have
therefore proven that <K>d(ff)   <K>(ff).

Remark 4.5. Perhaps a more conceptual way to prove Lemma 4.4 would be to
regard the K-Toda bracket as a dn in the AHSS for K with respect to the coskele*
 *tal
filtration, and the dual K-Toda bracket as a dn in the AHSS with respect to the
skeletal filtration. There is a comparison of these spectral sequences which is*
 * an
isomorphism on E1-terms, hence the spectral sequences are isomorphic, and in
particular they have the same differentials dn.  See, for instance, Appendix B *
 *of
[15].

  We need to extend our K-Toda brackets to operations on the Er term of the
E-ASS. Suppose that the attaching map f :  -1K1 ! S0 has E-Adams filtration
d.

Definition 4.6 (K-Toda brackets on Er(X) ). Suppose that ff is an element of
Es,tr(X). Choose a lift of ff to eff2 sst-s(Wss+r-1(X)). Let ef:  -1K1 ! Wd(S0)
be a lift of f. Then we have
                                                fe*          s+d+r-1
  sst-s(Wss+r-1(X)) - *sst-s+n(Wss+r-1(X ^ K1)) -! sst+n-1(Ws+d    (X)).

If there exists a lift effwhich is in the image of  *, then we say that the K-T*
 *oda
bracket is defined. Take fl 2 sst-s+n(Wss+r-1(X)) such that  *(fl) = eff. Then *
 *the
image of ef*(fl) in Es+d,t+d+n-1r(X) is in the K-Toda bracket <K>(ff). We define
the K-Toda bracket to be the set of all such images for various choices of eff,*
 * fl, and
ef. We shall say that the K-Toda bracket has E-Adams degree d.




14                          MARK BEHRENS


Remark 4.7. If the K-Toda bracket of ff is defined, where ff 2 Es,t1(X), then i*
 *f ff
detects __ff, every element of <K>(ff) detects an element of <K>(__ff). In part*
 *icular, if
<K>(ff) contains 0, then there is an element of <K>(eff) of E-Adams filtration *
 *greater
than k + d.


                       5.Statement of results

  In this section we will state our main results concerning filtered root invar*
 *iants.
The proofs of many of the theorems are given in Section 6. Throughout this sect*
 *ion
let E be a ring spectrum, let X be a finite complex, and suppose ff is an eleme*
 *nt
of sst(X) of E-bifiltration (I, J) with

                        I= (k1, k2, . .,.kl)

                        J= (-N1, -N2, . .,.-Nl)


Theorem 5.1 (Relationship to homotopy root invariant ). Suppose_that R[ki]E(ff)
contains a permanent cycle fi.  Then there exists an element fi2 ss*(X) which
fi detects such that the following diagram commutes up to elements of E-Adams
filtration greater than or equal to ki+1.

                          St ___fi//_ -Ni+1X

                         ff||          ||
                           fflffl|     ||

                          X            |
                           |           |
                           |           |
                           fflffl|     fflffl|
                          tX ____//_ P-Ni ^ X

  The proof of Theorem 5.1 is deferred to Section 6. If ki+1= 1, then i is equal
to l (the maximal index of the bifiltration), and we get the following corollar*
 *y.

Corollary 5.2. The top filtered root invariant R[kl]E(ff) is contained in E1 (X*
 *).
    __[kl]                                                 [kl]
Let RE  (ff) be the set of elements of ss*(X) detected by RE  (ff). There are t*
 *wo
possibilities:
                                    __[kl]
   (1) The image of the elements of RE  (ff) in ss*(P-Nl ^ X) under the inclusi*
 *on
       of the bottom cell is zero, and the homotopy root invariant lies in a hi*
 *gher
       stem than the kthlfiltered root invariant.
   (2) There is an equality (modulo indeterminacy)
                            __[kl]   \
                            RE  (ff) = R(ff)

Proof.Since kl = 1, the diagram in Theorem 5.1 commutes modulo elements of
infinite Adams filtration.  The E-Adams resolution_was assumed to converge, so
this means the diagram actually_commutes.  Let fibe the element described in
Theorem 5.1.  If the image of fiin sst-1(P-Ni) is zero, then the homotopy root *
 * __
invariant lies in a higher stem than the ithfiltered root invariant. Otherwise,*
 * fiis
an element of R(ff).

  Unfortunately, Corollary 5.2 is difficult to invoke in practice.  This is bec*
 *ause
given a filtered root invariant, one usually does not know whether it is the hi*
 *ghest
one or not. In practice we are in the situation where we have a permanent cycle*
 * in




            ROOT INVARIANTS IN THE ADAMS SPECTRAL SEQUENCE           15


a filtered root invariant and we would like to show that the diagram of Theorem*
 * 5.1
commutes on the nose. One strategy is to write out the Atiyah-Hirzebruch spectr*
 *al
sequence for ss*(P-Ni) and try to show there are no elements of higher E-Adams *
 *   __
filtration which could be the difference between the image of ff and the image *
 *of fi
in ss*(P-Ni). This method is outlined in Procedure 9.1.
  We now present some theorems which relate filtered root invariants to differe*
 *n-
tials and compositions in the Adams spectral sequence. Recall that rj = kj+1- k*
 *j.

Theorem 5.3 (Relationship to Adams differentials ). Suppose that the P--Ni+1Ni-
Toda bracket has E-Adams degree d and that d   ri+1.  Then the following is
true.

   (1) <P--Ni+1Ni>(R[ki+1]E(ff)) is defined and contains a permanent cycle.

   (2) R[ki]E(ff) consists of elements which are dr cycles for r < ri+ d.
   (3) There is a containment

                   dri+dR[ki]E(ff)   <P--Ni+1Ni>(R[ki+1]E(ff))

       where both elements are thought of as elements of E*,*ri+d(X).

  The proof of Theorem 5.3 is deferred to Section 6. We intend to use Theorem 5*
 *.3
in reverse: given the ithfiltered root invariant, we would like to deduce the (*
 *i + 1)st
filtered root invariant from the presence of an Adams differential. If the diff*
 *erential
in Theorem 5.3 is zero, we still may be able to glean some information from the
presence of a non-trivial composition.

Theorem 5.4 (Relationship to compositions ). Suppose that R[ki]E(ff) is a coset

of permanent cycles.  Let Re[ki]E(ff) denote the coset of all lifts of elements*
 * of

R[ki]E(ff) to sst+Ni-1(Wki(X)).  When defined, there are lifts of the Toda brac*
 *k-

ets <P--Nim>(Re[ki]E(ff)) to

                  <^P -Ni e[ki]
                     -m (>RE (ff))   sst+m-2(Wki+1(X)).

Let M be the minimal such m > Ni with the property that <^P--Nim>(Re[ki]E(ff))
contains a non-trivial element. Let d be the E-Adams degree of the Toda bracket
<P--Ni+1Ni>(-), and suppose that d   ri+1. Then the following is true.

   (1) Let
                         __[ki]
                         RE  (ff)   sst+Ni-1(X)

       denote the elements which are detected by the permanent cycles of R[ki]E*
 *(ff).
       Then the Toda bracket
                              __[ki]
                     <P--NiM>(R E (ff))   sst+M-2 (X)

       is defined.
   (2) The Toda bracket <P--Ni+1M>(R[ki+1]E(ff)) is defined in Eri+1(X), and co*
 *ntains
       a permanent cycle. We shall denote the collection of all elements which *
 *are
       detected by these permanent cycles by
                   __________________
                   <P--Ni+1M>(R[ki+1]E(ff))  sst+M-2 (X).

   (3) There is an equality (modulo indeterminacy)
                          __[ki]    \ __________________-Ni+1[ki+1]
                 <P--NiM>(R E (ff)) = <P-M   >(RE   (ff)).




16                          MARK BEHRENS


Remark 5.5. The hypothesis d   ri+1 in Theorems 5.3 and 5.4 is a necessary
technical hypothesis to make the proofs work.  In practice, d is often equal to*
 * 1.
Since rj is always positive, the hypothesis is satisfied in this case.

  Finally, we give a partial description of the first filtered root invariant. *
 *We begin
with a simple observation.

Lemma 5.6. If filtE (ff) = k, then R[s]E(ff) is trivial for s < k.

Proof.We must show that in the E-bifiltration of ff, k1   k. Consider the follo*
 *wing
diagram.

                     sst(X)_______//_sst(W0k-1(X))

                        |               |
                        |               |
                        fflffl|         fflffl|
                     sst(tX)____//sst-1(W0k-1(P-1 ))

Since filtE (ff) = k, the image of ff in sst(W0k-1(X)) is trivial. Therefore th*
 *e image
of ff in sst-1(W0k-1(P-1 )) is trivial. By the maximality of k1, we have k1   k.

  If filtE (ff) = 0, (i.e.  when ff has a non-trivial Hurewicz image) then we c*
 *an
sometimes identify the first non-trivial filtered root invariant with the E-roo*
 *t in-
variant.

Proposition 5.7 (Relationship to the E-root invariant ). The E-root invariant

RE (ff) is non-trivial if and only if R[0]E(ff) is non-trivial. If this is the *
 *case, regarding
RE (ff) as being contained in E0,*1(X), we have

                           R[0]E(ff)   RE (ff).

Proof.This is immediate from the definitions.

  If filtE (ff) = 1, then the E-root invariant is trivial and Lemma 5.6 implies*
 * that
the zeroth filtered root invariant is trivial._We can however sometimes compute*
 * the
first filtered root invariant using the E ^ E-root invariant.
                                       __
Proposition 5.8 (Relationship to the E^E -root invariant ). Suppose_that ff has*
 * E-
Adams filtration 1. Then there exists an element eff2 sst(E ^E ^X^p) = E1,t+11(*
 *X^p)
which detects ff in the E-ASS, and such such that RE^__E(eff) is trivial if and*
 * only if

R[1]E(ff) is trivial. There is a containment

                          R[1]E(ff)   RE^__E(eff).

  In practice, we will not know what choice of detecting element effto choose, *
 *so
the following corollary will prove useful.
                                                            __
Corollary 5.9. Suppose that filtE (ff) = 1. If eff2 sst(E ^ E ^ X) is any eleme*
 *nt
which detects ff in the E-ASS, then

                        R[1]E(ff)   RE^__E(eff) + A.

Here A is the image of the map

             sst(W0(0,1)(P(-N,-N+1)^ X)) @-!sst-1(W11( -N X))

where @ is the boundary homomorphism associated to the cofiber sequence

          W11( -N X) '-!W01(P-N ^ X) ! W0(0,1)(P(-N,-N+1)^ X)




            ROOT INVARIANTS IN THE ADAMS SPECTRAL SEQUENCE           17

            __
and the E ^ E-root invariant of effis carried by the -N-cell of P-1 .

  The proof of Proposition 5.8 and Corollary 5.9 is deferred to Section 6. When
E = HFp = H, we can identify the first filtered root invariant with the algebra*
 *ic
root invariant.

Theorem 5.10 (Relationship to the algebraic root invariant ). If E is the Eilen*
 *berg-
MacLane spectrum HFp = H and ff has Adams filtration k, then k1 = k.  Fur-

thermore, the filtered root invariant R[k]H(ff) consists of d1 cycles which det*
 *ect a
                              __[k]      k,t+k+N -1
coset of non-trivial elements RH (ff)   E2      1  (X), and there exists a choi*
 *ce
of eff2 Ek,t+k2(X) which detects ff in the ASS such that
                           __[k]
                           RH (ff)   Ralg(eff).

  The proof of Theorem 5.10 is deferred to Section 6.  We have given a partial
scenario as to how the filtered root invariants can be used to calculate root i*
 *nvariants
using the E-ASS. One first calculates the zeroth filtered root invariant as an *
 *E-root
invariant or the first non-trivial root invariant as an algebraic root invarian*
 *t. Then
the idea, while only sometimes correct, is that "if a filtered root invariant d*
 *oes not
detect the root invariant, then it either supports a differential or a composit*
 *ion that
points to the next filtered root invariant." The last filtered root invariant t*
 *hen has
a chance of detecting the homotopy root invariant. We stress that many things c*
 *an
interfere with this actually happening.

                   6.Proofs of the main theorems

  In all of the proofs below, we shall assume that our finite complex X is actu*
 *ally
S0. The general case is no different, but smashing everything with X complicates
the notation.

Proof of Theorem 5.1.Let fi be an element of R[ki]E(ff). Then there exists a li*
 *ft fff
such that fi is the image of fffunder the collapse map

                  sst-1(WI(P J)) ! sst-1(Wkki+1-1i(S-Ni)).

Consider the following diagram.

            St-1____SSSS
             ____ff|______SSSSS
             ____fflffl|____SfffSSSS
            ____S_____-1     SSSS
            ____| ____           SSSS
           ____fflf____fl|          SSSS))
         _fi_______P_efi_____            W (P J)
          ____      -______1              I
         _____|       _____                 |
         ____ fflffl|  ____                 fflffl|
        ____P-Ni oo____________________Wki(P-Ni) oo___Wki+1(P-Ni)
        ____;;v         _____         jj55
       _____vvv          ____    jjjjj      |
       ____'vv            ,,__  jj '        |
    S-Ni oo___________Wki(S-Ni)            j||
                           |                |
                           ||               fflffl|
                          j|         W ki+1-1(P-N )
                           |          ki55k      i
                           |       kkkk
                           fflffl|k'kk
                    Wkki+1-1i(S-Ni)




18                          MARK BEHRENS


In the above diagram,   is induced from the composite

                                    0
                   WI(P J) ! WI0(P-JNi) ! Wki(P-Ni)



where I0 = (ki, . .,.kl) and J0 = (-Ni, . .,.-Nl).  Since R[ki]E(ff) contains a*
 * per-

manent cycle fi, there exists a map efi(as above) such that jfeiprojects to fi,*
 * and
such_that 'jfei= j fff.  If ffi = 'efi-  fff,_then ffi lifts to sst-1(Wki+1(P-N*
 *i)).  Let

fibe the_map induced by efi, and denote by ffi2 sst-1(P-Ni) the image of ffi. T*
 *hen
filtE (ffi)   ki+1and have the following formula.

                            _'O __fi= __O ff + _ffi



This is precisely what we wanted to prove.



Proof of Theorem 5.3.Fix a lift fff2 sst-1(WI(P J)) of ff. Define a spectrum


                   U = W(ki,ki+1,ki+2)(P-(-Ni,-Ni+1,1)Ni).


There is a natural map

                             WI(P J) ! U


and let fl 2 sst-1(U) be the image of fffunder this map.
  Since we have assumed that the P--Ni+1Ni-Toda bracket has E-Adams degree d,
there is a lift of the attaching map


                         f :  -1P--Ni+1Ni+1! S-Ni


to a map

                      fe:  -1P--Ni+1Ni+1! Wd(S-Ni)



Define a filtered stunted projective space (P--Ni+1Ni)[d]by the following cofib*
 *er se-
quence.

                           fe        -N        -Ni+1
                  -1P--Ni+1Ni+1-!Wd(S  i) ! (P-Ni  )[d]


The spectrum U is given by the homotopy pushout

    `                                        '

     Wki+2(P-Ni) tWk     -Ni+1Wki+1(P--Ni+1N)tWk  (S-Ni)Wki(S-Ni).
                    i+2(P-Ni  )             i   i+1


Since we have assumed that d   ri+1 = ki+2- ki+1, the spectrum U admits the
equivalent description as

 `                                              '

   Wki+2(P-Ni) tWk     -Ni+1Wki+1((P--Ni+1N)[d])tWk   +d(S-Ni)Wki(S-Ni).
                  i+2(P-Ni  )              i       i+1




            ROOT INVARIANTS IN THE ADAMS SPECTRAL SEQUENCE           19


  Consider the following commutative diagram.


                  sst-2(Wki+2+d-1ki+1+d(S-Ni))
                        99:zOO ffNNN
                     <_;-;- |     NNjNN
                    >"="    |        NNN
                  @?"       ||          NN
                 A
       -Ni+1    CB          ef|     sst-2(Wki+1+d(S-Ni))
     <P-Ni  >(-)ED          |               OO|MM
              FF            |               |    MMM
             G              |               |      MMM
             HH             |               |        MMM
            I               |               |          MM&&
            JJ    sst-1(Wki+2-1ki+1(P-Ni+1-Ni+1))@tosst-2(Wki+1(S-Ni))t|
           KffKff      p       ffMMM        |               OO|
           LfiLfi   pppp         MMMM       |               |
          Mfl   ppppp             g2 MMMM   ||              @|
          Mfl wwppp                     MM  |               |
 sst-1(Wki+2-1ki+1(S-Ni+1))oo____________sst-1(U)g3//_sst-1(Wki+1-1ki(S-Ni))


Here g3 is obtained by collapsing out the first and second factors of the homot*
 *opy pushout
U, and similarly g2 is obtained by collapsing out the first and third factors. *
 *The map @tot
is the boundary homomorphism of the Meyer-Vietoris sequence, and may be thought*
 * of
as collapsing out all of the factors of the homotopy pushout. The wavy arrow in*
 *dicates
that the P-Ni+1-Ni-Toda bracket is taken by taking the inverse image under  , f*
 *ollowed by
application of ef.
  Our element fl 2 sst-1(U) has compatible images in all of the other groups in*
 * the
diagram. The image of fl under g3 projects to an element of R[ki]E(ff) in Eri. *
 *Following
the outside of the diagram from the lower right-hand corner to the top corner c*
 *ounter-
clockwise amounts to taking dri+din the E-ASS. Following from the lower right-h*
 *and
corner to the top clockwise applies the Toda bracket to the ksti+1filtered root*
 * invariant.
Thus dri+dR[ki]E(ff) and <P-Ni+1-Ni>(R[ki+1]E(ff)) have a common element.


Proof of Theorem 5.4.Fix a lift fff2 sst-1(WI(P J)) of ff. Let Vm be defined by


                Vm = W(0,ki,ki+1,ki+2)(P-(-Ni-1,-Ni,-Ni+1,1)m)


Then Vm is defined by the following homotopy pushout.


               P--Ni-1mtWk   -Ni-1 Wki(P--Nim)t       -Ni
                          i(P-m   )            Wki+1(P-m )

                Wki+1(P--Ni+1m) tWk     -Ni+1Wki+2(P-m )
                                   i+2(P-m   )

There is a natural map

                            WI(P J) ! Vm

and we define flm 2 sst-1(Vm ) to be the image of fffunder this map. We need to
lift flm a little more. The composite


           sst-1(Vm ) @-!sst-2(Wki+1(P--Nim)) ! sst-2(Wki+1(S-Ni))


sends flm to zero, since it carries the E-Adams differential of an element in R*
 *[ki]E(ff),
which was hypothesized to be zero.  Here @ is a Meyer-Vietoris boundary homo-
morphism. Thus our element flm lifts to an element eflm2 sst-1(eVm) where eVmis




20                          MARK BEHRENS


the following spectrum.

              P--Ni-1mtWk   -Ni-1 Wki(P--Nim)t       -Ni-1
                         i(P-m   )            Wki+1(P-m   )

                Wki+1(P--Ni+1m) tWk     -Ni+1Wki+2(P-m )
                                   i+2(P-m   )

The following diagram explains the lifted bracket <^P--Nim>(Re[ki]E(ff)).


(6.1)     sst-1(eVm)__@___//sst-2(Wki+1(P--Ni-1m))osst-2(Wki+1(S-mo))_

            p2||                    ||                     ||
              fflffl|   g           fflffl|                fflffl|
      sst-1(Wki(S-Ni))_____//_sst-2(Wki(P--Ni-1m))ooss44t-2(Wki(S-m_))
                   *j*j+k+k                         4t4t
                         ,l,l-m-m-m.n.n/o/o/o0p0p1q1q1q2r2r3s3s

                                <P-Ni-1-m>(-)


Here p2 is projection onto the second factor of the homotopy pushout eVm, @ is a
Meyer-Vietoris boundary, and g is the attaching map of the -Ni-cell to P--Ni-1m.
Assuming (inductively) that for every Ni< m0< m,

                          ^<P -Ni e[ki]
                             -m0(>RE (ff)) = 0

we there is a lift of @(eflm) to sst-2(Wki+1(S-m ). The set of all such lifts i*
 *s defined
to be
                  <^P -Ni e[ki]                   -m
                     -m (>RE (ff))   sst-2(Wki+1(S   ))
Diagram 6.1 implies that this set of lifts is indeed a lift of the Toda bracket

                   <P--Nim>(Re[ki]E(ff))   sst-2(Wki(S-m )).


Let M be the first m such that <^P--Nim>(Re[ki]E(ff)) contains a non-trivial el*
 *ement.

Let efl= eflMand let eV= eVM. The image of eflunder the composite

          sst-1(eV) @-!sst-2(Wki+1(P--Ni-1M)) ! sst-2(Wki+1(P--Ni-1M+1))


is zero, and efllifts even further, to an element eefl2 sst-1(eeV), where eeVis*
 * defined to
be the following spectrum.

               P--Ni-1MtWk   -Ni-1 Wki(P--NiM)tWk  (S-M )
                          i(P-M   )              i+1

                Wki+1(P--Ni+1M) tWk     -Ni+1Wki+2(P-M )
                                   i+2(P-M   )
Let f be the attaching map

                        f :  -1P--Ni+1M+1! S-M .

Since <P--Ni+1M> has E-Adams degree d, there is a lift

                      fe:  -1P--Ni+1M+1! Wd(S-M ).




ROOT INVARIANTS IN THE ADAMS SPECTRAL SEQUENCE           21

                                                                               *
 *                                                           <P-Ni-M>(-)
                                                                               *
 *                                    sst-2(S-MP)ooo/o/o/o/o/o/o/o/o/o/o/o/o/ss*
 *t-1(S-Ni)o/o/o/o/o/o/o/OOOO|

                                                                               *
 *                                         |   PPPP                          nn*
 *nnnn   |
                                                                               *
 *                                         ||      PPPP                   gnnnn*
 *       |
                                                                               *
 *                                         |          PPPP             nnnn    *
 *       |
                                                                               *
 *                                         |              PP''P     vvnnn      *
 *       ||
                                                                               *
 *                                         ||             sst-2(P--Ni-1M)      *
 *       |
                                                                               *
 *                                         |                    OO|            *
 *       ||
                                                                               *
 *                                         |                    |              *
 *       |
                                                                               *
 *                                         |                    |              *
 *       |
                                                                               *
 *                                         |                    |              *
 *       |
                                                                               *
 *                                         |                    |              *
 *       |
                                                               sst-2(Wkki+2+d-1*
 *i+1+d(S-Mo))o___________________sst-2(Wki+1+d(S-M ))_//_sst-2(Wki(P--Ni-1M)) *
 *       |
                                                                         OO   h*
 *hQQQ                                     OO|                ??"   ggOOO      *
 *       |
                                                                         OfflOf*
 *fQQQQQefQl                               ||               """        OOOOgO  *
 *       ||
                                                                         Offl  *
 *       QQQQ                              |              ""               OOO *
 *       |
                                                                         Offl  *
 *          QQQQ                           |             ""                  OO*
 *OOO    |
                                                                         Offl  *
 *                                         |           """                     *
 *   O   |
                                                               <P-Ni+1-M>(-)Off*
 *lOffl   sst-1(Wkki+2-1i+1(P--Ni+1M+1))   |@|       """"               sst-1(W*
 *ki(S-Ni))
                                                                         Offl  *
 *             m         hhPPP             |       ""                     ggggg*
 *33g    |
                                                                         Offl  *
 *         mmmm            PPPP            |     """               ggggggg     *
 *       |
                                                                         OfflOf*
 *fl    mmmm                   PPPP        |    ""           ggggggg           *
 *       |
                                                                         Offl  *
 *  mmmm                      p3  PPPP     |   ""     ggggggg   p2             *
 *       |
                                                                              v*
 *vmmmm                              PP    | ""gggggggg                        *
 *       fflffl|
                                                               sst-1(Wkki+2-1i+*
 *1(S-Ni+1))oo__________________________sst-1(eeV)__________________//sst-1(Wkk*
 *i+1-1i(S-Ni))



                                                                               *
 *         Figure 3. The main diagram for the proof of Theorem 5.4




22                          MARK BEHRENS


  Let (P--Ni+1M)[d]be the cofiber of ef. Our hypothesis that d   ri+1 implies t*
 *hat
eeVhas the following equivalent description as a homotopy pushout.


              P--Ni-1MtWk   -Ni-1 Wki(P--NiM)tWk   +d(S-M )
                         i(P-M   )              i+1

              Wki+1((P--Ni+1M)[d]) tWk     -Ni+1Wki+2(P-M )
                                      i+2(P-M   )
Our proof is reduced to chasing the diagram displayed in Figure 3.
  In Figure 3, efand g are attaching maps, p2 is projection onto the second fac*
 *tor

of the homotopy pushout eeV, p3 is projection onto the third factor followed by
projection onto the -Ni-cell, and @ is the Meyer-Vietoris boundary.  The wavy

arrows represent Toda brackets. The element eefl2 eeVmaps to compatible element*
 *s of
every group displayed in Figure 3. The image of eeflin sst-1(Wkki+1-1i(S-Ni)) p*
 *rojects

to an element of R[ki]E(ff). The image of eeflin sst-1(Wkki+2-1i+1(S-Ni+1)) pro*
 *jects to

an element of R[ki+1]E(ff).  Let fi be the image of eeflin sst-2(S-M ).  Follow*
 *ing the
outside of Figure 3 from the lower left hand corner clockwise reveals that
                             __________________
                        fi 2 <P--Ni+1M>(R[ki+1]E(ff)).

Following the outside of Figure 3 from the lower right-hand corner counterclock*
 *wise
gives
                                       __[ki]
                         fi 2 <P--NiM>(R E (ff)).

Thus fi is a common element of the two groups.

Proof of Proposition 5.8.Suppose that the first filtered root invariant of ff i*
 *s carried
by the -N-cell of P-1 . If the first filtered root invariant is trivial, then N*
 * = 1.
Consider the following diagram.

                              0R[1]E(-)p0p/o/o/o.n.n
                       3s3s2r2r1q1q    -m-m,l,l+k+k*j*j
                    5u4t4t                      )))i
            sst(tS0)oo_____sst-1(WI(P_J))____//_sst-1(W11(S-N ))
               OO            m   |                   ||
               |         mmmm    |                   ||
               |      mmmm       |                   ||
               |   vvmmmm        |                   ||
          sst-1(W1(P ))          |                   ||
                                 |                   ||
               |                 |                   ||
               |                 |                   ||
               fflffl|           fflffl|             ||
         sst-1(W11(P ))oo_sst-1(W11(P -N))___//_s55st-1(W11(S-N ))
                    )i*j*j+k                   4t5u
                        +k,l,l-m-m.n.n/o/o/o0p0p1q1q2r2r3s3s4t

                               RE^__E(-)

Let fffbe a lift of the image of ff in sst-1(tS0) to sst-1(WI(P J)).  Then fffh*
 *as
compatible images in every group in the diagram._The detecting element effis the
image of fffin sst-1(W11(P-1 )) = sst-1( -1E ^ E^p).

Proof of Corollary 5.9.First observe that if the first filtered root invariant *
 *is carried
by the -M-cell, then M   N. Let eff0be a `preferred' detecting element such as
the one described in Proposition 5.8. It suffices to show that if fi 2 RE^__E(e*
 *ff) and
fi02 RE^__E(eff0) (or fi0= 0 if RE^__E(eff0) lies in a higher degree than RE^__*
 *E(eff)), then




            ROOT INVARIANTS IN THE ADAMS SPECTRAL SEQUENCE           23


fi -fi02 A. The elements fi and fi0are the images of elements fl and fl0, respe*
 *ctively,
under the collapse map

                   sst-1(W11(P -N)) ! sst-1(W11(S-N )).

where fl and fl0 both map to the image of ff under the homomorphism


                    sst-1(W11(P -N)) '-!sst-1(W01(P )).


Therefore there is an element ~ 2 sst(W0(0,1)(P(-1,-N+1))) such that fl-fl0= @1*
 *(~),
where @1 is the connecting homomorphism of the long exact sequence associated to
the cofiber sequence


               W11(P -N) ! W01(P ) ! W0(0,1)(P(-1,-N+1)).

A comparison of cofiber sequences shows that there is a commutative diagram

                                     @1            1  -N
              sst(W0(0,1)(P(-1,-N+1)))__//_sst-1(W1(P   ))

                         ||                     ||
                         fflffl|                fflffl|
               sst(W0(0,1)(P(-N,-N+1)))@//_sst-1(W11(S-N ))


Thus we see that

                 fi - fi0=  (fl - fl0) =   O @1(~) = @ O  (~)

so fi - fi0 is contained in A.


Proof of Theorem 5.10.We shall refer to the appropriate Ext groups as as the E2
terms of the ASS. Observe that Lemma 5.6 implies that k1   k. We first will pro*
 *ve
that k1 = k. Suppose that k1 > k. Let eeffbe a lift of effto sst(Wk(S0)). Consi*
 *der
the following diagram.


        Ek,t+k-12(S-1)oo___sst-1(Wk(S-1))

             f||                  ||
              |fflffl             fflffl|
        Ek,t+k-12(P-1 )oo__sst-1(Wk(P )-1o)o__ sst-1(Wk1(P )-1 )


The element eeffmaps to eff2 Ek,t+k-12(S-1). By the definition of k1, the image*
 * of
eeffin sst-1(Wk(P )-1 ) lifts to an element of sst-1(Wk1(P )-1 ).  This implies*
 * that

f(eff) = 0. However, effis non-zero, and the algebraic Segal conjecture implies*
 * that
f is an isomorphism. We conclude that k1 = k.
  The filtered root invariant R[1]H(ff) is a subset of Ek,t+N1+k-11(S0).  All o*
 *f the
attaching maps of P-1  are of positive Adams filtration, so Theorem 5.3 im-

plies that R[1]E(ff) consists of d1-cycles.  Let fff 2 sst-1(WI(P J)) be a lift*
 * of

ff, and let fi 2 sst-1(Wkk(S-N1 )) be the corresponding element in R[k]H(ff).  *
 *Let
fff12 sst-1(Wk(P )-1 ) be the image of fffunder the natural map


                         WI(P J) ! Wk(P )-1 .




24                          MARK BEHRENS


Consulting the diagram,


                          sst-1(Wk(P )-1 )

                                 |
                                 |
                                 fflffl|
      Ek,t+k-12(P-N1o)o__sst-1(Wkk+1(P-N1 ))oo_sst-1(Wkk+1(S-N1 ))

                                 |                     |
                                 |                     |
                                 fflffl|               fflffl|
                          sst-1(Wkk(P-N1 ))oo___sst-1(Wkk(S-N1 ))


both fff1and fi have the same image in sst-1(Wkk(P-N1 )). Let fl be the image o*
 *f fff1
in sst-1(Wkk+1(P-N1 )). Since fi is a d1-cycle, it lifts to efi2 sst-1(Wkk+1(S-*
 *N1 )). Let

eflbe the image of efiin sst-1(Wkk+1(P-N1 )). It not necessarily the case that *
 *fl = efl,
but it is the case that the difference fl - eflvanishes in Ek,t+k-12(P-N1 ).
  Consider the following diagram.

                              __R[k](-)
     sst(tS0)//o//o///o//o///o//H///o//o///o//o////ooooooooooooooooosst-1(Wkk+1*
 *(S-N1|))|
        OO|ggOOO                                    mm    |        ED|
        |     OOO                                mmmm     |          |
        |       OOOO                          mmm         |          |
        |           O                       vvmmm         |          |
        |                    J__//       k -N1            |          |
        |         sst-1(WI(PO )) sst-1(Wk(S    ))         |          |
        |            ""     OOOO                          |          |
        |          ""          OOO      |                 |          |
        |         ""              OOO   |                 |          |
        |        ""                 ''  fflffl|           |          |
        |      """               sst-1(Wkk(P-N1 ))        |          |
        |    ""                             hhQQQ         |          |
        |   ""                                QQQQ        |          |
        |  ""                                     QQQQ    |          |
        |""""                                        Q    fflffl|    |
 ss   (W (P )  _)________________________________//       k+1        |
   t-1  k    -1                                   sst-1(Wk  (P-N1 )) |
                                                                     |
        |                                                 |          |
        |                                                 |          |
        fflffl|                                           fflffl|    |
 Ek,t+k-1(P-1 )____________________________________//Ek,t+k-1(P-N  ) |
  2     OO                       N1                   2   OO     1   |
        |                                                 |          |
      ~=f|                                                |          |
        |                                                 |        BC|
  Ek,t+k-12(S-1)//o///o//o///o//o///o//o///o//o///o//oooooooooooooo_oooEk,t+k-1*
 *(S-N1 )
                              Ralg(-)                                   2

                                     __[k]       k,t+k-1
The element efimaps to an element of RH (ff) in E2     (S-N1 ). The map f is
an isomorphism, so the image of fff1lifts to an element eff2 Ek,t+k-12(S-1). Th*
 *is is
the choice of effwhich detects ff that we appeal to in the statement of Theorem*
 * 5.3.
Recall that the image of fff1in sst-1(Wkk+1(P-N1 )) was fl while the image of f*
 *i is efl.
Since the difference fl - eflmaps to zero in Ek,t+k-12(P-N1 ), we may conclude *
 *that
the image of efiin Ek,t+k-12(S-N1 ) is also an element of the algebraic root in*
 *variant,
provided the algebraic root invariant does not lie in a higher stem.
  We have proven that either what is claimed in Theorem 5.10 holds, or the al-
gebraic root invariant lives in a larger stem than R[1]H(ff). We shall now show*
 * that
this cannot happen. Let M be maximal such that the image of effin Ek,t+k-12(S-1)




            ROOT INVARIANTS IN THE ADAMS SPECTRAL SEQUENCE           25


maps to zero under the composite


                 Ek,t+k-12(S-1) -M-1Of----!Ek,t+k-12(P-M+1 ).

By the algebraic Segal conjecture, such a finite M exists. We wish to show that
N1 = M. So far we know that N1   M. In light of the definition of N1, we simply
must show that the ff is sent to zero under the composition

                 sst-1(S-1) ! sst-1(W0(k-1,k)(P(-M+1,1))).

To this end consider the following diagram.


                          sst(W0k-1(P )-1o)o________sst(Wkk-1-1(P )-1 )


                                @||                        ||
                                 |fflffl                   fflffl|
  sst-1(Wk(P )-1 )_______//_sst-1(Wkk(P-M+1o))d1o__sst(Wkk-1-1(P-M+1 ))

         |                       |
         |                       |
         fflffl|                 |fflffl
      sst(tS0)_______//_sst-1(W0(k-1,k)(P(-M+1,1)))


The central vertical column corresponds to part of the long exact sequence for a
cofibration. The element fff12 sst-1(Wk(P )-1 ) maps to the image of ff in sst(*
 *tS0).
Let g be the image of fff1in sst-1(Wkk(P-M+1 )). By our choice of M, g must van*
 *ish
in E2, that is to say, there must be an element h 2 sst(Wkk-1-1(P-M+1 )) such t*
 *hat
d1(h) = g. The map   is surjective, because we are dealing with ordinary homolo*
 *gy,
and the map

                          H*(P )-1  ! H*(PN )

is surjective for every N. Thus h lifts to an element eh2 sst(Wkk-1-1(P )-1 ). *
 *Applying

@ to the image of ehin sst(W0k-1(P )-1 ), we get g, so g is in the image of @. *
 *Thus, by

exactness of the vertical column, the image of g in sst-1(W0(k-1,k)(P(-M+1,1)))*
 * is

trivial. Thus the image of ff in sst(tS0) maps to zero in sst-1(W0(k-1,k)(P(-M+*
 *1,1))),
which is what we were trying to prove.


                          7. bo resolutions

  The bo resolution for the sphere was the motivating example for the theorems
in Section 5.  The d1's on the v1-periodic summand of the 0 and 1-lines of the
bo-resolution reflect the root invariants of the elements 2i at the prime 2.  We
do not rederive these root invariants, but satisfy ourselves in explaining how *
 *the
theorems of Section 5 play out in this context. Alternatively, one might interp*
 *ret
this section as explaining how to use bo to compute the root invariants in the *
 *K(1)-
local category.  This section is furthermore meant to explain how the techniques
of this paper, when applied to the small resolution of LK(2)(S0) described in [*
 *13],
could be used to compute root invariants in_the K(2)-local category.
  Recall that  4bsp is a summand of bo ^  bo, hence its homotopy is a stable
summand of the 1-line of the E1-term of the bo-ASS. The composite of the d1 of
this spectral sequence with the projection onto the summand
                                         __           4
                  sst(bo) d1-!sst-1(bo ^ bo) ! sst-1(  bsp)




26                          MARK BEHRENS

            Figure 4. The v1-periodic summand of the bo-ASS.



is the map _3 - 1 (up to a unit in Z(2)). The fiber of _3 - 1 is the 2-primary J
spectrum. Figure 4 shows the v1-periodic summand of the bo-ASS with differentia*
 *ls.
In this figure, dots represent copies of F2 and circles represent copies of Z. *
 *The d1
differentials are (up to a 2-adic unit) multiplication by the power of 2 indica*
 *ted.
What survives are the v1-periodic elements in ssS*at p = 2.
  Our theorems explain the relationship between Rbo(2k) and R(2k) for all k > 0.
It is quite straightforward to calculate Rbo(2k) (see [25]). Let j, ff, and fi *
 *be the
multiplicative generators of ss*(bo) in dimensions 1, 4, and 8, respectively. T*
 *he we
have the following bo-root invariants.
                                8
                                >>>fii k = 4i
                                <jfii  k = 4i + 1
                      Rbo(2k) = >   2  i
                                >>:j fik = 4i + 2
                                 fffii k = 4i + 3


These are the filtered root invariants R[0]bo(2k), by Proposition 5.7.  For k  *
 * 1, 2

(mod  4), the elements R[0]bo(2k) are in the Hurewicz image of bo, hence they a*
 *re
permanent cycles. They detect elements in R(2k).
  For k   3, 4 (mod  4), the elements Rbo(2k) are not in the bo-Hurewicz image.

Therefore, the elements R[0]bo(2k) support differentials. We have

                           d1R[0]bo(2k) = 2 . effk

where effksurvives to a v1-periodic element of order 2 in dimension 4k - 1. In *
 *the
bo-bifiltration of 2k, the -N1 cell carries the zeroth filtered root invariant,*
 * where
N1 is given by              (


                       N1 =  8i + 5 k = 4i + 3
                             8i + 1 k = 4i.

The first cell to attach to the -N1 cell in P-N1 is the -N1+1 cell, and the att*
 *aching
map is the degree 2 map. Therefore, we may deduce from Theorem 5.3 that effkis
an element of R[1]bo(2k). The element effkdetects the homotopy root invariant R*
 *(2k).


       8. The algebraic Atiyah-Hirzebruch spectral sequence

  As discussed after the statement of Corollary 5.2, it is often the case that *
 *one
may know R[k]E(ff) contains a permanent cycle in the E-Adams spectral sequence,
but nothing else. One would want to conclude that this permanent cycle detects
the homotopy root invariant R(ff), but Theorem 5.1 says this is only true modulo




            ROOT INVARIANTS IN THE ADAMS SPECTRAL SEQUENCE           27


obstructions in higher E-Adams filtration. However, it is sometimes the case th*
 *at
brute force computations in the E2-term of the E-Adams spectral sequence for PN
will yield enough data to eliminate such obstructions. This section is concerne*
 *d with
computations of the E2-term E2(PN ). Specifically, we shall assume that E = BP
and describe the algebraic Atiyah-Hirzebruch spectral sequence (AAHSS), which
computes Ext(BP*PN ) from Ext(BP*).  In Section 9, we describe a procedure
(Procedure 9.1) that explains how a limited knowledge of differentials in the A*
 *AHSS
can be exploited to compute homotopy root invariants from filtered root invaria*
 *nts.
  The AAHSS is the spectral sequence obtained by applying Ext to the skeletal
filtration of a complex. The AAHSS has appeared in various forms in the literat*
 *ure.
In [22], Mahowald uses ordinary cohomology, and uses this spectral sequence to
compute Ext of various stunted projective spaces. In [35], Sadofsky uses the BP
version that is the subject of this section. There is some overlap with this se*
 *ction
and the computations in [35].
  We will first describe the AAHSS for computing the Adams-Novikov E2-term
E2(PN ), where N   -1 (mod  q). Let F be the universal p-typical formal group
law associated to BP , and write the p-series of F as

                              XF     i  X
                   [p]F (x) =    vixp =    cjx(p-1)j+1
                              i 0        j

where vi are the Araki generators. Then we can say the following about the coef*
 *fi-
cients ci.

                     c0= p

                     c1= v1

                     cj  0  (mod  p) for 1 < j < p + 1

                   cp+1  v2  (mod  p)

There are short exact sequences (compare with [35, 2.3])


(8.1)        0 ! BP*(Skq-1) OE-!BP*(PNkq-1) ! BP*(PNkq) ! 0

(8.2)      0 ! BP*(PN(k-1)q) ! BP*(PNkq-1) -!BP*(Skq-1) ! 0

and these give rise to long exact sequences upon applying Ext.  They have the
following boundary homomorphisms.

               ffi1: Exts,t(BP*PNkq) ! Exts+1,t(BP*Skq-1)

               ffi2: Exts,t(BP*Skq-1) ! Exts+1,t(BP*PN(k-1)q)

For l   k, BP*(Plkqq-1) is generated as a BP*-module by elements ejq-1in dimens*
 *ion
jq - 1. The map OE in the short exact sequence 8.1 is given by
                                    X
                       OE('kq-1) 7!    cieq(k-i)-1.
                                     i

Splicing these sequences together creates an exact couple, and the resulting sp*
 *ectral
sequence is the AAHSS. We shall index it just as one indexes the stable EHP spe*
 *ctral
sequence [33, 1.5]. In fact, if N = q-1 then this is precisely the BP -algebrai*
 *c stable
EHP spectral sequence. Thus we have a spectral sequence

                       Ek,n,s1) Exts,s+k(BP*PN )




28                          MARK BEHRENS


where the E1 term is described below.

                     Ek,2m,s1= Exts,s+k(BP*Smq-1 )

                   Ek,2m+1,s1= Exts+1,s+k(BP*Smq-1 )

The indexing works out so that

                       dr : Ek,n,sr! Ek-1,n-r,s+1r.

  If we wish to compute Ext(BP*PN ) for N = lq, simply truncate the AAHSS for
Ext(BP*PN-1 ) by setting Ek,2l,s1= 0 for all k and s.
  We shall refer to an element in the E1-term of the AAHSS by its name in
Ext(BP*) and the cell that is borne on. Thus if fl 2 Ext(BP*) is in Ek,2m,s1, t*
 *hen
we shall refer to it as fl[mq - 1]. Likewise, if fl is in Ek,2m+1,s1, we shall *
 *refer to it
as fl[mq].
  In order to implement Procedure 9.1 we shall need to know how to explicitly
compute differentials in the filtered spectral sequence. It is useful to use th*
 *e diagram
below. In this diagram and the remainder of this section we will drop BP* from
our Ext notation for compactness.


                           Exts+1,s+k(PNlq-1)_*__//Exts+1,s+k(Slq-1)

                                   |                     ||
                                   |                Ek-1,2l,s+1
                                   |                 1
                                   fflffl|
                            Exts+1,s+k(PNlq)_ffi1//_Exts+2,s+k(Slq-1)

                                   |                     ||
                                   |               Ek-1,2l+1,s+1
                                   |                1
                                   fflffl|
                                   ..
                                   .
                                   |
         Ek,2m,s                   |
          1                        |
            ||                     fflffl|
     Exts,s+k(Smq-1 )_ffi2//_Exts+1,s+k(PN(m-1)q)

                                   |
        Ek,2m+1,s                  |
         1                         |
            ||                     fflffl|
    Exts+1,s+k(Smq-1 )_OE*//_Exts+1,s+k(PNmq-1)


Suppose fl[n] is an element of E1, where n = mq - ffl, ffl = 0, 1. Let fl1 be t*
 *he image

of fl in Exts+1,s+k(PNmq-1) or Exts+1,s+k(PN(m-1)q), depending on the value of *
 *ffl.
Then lift fl1 as far as possible up the tower in the center of the above diagra*
 *m.
Suppose that fl1 lifts to fl2 2 Exts+1,s+k(PNlq-ffl1), where ffl1 = 0, 1.  Then*
 * if fl3 is
the image of fl2 under  * or ffi1 (depending on the value of ffl1), there is an*
 * AAHSS
differential
                            dr(fl[n]) = fl3[n0]

where n0= lq - ffl.
  The above description makes the following proposition clear. In the statement*
 * of
this proposition, and in what follows, when we say that differentials are compu*
 *ted
modulo In, we mean is that we are considering their images in Ext(BP*(-)=In).




            ROOT INVARIANTS IN THE ADAMS SPECTRAL SEQUENCE           29


Proposition 8.1. Modulo In, we have the differential

                  d2pn-1(ff[mq]) = vn . ff[mq - 2pn + 1].

Proof.Take ff 2 Exts+1,s+k(Smq-1 ). Clearly, it is the case that

                 OE*(ff)         vnff[mq - 2pn + 1] + . . .
                          (mod In)

lifts to Exts+1,s+k(P mq-2pn+1), and the image of this lift under the map  * is
                                                n+1
                     vn . ff 2 Exts+1,s+k(Smq-2p   ).



  In order to compute ffi1 and ffi2, we must know something of the BP*BP comodu*
 *le
structure of BP*(PN ).  It suffices to understand the BP*BP comodule structure
of BP*(BZ=p), since P0 is a stable summand of BZ=p+ . The BP*BP comodule
structure of BP*(PN ) may then be deduced from James periodicity. In [35], it is
proven that there is a cofiber sequence
                                    p
                      CP 1 ! (CP 1), !  BZ=p

(where , is the canonical complex line bundle on CP 1) which induces the short
exact sequence below.
                                     p
             0 ! BP*CP 1 ! BP*(CP 1), ! BP* BZ=p ! 0

Since BP*(CP 1),psurjects onto BP* BZ=p, it suffices to understand the BP*BP -
comodule structure of the former.
  Recall that BP *(CP 1) = BP *[[x]] where x is the Euler class of ,. Therefore,
the image of the inclusion
                                 p     *   1
                      BP *(CP 1), ! BP (CP  )

is the ideal
                     [p]F (x) . BP *[[x]]   BP *(CP 1).

We shall identify BP *(CP 1),p with this ideal.p
  Let y2k be the generator in BP2k((CP 1), ) dual to [p]F (x) . xk-1. Let f(x) *
 *be
the formal power series over BP*BP whose inverse is given by

                                     XF     i
                           f-1 (x) =    tixp .
                                     i o

The power series f(x) is the universal strict isomorphism of p-typical formal g*
 *roup
laws.

Proposition 8.2. The BP*BP coaction on y2k is given below.
                        X   `f([p]F (x))       '
(8.3)         _(y2k) =       _________(f(x))j-1      y2j
                       i+j=k   [p]F (x)         k-1

Here the subscript k - 1 indicates the coefficient of xk-1.

Proof.We digress for a moment on cooperations. For a spectrum X, the left unit
jL : BP ! BP ^ BP induces the BP*BP coaction


        _ : BP*(X) (jL)*---!(BP ^ BP )*(X) ~=BP*BP  BP* BP*(X).




30                          MARK BEHRENS


It is sometimes convenient to consider the dual coaction on cohomology


                   _* : BP *(X) (jL)*---!(BP ^ BP )*(X).

  In Theorem 11.3 of Part II of [1], Adams describes the MU*MU coaction on
MU*(CP 1). When translated into BP -theory, the coaction formula reads
                               X
                     _(e2k) =      (f(x))jk   e2j.
                              i+j=k

Here, e2k is the generator of BP2k(CP 1) dual to xk 2 BP 2k(CP 1). Upon dual-
ization, we get the following formula for the dual coaction on an element h(x) *
 *in
BP *(CP 1).

                         _*(h(x)) = f*h(f(x))

The polynomial f*h(x) is the polynomial obtained from applying the right unit to
the coefficientspof h(x). We deduce the dual coaction on an element [p]F (x) . *
 *h(x)
in BP *((CP 1), ), regarded as the ideal ([p]F (x)) contained in BP *[[x]].

             _*([p]F (x) . h(x))= f*[p]F (f(x)) . f*h(f(x))

                            = [p]f*F(f(x)) . f*h(f(x))

                            = f([p]F (x)) . f*h(f(x))


                            = [p]F (x) . f([p]F_(x))_[p]. f*h(f(x))
                                                        F (x)

Here, f*F is the pushforward of the formal group law F under the map f.  Let-
ting h(x) = xj-1pand dualizing, we have the desired formula for the coaction on
BP *((CP 1), ).


  We wish to use the above coaction formula to compute differentials in the AAH*
 *SS
with p   3.  The first few terms of the relevant power series are listed below.
Everything in what follows is written in terms of Hazewinkel generators.

(8.4)           [p]F (x) = px + (1 - pp-1)v1xp + O(xp + 1)

(8.5)          f(x) = x - t1xp + (pt21+ v1t1)x2p-1+ O(x2p)
(8.6) f([p]F_(x))_[p]= 1 - pp-1t1xp-1+
                     F (x)
                    pp-2(pp+1t21+ (1 - p - pp-1 + 2p)v1t1)x2p-2+ O(x2p-1)


We may compute these series further if we work modulo the ideal I2 = (p, v1).

                                 2     p2
(8.7)             [p]F (x)   v2xp + O(x  + 1)  (mod  I2)
                                         2
(8.8)              f(x)   x - t1xp + O(xp )  (mod  I2)

(8.9)              f([p]F_(x))_  1 + O(xp2-1)  (mod  I )
                     [p]F (x)                         2

These formulas give rise to the following proposition.




            ROOT INVARIANTS IN THE ADAMS SPECTRAL SEQUENCE           31


Proposition 8.3. In the filtered spectral sequence, the following formulas for *
 *dif-
ferentials hold up to multiplication by a unit in Z(p).

                   (

  d2(ff[kq - 1]) .= ff1 . ff[(k - 1)q -k1]6  0  (mod  p)
                    0                 k   0  (mod  p)
                     (

  d2(ff[kq])           ff1 . ff[(k - 1)q]k 6  1  (mod  p)
              (mod I1) 0             k   1  (mod  p)
                         8
                         ><<ff, ff1, . .,.ff1>[(k - p +k1)q --1]1  (mod  p)
                                ____-z___"
  dq(ff[kq - 1])    2             p-1
                  (mod I2)>:
                           0                           k 6  -1  (mod  p)
                     8
                     ><<ff, ff1, . .,.ff1>[(k - pk+ 1)q]0  (mod  p)
                            ____-z___"
  dq(ff[kq])    2             p-1
              (mod I2)>:
                       0                        k 6  0  (mod  p)


Proof.We shall begin by computing d2(ff[kq -1]). In light of the coaction equat*
 *ion
8.3, we compute from 8.5 and 8.6


           f([p]F_(x))_(f(x))r= xr - (r + pp-1)t xr+p-1+ O(xr+p).
             [p]F (x)                           1


We may conclude that the coaction formula on the generator ekq-1 of BP*(P01) is
given by


(8.10)     _(ekq-1) = 1   ekq-1+ ((k - 1)(p - 1) - 1)t1   e(k-1)q-1+ . . .


Now t1 represents ff1 in the cobar complex, and p . ff1 = 0.
  We now compute d2.


          ffi2(ff[kq - 1]) = ((k - 1)(p - 1) - 1)ff|t1[(k - 1)q - 1] + . . .


and it follows immediately that


            d2(ff[kq - 1])= ((k - 1)(p - 1) - 1)ff1 . ff[(k - 1)q - 1]

                         -kff1 . ff[(k - 1)q - 1]  (mod  p)


Of course, since p . ff1 = 0, one needs only to compute this differential modul*
 *o p.
  Now we shall deal with d2(ff[kq]). We shall work modulo I1.


                   OE*ff[kq]   v1 . ffv1[(k - 1)q - 1] + . . .



Therefore OE*ff[kq] lifts to Ext(P0(k-1)q). So


                 d2(ff[kq]) = ffi1(v1 . ff[(k - 1)q - 1] + . .)..




32                          MARK BEHRENS


The boundary homomorphism ffi1 is computed below on the level of cobar complexe*
 *s.

                                '*           *   (k-1)q
            C*(P0(k-1)q-1)________________//C (P     )
                        VV
                          VVVVVVdVV
                                  VVV
                                OE* VVVVV++
            C*(S(k-1)q-1)________________//C*(P0(k-1)q-1)
        v1 . ff[(k - 1)q - 1]O+_._._.//_v1 . ff[(k - 1)q - 1] + . . .
                        WWWWWW
                             WWW
                                WWWWWW
                                     WWWW++W
       -(k - 1)ff|t1[(k - 1)qO-_1]//-(k - 1)v1 . ff|t1[(k - 2)q - 1] + . . .

Here, C*(-) represents the cobar complex. We conclude that

      ffi1(v1 . ff[(k - 1)q - 1] + . .).  (1 - k)ff|t1[(k - 1)q - 1]  (mod  I1)

  The dq's are proved similarly.  The relevant information to be gathered from
Equations 8.8 and 8.9 is
                               ` '
  f([p]F_(x))_(f(x))r     xr +  r (-t )xr+(p-1)+ . . .
    [p]F (x)       (mod I2)     1    1
                              `    '
                                                     2    r+(p-1)2+1
                            +  p r- 1(-t1)p-1xr+(p-1)+ O(x         )


The Massey product
                             <ff, ff1,_._.,.ff1_-z___">

                                    p-1

corresponds (up to multiplication by a unit) to

                             ff|tp-11+ . . .

in the cobar complex. We have that
                               `     '

        dq(ff[kq - 1])    2       r   <ff, ff1, . .,.ff1>[(k - p + 1)q - 1].
                        (mod I2)p - 1      ____-z___"
                                             p-1

where r = (k - p + 1)(p - 1) - 1. Consider the coefficient
                   `     '
                      r    = r_._(r_-_1)_._.(.r_-_p.+ 2)
                     p - 1          (p - 1)!

Since p 6 | p - 1, in order for p to not divide this binomial coefficient, it m*
 *ust be the
case that r   -1 (mod  p). In terms of k, this translates to k   -1 (mod  p).
  Just like in the calculation of d2ff[kq], the map ffi1 introduces an offset i*
 *n coeffi-
cients which results in the claimed formula for dqff[kq].

  It is interesting to note that these differentials, combined with the image o*
 *f J
differentials, are enough to compute the AAHSS completely through a certain ran*
 *ge
at odd primes.
  In Ext(BP*), many elements live in an ff1-fi1 tower, that is, a copy of P [fi*
 *1]
E(ff1). The following general observation says that elements of the E1-term of *
 *the




            ROOT INVARIANTS IN THE ADAMS SPECTRAL SEQUENCE           33


AAHSS which lie within an ff1-fi1 tower typically either support differentials *
 *are
the target of differentials.


Proposition 8.4. Let fl 2 Ext(BP*). Suppose furthermore that there exist ele-
ments ~ and ~ in the E2 term of the Adams-Novikov spectral sequence such that
either


                               ff1~ = fl

                           <fl, ff1,_._.,.ff1_-z___"> = ~

                                  p-1

or


                           <~, ff1,_._.,.ff1_-z___"> = fl

                                 p-1
                               ff1fl = ~.


(Here we insist that the Massey products have no indeterminacy.) Assume further-
more that p and v1 do not divide any of ~, ~, or fl. Then, in the AAHSS, fl[n] *
 *is
either killed by a differential, or supports a non-trivial differential.


Proof.The condition on p and v1 not dividing any of the elements ~, ~, or fl is*
 * just
to ensure that we may use the formulas in Proposition 8.3. Let us suppose that *
 *fl
satisfies the first set of conditions.

    Case 1: n   -1 (mod  q):Write n = mq - 1.
        Subcase (a): m   -1 (mod  p):Then dqfl[mq - 1] = ~[(m - p + 1)q - 1]
           or difl[mq - 1] 6= 0 for some i < q.
        Subcase (b): m 6  -1 (mod  p):Then d2~[(m + 1)q - 1] = fl[mq - 1].
    Case 2: n   0 (mod  q):Write n = mq.
        Subcase (a): m   0 (mod  p):Then either dqfl[mq] = ~[(m - p + 1)q] or
           we have difl[mq] 6= 0 for some i < q.
        Subcase (b): m 6  0 (mod  p):Then d2~[(m + 1)q] = fl[mq].

Suppose now that fl satisfies the second set of conditions.

    Case 1: n   -1 (mod  q):Write n = mq - 1.
        Subcase (a): m   0 (mod  p):Then dq~[(m + p - 1)q - 1] = fl[mq - 1].
        Subcase (b): m 6  0 (mod  p):Then d2fl[mq - 1] = ~[(m - 1)q - 1].
    Case 2: n   0 (mod  q):Write n = mq.
        Subcase (a): m   1 (mod  p):Then dq~[(m + p - 1)q] = fl[mq].
        Subcase (b): m 6  1 (mod  p):Then d2fl[mq] = ~[(m - 1)q] or d1fl[mq] 6=
           0. (This is the case of p . fl 6= 0.)





  It is also useful to record all of the differentials supported by the v1-peri*
 *odic
elements of Ext(BP*) for p > 2 in the AAHSS. These results are used in executing
Step 4 of Procedure 9.1.




34                          MARK BEHRENS


Proposition 8.5. In the AAHSS, the differentials supported by ImJ elements
whose targets are ImJ elements are given by

              d1(1[kq])= p[kq - 1]

           d1(ffi=j[kq])= ffi=(j-1)[kq - 1] for j > 1

          d2j(1[kq - 1]).="ffj[(k - j)q - 1] where  p(k) = j - 1.

          d2j+1(ffi[kq]).="ffi+j[(k - j)q - 1] where  p(k + i) = j - 1.

(Here "fflis ffl=m for m =  p(l) + 1. It is the additive generator of imJ in th*
 *e lq - 1
stem.)

Proof.These differentials detect the corresponding differentials in the AHSS wh*
 *ich
converge to the stable homotopy of PN . These differentials were computed for P0
by Thompson in [38] and are summarized in [33].

  There are still v1-periodic elements of Ext(BP*) which are eligible to support
differentials in the filtered spectral sequence (or subsequently in the Adams-N*
 *ovikov
spectral sequence for PN ) whose targets are elements which are not v1-periodic.
These elements are described below.

Corollary 8.6. The only elements in imJ in the AAHSS for P-1  which are neither
targets or sources of the differentials described in Proposition 8.5 are

                              f"fi[kq - 1]

where pi|(k + i).

  We obtain the following consequence which will become quite relevant when we
execute Step 4 of Procedure 9.1.

Proposition 8.7. None of the elements fii=j[kq] are the target of differentials*
 * in
the AAHSS or subsequently in the Adams-Novikov spectral sequence for ss*(PN ).

Proof.The only elements which could kill fii=j[kq] in the filtered spectral seq*
 *uence
are of the form ffn[mq] and 1[mq - 1]. But these cannot by Corollary 8.6. There
are no elements in the E2 term of the Adams-Novikov spectral sequence which can
kill fii=j[kq]. Such elements would have to lie in Adams-Novikov filtration 0.


 9. Procedure for low dimensional calculations of root invariants

  In this section, we let E = BP , and concentrate on BP -filtered root invaria*
 *nts.
The following procedure is used in later sections to compute homotopy root in-
variants from filtered root invariants using Theorem 5.1. It only has a chance *
 *to
work through a finite range, and is very crude. We state it mainly to codify ev*
 *ery-
thing that must be checked in general to see that a filtered root invariant det*
 *ects a
homotopy root invariant.

Procedure 9.1. Suppose we are in the situation where we know fi is an element
of R[k]BP(ff), and we know that fi is a permanent cycle. Let '-N  be the inclus*
 *ion
of the -N cell of P-N , where -N = -Ni is the index of the bifiltration of ff t*
 *hat
corresponds to the cell that bears the kth filtered root invariant. Let  -N ff *
 *be the
image of ff in ss*(P-N ).  Then Theorem 5.1 tells us that  -N ff = '-N fi modulo
elements in higher Adams-Novikov filtration.




        ROOT INVARIANTS IN THE ADAMS SPECTRAL SEQUENCE           35


Step 1: Make a list

                            fli[-ni]

   of additive generators in the E1 term of the AAHSS for Ext(P-N ), where
   flilies in Adams-Novikov filtration siand stem ki, satisfying the following.
   (1) The homological degree si satisfies
                     (

                 si>  k      ni  1  (mod  q)
                      k + 1  ni  0  (mod  q)

   (2) The stem ki is greater than p . |ff|, and less than |fi|.
   (3) We have ni= ki- |ff| + 1, and ni  0, 1 (mod  q).
   These are precisely the conditions required for fli[ni] to be a candidate to
   survive in the AAHSS to an element in the same stem, but higher Adams-
   Novikov filtration, as '-N fi. Condition (3) is a consequence of P-N  having
   cells in dimensions congruent to 0, -1 modulo q.  We apply the first in-
   equality in (2) because Jones's theorem [29] will_preclude the possibility of
   these surviving to detect the difference of '-N fiand  -N ff. Thus we have
                 X
-N ff = fi[-N] +    aifli[-ni] + ( terms born on cells < -(p - 1)|ff|)+.1
                  i

Stem 2.: Attempt to determine which fli[ni] are killed in the filtration spec-
   tral sequence, or subsequently, in the Adams-Novikov spectral sequence for
   computing ss*(P-N ).  This_will limit the possibilities for what can detect
   the difference of '-N fiand  -N ff.
Step 3: First eliminate those fliwhich are not permanent cycles in the ANSS.
   These cannot be root invariants. Then, attempt to show that every remain-
   ing non-trivial linear combination of the elements fli[ni] which are not kil*
 *led
   in the AAHSS actually supports a non-trivial differential in_the_AHSS for
   computing ss*(P-1 ). Suppose this is the case. Then, if '-N fi- -N ff is non-
   trivial, and is born on cells above the -N-cell, it will project non-trivial*
 *ly
   to ss*(P-M ) for M < N, and will represent the image of the homotopy
   root invariant. The homotopy root invariant is a permanent cycle_in the
   AHSS for ss*(P-1 ). Thus we may assume that the difference '-N fi- -N ff
   is actually born on cells above the -(p - 1)|ff| + 1 cell. If this difference
   was non-trivial, we would violate_Jones's theorem [29].  The difference is
   therefore trivial, and '-N fi=_ -N ff.                     __
Step 4: We will have shown that fi2 R(ff) if we know that '-N fi6= 0.  It
   would suffice to show that fi[-N] is not killed in the AAHSS, and also
   survives to a non-zero element in the ANSS for computing ss*(P-N ). Form
   a list of all of the elements

                            ji[ni]

   in the E1-term of the AAHSS where ji is in Adams-Novikov filtration si
   and stem(ki such that:

   (1) si   k      -N   -1 (mod  q)
            k - 1  -N   0 (mod  q)
   (2) ki  |fi|
   (3) ni= -N + |ff| - ki+ 1   0, -1 (mod  q)




36                          MARK BEHRENS


       These are precisely the elements which have a chance of killing fi[-N] i*
 *n the
       AAHSS, or subsequently in the ANSS. Next, show these elements actually
       do not kill fi[-N].

Remark 9.2. By comparing the definition of differentials in the AAHSS with the
differentials in the AHSS, it is clear that if fl detects __flin the ANSS for c*
 *omputing
ssS*, then if fl[n] supports a non-trivial AAHSS differential

                            dr(fl[n]) = j[m]

where j is a permanent cycle in the ANSS, then either __flsupports a non-trivial
AHSS di for i < r, or there is an AHSS differential

(9.1)                       dr(__fl[n]) = __j[m].

where j detects __j. However, the differential given in Equation 9.1 may be tri*
 *vial
in the AHSS if __jis the target of a shorter differential.
__Since in Step 3 of Procedure 9.1 we are only concerned with whether the eleme*
 *nts
fli[ni] support non-trivial differentials in the AHSS, it suffices to show that*
 * the
elements fli[ni] support non-trivial differentials in the AAHSS. One must then *
 *make
sure that the targets of the AAHSS differentials are not the targets of shorter
differentials in the AHSS.


   10.BP -filtered root invariants of some Greek letter elements

  In this section we compute the first two BP -filtered root invariants of some
chromatic families. If ff = ff(n)i=j1,...,jkis the nthalgebraic Greek letter el*
 *ement of the

ANSS which survives to a non-trivial element of ssS*, then one might expect tha*
 *t it
should be the case that
                                          [n+1]  (n)
(10.1)                ff(n+1)i=j1,...,jk2RBP  (ffi=j1,...,jk)

The purpose of this section is to show that Equation 10.1 holds for n = 0 at all
primes and for n = 1 at odd primes. Modulo an indeterminacy group which we do
not compute, we also show that the n = 1 case of Equation 10.1 is true_at the p*
 *rime
2. Throughout this section, for a ring spectrum E, we shall let eE'  E denote t*
 *he
cofiber of the unit. Also, whereas in Section 8 we used Hazewinkel generators, *
 *in
this section we always use Araki generators. This is because the p-series is mo*
 *re
naturally expressed in the Araki generators.
  For appropriate i and j, the elements which generate the 1-line of the ANSS a*
 *re
ffi=j2 BP*BP and are given by

                                        i- vi
                          ffi=j= jR_(v1)____1pj.


Proposition 10.1. The first two filtered root invariants of pi are given by

                            (-v1)i2 R[0]BP(pi)

                          (-1)i. ffi2 R[1]BP(pi)

  Define, for appropriate i and j,
                                  p    p   i   i
                 efii=j= (v2_+_v1t1_-_v1t1)_-_v22 BP*BP.
                                  vj1




            ROOT INVARIANTS IN THE ADAMS SPECTRAL SEQUENCE           37


Observe that the image of efii=jin BP*BP=I1 coincides with

                             jR_(v2)i-_vi2_.

                                  vj1

It follows from the definition of the algebraic Greek letter elements that, in *
 *the
cobar complex, d1(efii=j) = p . fii=junless p = 2 and i = j = 1, in which case
efi1= ff2=2, which is a permanent cycle that detects  .

  In order to compute the first filtered root invariants of the elements ffi=j,*
 * we

first compute_the_BP ^ gBP-root invariants, and then invoke Corollary 5.9. Note
that gBP_=  BP , so the BP ^ gBP-root invariant is just the suspension of the
BP ^ BP -root invariant. The following result holds for any prime.

Proposition 10.2. The BP ^ gBP-root invariant of ffi=jis given by

                RBP^gBP(ffi=j) = (-1)i-jefii=j+ pefii=jBP*gBP

Corollary 10.3. Suppose the Greek letter element ffi=jexists and is a permanent
cycle in the ANSS. Let __ffi=j2 ssS*be the element that ffi=jdetects. Then the *
 *zeroth
filtered root invariant of __ffi=jis trivial. If p is odd, the first filtered r*
 *oot invariant
of __ffi=jis given by

                  (-1)i-jefii=j+ c . effi(p+1)+j2 R[1]BP(__ffi=j)

where c is some constant. The second filtered root invariant is given by

                        (-1)i-jfii=j2 R[2]BP(__ffi=j).

  Deducing the first filtered root invariant in Corollary 10.3 from Proposition*
 * 10.2
amounts to computing the indeterminacy group A described in Corollary 5.9. The
second filtered root invariant then follows from Theorem 5.3. We describe a spe*
 *ctral
sequence which computes A (10.8).  We fully compute A for odd primes p and
describe some of the 2-primary aspects of this computation in Remark 10.6.
                            [1]
  We have shown that ffk 2 RBP (pk). It is shown in [25], [35] that these eleme*
 *nts
survive in the Adams-Novikov spectral sequence to elements of R(pk). Similarly,*
 * we
                       [2] __
have shown that fik 2 RBP (ffk). Again, in [25], [35] it is shown that these el*
 *ements
survive to elements of R(__ffk) for p   5. Sadofsky goes further in [35] to sho*
 *w that
fip=22 R(__ffp=2) for p   5.
                                                          [2]
  In general, we have shown that if pk-1|s, then fis=k 2 RBP (ffs=k).  According
to the summary presented in [33, 5.5], the elements fis=kexist and are permanent
cycles in the ANSS for p   5 for these values of k and s. However, without any
additional information, we can only deduce the conclusions of Theorem 5.1.  In
Section 12 we indicate what these filtered root invariant calculations mean for*
 * the
computation of homotopy root invariants in low dimensions at the prime 3.
  In proving Propositions 10.1 and 10.2, we shall need to make use of the follo*
 *wing
well known computation (see Lemma 2.1 of [2]).

Lemma 10.4. Let E be a complex oriented spectrum whose associated p-series
[p]E (x) is not a zero divisor in E*[[x]]. Then the coefficient ring of the Tat*
 *e spectrum
is given by

            ss*(tE) = (E*[[x]]=([p]E (x)))[x-1] = E*((x))=([p]E (x))




38                          MARK BEHRENS


where the degree of x is -2. Furthermore, the inclusions of the projective spec*
 *tra
are described as the inclusion of the fractional ideal

   xN E*[[x]]=([p]E (x)=x) = E*( (BZ=p)-2N ) ! ss*(tE) = E*((x))=([p]E (x)).

  According to Appendix B of [15], the skeletal and coskeletal filtrations of (*
 *E ^
P )-1  give rise to the same notions of the E-root invariant. It follows that E*
 *-root
invariants are quite easy to compute for complex orientable spectra E provided *
 *one
has some knowledge of the p-series of E. The method is outlined in the following
corollary.

Corollary 10.5. Suppose that ff is an element of E*. Viewing __ffas the image of
the constant power series ff in E*((x))=([p]E (x)), suppose that n is maximal s*
 *o that
                          __ff= a  n      n+1
                                 nx  + O(x   )

with an 2 E* nonzero. Then an is an element of RE (ff).

Proof of Proposition 10.1.We first compute RBP (pi). The p-series [p]F (x) of t*
 *he
universal p-typical formal group F is given by
                                              2
                   [p](x) = px +F v1xp +F v2xp +F . . .

where vi are the Araki generators of BP*. In tBP*, we have the relation

                         p = -v1xp-1 + O(xp)

which gives, upon taking the ithpower,

                    pi= (-v1)ix(p-1)i+ O(x(p-1)i+1).

Any other expression of pi in terms of x(p-1)iand higher order terms will have
a leading coefficient that differs by an element of the ideal pBP*   BP*.  Using
Corollary 10.5 and Proposition 5.7, we may conclude that

                  (-v1)i+ pBP* = RBP (pi)   R[0]BP(pi).

We calculate the Adams-Novikov d1 on this coset as

                          d1(vi1) = jR (vi1) - vi1.

The algebraic Greek letters ffi are defined by
                                     i) - vi
                           ffi= jR_(v1____p1


so we are in the situation where

                 d1(R[0]BP(pi))   p . (-1)iffi+ p2ffiBP*BP.

By Theorem 5.3, we may conclude that the first filtered root invariant R[1]BP(p*
 *i) is
contained in the coset (-1)iffi+ pffiBP*BP.

Proof of Proposition 10.2.The proof consists of two parts. In Part 1 we prove t*
 *he
proposition with v1 inverted, and in Part 2 we prove that the v-11BP ^ gBP-root
invariant can be lifted to compute the BP ^ gBP-root invariant.


Part 1: computing the v-11BP ^ gBP-root invariant




            ROOT INVARIANTS IN THE ADAMS SPECTRAL SEQUENCE           39


  We may calculate tBP ^BgP *from tBP*BP using the split short exact sequence

               0 ! tBP* ! tBP ^ BP* ! tBP ^ gBP*! 0.

Applying Lemma 10.4 to the left and right complex orientations of BP*BP , we get

                 tBP ^ BP* = BP ^ BP*((xL))=([p]L(xL))

                          BP ^ BP*((xR ))=([p]R (xR )).

Here, the p-series [p]L(x), [p]R (x), and the coordinate xR are given by the fo*
 *rmulas
                                           2
              [p]L(x)= px +FL v1xp +FL v2xp +FL . . .
                                                     2
             [p]R (x)= px +FR jR (v1)xp +FR jR (v2)xp +FR . . .
                                          2
                 xL = xR +FL t1xpR+FL t2xpR+FL . ...

The formal group laws FL and FR are the p-typical formal group laws over BP*BP
induced by left and right units jL and jR , respectively.  Consider the followi*
 *ng
computation in tBP ^ gBP*.
                          i                       2      j
              pxR= [-1]FR  jR (v1)xpR+FR jR (v2)xpR+FR . . .
(10.2)                                   2     p2+1
                 = -jR (v1)xpR- jR (v2)xpR+ O(xR   )

The last equality for p > 2 follows from the fact that for any p-typical formal*
 * group
law G where p > 2, the -1-series is given by

                            [-1]G (x) = -x.

For p = 2 this is not true, but it turns out that the last equality of Equation*
 * 10.2
still holds up to the power indicated in tBP*gBP. The two expressions we want to
be equal are written out explicitly below.
                       2             4
(10.3) [-1]FR  jR (v1)xR +FR jR (v2)xR +FR . . .

           = (2t1 - v1)x2R+ (14t2 - 4t31- v1t21- 3v21t1 - v2 + v31)x4R+ O(x5R)


(10.4) - jR (v1)x2R- jR (v2)x4R+ O(x5R)

              = (2t1 - v1)x2R+ (14t2 + 4t31- 13v1t21+ 3v21t1 - v2)x4R+ O(x5R)

The coefficient of x2Ris the same in Equations 10.3 and 10.4. Since we are work*
 *ing in

tBP*gBP, we are working modulo the 2-series, and thus we only need the coeffici*
 *ents
of x4Rto be equivalent modulo 2. However, the coefficient of x4Rin Equation 10.3

has an extra v31, but since we are working in the reduced setting of tBP*gBP, we
have v1x4L= 0. We may switch to xR since x4L= x4R+ O(x5R).
  Returning to our manipulations of the p-series, upon dividing Equation 10.2 by
xR , we get                                  2        2

                 p = -jR (v1)xp-1R- jR (v2)xpR-1+ O(xpR)

which implies                          2        2

                   jR (v1) = -jR (v2)xpR-p+ O(xpR-p+1)

since the image of p is zero in BP*gBP. Taking the ith power and exploiting the
fact that the image of vi1in BP*gBP is also zero, we may write
                                                   2-p)    i(p2-p)+1
(10.5)      pjffi=j= jR (v1)i- vi1= (-jR (v2))ixi(pR  + O(xR       ).




40                          MARK BEHRENS


Since we are modding out by the p-series in xL, we have the relation
                              i             2      j
                 pxL = [-1]FL  v1xpL+FL v2xpL+FL . . .
                                   2
                     = -v1xpL+ O(xpL)

or, dividing by xL and taking the jth power,
                                                   2-p
(10.6)             pj = (-v1)jxj(p-1)L+ O(xj(p-1)+pL )

Upon combining Equation 10.6 with Equation 10.5, and using the fact that xkR=
xkL+ O(xk+1R), our original expression for ffi=jbecomes the following.

                        2-p
  (-v1)jffi=j+ ffi=jO(xpR  )
                                           2-p)-j(p-1)   i(p2-p)-j(p-1)+1
                 = (-1)i(jR (v2)i- vi2)xi(pR        + O(xR            )

Explicit formulas (see, for instance, [33, 4.3.21]) reveal that

                      jR (v2) = v2 + v1tp1- vp1t1 + py

for some y 2 BP*BP .  Using this formula and Equation 10.6 to write elements
which are divisible by p in terms of higher order elements, we have

                             2-p
(10.7) (-v1)jffi=j+ ffi=jO(xpR  )
                                           2-p)-j(p-1)   i(p2-p)-j(p-1)+1
                       = (-1)ivj1"fii=jxi(pR        + O(xR            )

In order to solve for ffi=j, we would like to divide by vj1. Therefore, we shal*
 *l finish

our algebraic manipulations in tv-11BP ^ gBP*. We will then show that we can pu*
 *ll
back our results to results in tBP ^ gBP*. Taking the image of Equation 10.7 in
tv-11BP ^ BP* and dividing by (-v1)j, we get the expression
                            2-p)-j(p-1)        p2-p       i(p2-p)-j(p-1)+1
    ffi=j= (-1)i-jefii=jxi(pR        + ffi=jO(xR   ) + O(xR            )

By successively substituting the left hand side of this expression into the rig*
 *ht hand
side, we obtain the expression below.
                                   2-p)-j(p-1)
           ffi=j= (-1)i-j"fii=jxi(pR        + higher order terms.

We may conclude that Rv-11BP^gBP(ffi=j) = efii=j, or that the v-11BP ^ gBP-root
invariant lives in a higher stem.


Part 2: lifting to the BP ^ gBP-root invariant


  We claim that the localization map

                 BP ^ gBP*(P-N ) ! v-11BP ^ gBP*(P-N )

is an inclusion. In order to see this, chase the following diagram.

       BP*(P-N")________//`BP ^ BP*(P-N")______//_`BP ^ gBP*(P-N )

           |                    |                      |
           |                    |                      |
           fflffl|              fflffl|                fflffl|
     v-11BP*(P-N )_____//v-11BP ^ BP*(P-N_)___//v-11BP ^ gBP*(P-N )




            ROOT INVARIANTS IN THE ADAMS SPECTRAL SEQUENCE           41


The relevant observations are that the other two localization maps in the diagr*
 *am
are inclusions (since there is no v1 torsion in these groups), and the top and *
 *bottom
sequences are compatibly split cofiber sequences.
  We wish to deduce something about the BP ^ gBProot invariant from the com-
putation of the v-11BP ^ gBProot invariant. We refer to the following diagram.

                           Rv-11BP^gBP(-)
 v-11BP ^ gBP* //o//o///o//o///o//o///o//o///o////ooooooooooooooov-11BP ^ gBP*(*
 *S-N+1 )
       |  ffMMM                                    kkk55k  |
       |     MMM                               kkkk        |
       |       MMMM                         kkkk           |
       |                 R   g (-)        kk               |
       |          g     //BP^BP//o////ooo/o/o/oooBP ^ gBP (|S-N+1 )
       |     BP ^ BP *                                   * |
       |             |                |                    |
       |             |                '|                   '|
       |             fflffl|          fflffl|              |
       |                                                   |
       |     tBP ^ gBP*_____________//BP ^ gBP*( P-N )     |
       |         qq                       SSS              |
       |      qqq                            SSSSS         |
       |    qqq                                  SSSS      |
       fflffl|xxqq                                  SS))   fflffl|
tv-11BP ^ gBP*___________________________________//v-11BP ^ gBP*( P-N )


Here N equals 2i(p2 - p) - 2j(p - 1) + 1. We have shown that '((-1)i-jefii=j) =
 (ffi=j) after v1 is inverted.  Because the v1-localization map is an inclusion*
 *, we

may conclude that '((-1)i-jefii=j) =  (ffi=j) in BP ^ gBP*( P-N ). Therefore, we

have computed the BP*gBP-root invariant

                RBP^gBP(ffi=j) = (-1)i-jefii=j+ pefii=jBP*gBP

or it lies in a larger stem.  The latter cannot be the case, however, as '(efii*
 *=j) is
non-zero.

  We wish to apply Corollary 5.9 to Proposition 10.2 and conclude that

               R[1]BP(__ffi=j)   (-1)i-jefii=j+ pefii=jBP*gBP + A.

Here A is the image of the boundary homomorphism

       ssiq-1(W0(0,1)(P(-(ip-j)q-1,-(ip-j)q))) @-!ssiq-2(W11(S-(ip-j)q-1)).

Let N be (ip - j)q + 1.

  The group ss*(W0(0,1)(P(-N,-N+1))) may be computed from the AHSS associated
to the filtration

        W00(S-N )   W0(0,1)(P(-N+1-N,-N+1))   W0(0,1)(P(-N+2-N,-N+1))   . ...

The resulting spectral sequence takes the form
                   8
                   ><0            l 6  0, -1  (mod  q)

(10.8)      Ek,l1= >ssk(W01(Sl))  l   0, -1  (mod  q), l > -N
                   :ssk(W 0 -N
                         0(S   )) l = -N

and converges to ssk(W0(0,1)(P(-N,-N+1))). The groups ss*(W01(Sl)) in the E1-te*
 *rm
of the spectral sequence (10.8) may be computed by taking the 0 and 1-lines of *
 *the
E1-term of the ANSS for the sphere, and taking the cohomology with respect to




42                          MARK BEHRENS


the d1 from the 0-line to the 1-line.  The differentials in spectral sequence (*
 *10.8)
are a restriction of the differentials in the AAHSS.
  The image of the map @ is generated by the image of Adams-Novikov d1's sup-
ported on the 0-line, the subgroup pW11(S-N ), and the images of the higher dif-
ferentials in the AAHSS whose sources are permanent cycles in spectral sequence
(10.8) and whose targets are elements in Adams-Novikov filtration 1 that are ca*
 *rried
by the -N-cell.
  We remark that for 0 6= x 2 ss*(W11(S0)) which is not a permanent cycle, we
have a non-trivial differential

                         d1(x[kq]) = px[kq - 1].

The reason the differential must be non-trivial is that there is no torsion in *
 *the
E1-term of the ANSS, and if px were the target of a d1 in the ANSS, then x would
have to be a d1-cycle.  Thus, if we are looking for longer differentials in spe*
 *ctral
sequence (10.8) supported by kq-cells, we may restrict our search to those which
are actually d1-cycles.
  We now use spectral sequence (10.8) to compute the indeterminacy group A for
odd primes p.

Proof of Corollary 10.3.Since filtBP (__ffi=j) = 1, Lemma 5.6 implies the filte*
 *red

root invariant R[0]BP(__ffi=j) is trivial. In Proposition 10.2, we found the BP*
 * ^ gBP-
root invariants RBP^gBP(ffi=j). We will now deduce the first filtered root inva*
 *riant

R[1]BP(__ffi=j) through the application of Corollary 5.8. We just need to compu*
 *te A
using spectral sequence (10.8).
  The indeterminacy group A is generated by the images of Adams-Novikov d1's,
the subgroup pW11(S-N ), and the higher AAHSS differentials.  We just need to
compute the latter. The only elements of spectral sequence (10.8) that could co*
 *n-
tribute to A are those of the form

                      ffk=l[(i - k)q],    k   ip - j.

Proposition 8.5 tells us that only one can contribute to ffi and that contribut*
 *ion is
given by

          @(ffi(p+1)-j-l[-(ip - j - l)q]) = effi(p+1)-j[-(ip - j)q - 1]

where l =  p(i)+1. Thus A is also spanned by the element eff-i(p+1)+j. We concl*
 *ude
that
                 (-1)i-jefii=j+ c . eff-i(p+1)+j2 R[1]BP(__ffi=j)

where c is some constant.
  To prove the second part of the proposition we appeal to Theorem 5.3. There
is an Adams-Novikov differential d1(efii=j) = p . fii=j.  The filtered root inv*
 *ariant

R[1]BP(ffi=j) is carried by the -N1-cell, where N1 = 2(i(p2 - p) - j(p - 1)) + *
 *1 =
(ip - j)q + 1. The first cell to attach nontrivially to this cell is the -(ip -*
 * j)q-cell,
and this is by the degree p map. We may conclude that

                        (-1)i-jfii=j2 R[2]BP(__ffi=j).



Remark 10.6. Computing the group A at the prime 2 requires a more careful
analysis. The AAHSS differentials don't follow immediately from the J-spectrum




            ROOT INVARIANTS IN THE ADAMS SPECTRAL SEQUENCE           43


AHSS differentials computed in [23], because the varying Adams-Novikov filtrati*
 *ons
of the v1-periodic elements. For instance, it turns out that

                      ff4=4= efi2=2+ x7 2 R[1]BP(__ff2=2)

where
                    x7   v2t1 + v1(t2 + t31)  (mod  2).

If i = j = 1, then efi1is a permanent cycle which represents the element ff2=2.*
 * If

i = j = 2, then efi2=2+ x7 is a permanent cycle which represents the element ff*
 *4=4.
Low dimensional calculations seem to indicate that in all other circumstances we
have
                     fii=j+ c . eff3i-j+1ff1 2 R[2]BP(__ffi=j)

where c is some constant which may or may not be zero and effkis equal to ffi=j
with j maximal. The anomalous cases with i = j = 1, 2 correspond to the existen*
 *ce
of the `extra' Hopf invariant 1 elements   and oe, which are detected in the AN*
 *SS
by ff2=2and ff4=4, respectively.  These filtered root invariants are thus consi*
 *stent
with the homotopy root invariant, which takes each Hopf invariant 1 element to
the next one, if it exists.


               11.Computation of R(fi1) at odd primes

  In this section we will compute the root invariant of fi1 at odd primes. This*
 * result
was stated, but not proved, in [25]. We do this by first computing the BP -filt*
 *ered
root invariants, and then by executing Step 4 of Procedure 9.1.

Proposition 11.1. For p > 2, the top filtered root invariant of fi1 is given by
R[2p]BP(fi1) .=fip1.


Sketch of Proof.We first wish to show that fip=p2 R[2]BP(fi1). In [33], it is s*
 *hown
that modulo the ideal I1, the representatives for fi1 and fip=pin the cobar com*
 *plex
are given by


                             X   `p'      p-i
                   fi1   -1_p        ti1|t1    (mod  I1)
                            0<i<p i
                            X    `p2'     p2-i
                 fip=p  -1_p         ti1|t1     (mod  I1)
                           0<i<p2  i
          p2
  Now,  p  i = 1 if and only if p|i. Therefore, the expression for fip=pmodulo *
 *I1
may be simplified.
                           X   `p2'  ip  p(p-i)
                fip=p  -1_p         t1 |t1      (mod  I1)
                          0<i<p ip
       p    p2
Since  i    pi (mod  p) for 0 < i < p, we see that in ExtBP*BP=I1(BP*), the

element fip=pis obtained from fi1 by application of the 0th algebraic Steenrod *
 *op-
eration. By computing the reduction map

                 ExtBP*BP(BP*) ! ExtBP*BP=I1(BP*=I1)

we may conclude that
                        P 0(fi1)   fip=p  (mod  p)




44                          MARK BEHRENS


in ExtBP*BP (BP*). By an algebraic analog of Jones's Kahn-Priddy theorem, this
coincides with the BP -algebraic root invariant, which corresponds to the first*
 * non-
trivial filtered root invariant. Thus we have the filtered root invariant

                            fip=p2 R[2]BP(fi1).

In the ANSS, the element fip=psupports the Toda differential.

                           dq+1(fip=p) .=fip1ff1

The -(p2-p-1)q -1 cell attaches to the -(p2-p)q -1 cell of P-1  with attaching
map ff1. Therefore, Theorem 5.3 tells us that
                                   [2p]
                            fip12 RBP (fi1).

There is no room for indeterminacy in the ANSS.

Corollary 11.2. For p > 2, the homotopy root invariant of fi1 is given by

                             R(fi1) .=fip1.

Proof.The element fip1lies on the Adams-Novikov vanishing line, so it must be t*
 *he
top filtered root invariant. Therefore, we may apply Corollary 5.2 to see that *
 *either
the image of the element fip1under the inclusion of the (-(p2 - p - 1)q - 1)-ce*
 *ll of
P-(p2-p-1)q-1is null, or fip1actually detects the homotopy root invariant. We m*
 *ust
therefore show that the element fip1[-(p2-p-1)q-1] in the AHSS for P-(p2-p-1)q-1
is not the target of a differential. We will actually show that fip1[-(p2-p-1)q*
 *-1] is
not the target of a differential in the AAHSS, and that there are no possible s*
 *ources
of differentials in the ANSS for P-(p2-p-1)q-1with target fip1[-(p2 - p - 1)q -*
 * 1].
  According to the low dimensional computations of the ANSS at odd primes
given in [33, Ch. 4], the only elements in the E1-term of the AAHSS which can k*
 *ill
fip1[-(p2 - p - 1)q - 1] in either the AAHSS or the ANSS are the elements

            fik[-(k - 1)(p + 1)q]1   k < p

             ffk=l[-(k - p)q - 1]1   k < p2, 1   l    p(k) + 1

as well as elements in ff1-fi1 towers, i.e. those that satisfy the hypotheses o*
 *f Propo-
sition 8.4. These latter elements cannot kill fip1[-(p2 - p - 1)q - 1] in the A*
 *AHSS
and cannot survive to kill anything in the ANSS by Proposition 8.4.  Some care
must be taken at p = 3, but in this low dimensional range there are no deviatio*
 *ns
from this pattern.
  By Proposition 8.3, the elements fik[-(k-1)(p+1)q] support non-trivial AAHSS
differentials
              d2(fik[-(k - 1)(p + 1)q]) .=ff1 . fik[-k(p + 1)q]).

According to Proposition 8.5, for l <  p(k) + 1, we have differentials in the A*
 *AHSS

                d1(ffk=l+1[-(k - p)q]) .=ffk=l[-(k - p)q - 1]

whereas for l =  p(k) + 1 and k   3 we have

               d5(ffk-2[-(k - p - 2)q]) .=ffk=l[-(k - p)q - 1].

For k = 2 we have [38]

                     d4(1[pq - 1]) = ff2[(p - 2)q - 1].

Finally, Proposition 8.3 implies that

                      dq(ff1[(p - 1)q - 1]) .=fi1[-1].




            ROOT INVARIANTS IN THE ADAMS SPECTRAL SEQUENCE           45


There are no elements left to kill fip1[-(p2 - p - 1)q - 1].

    12.Low dimensional computations of root invariants at p = 3

  The aim of this section is to use knowledge of the ANSS for ssS*in the first *
 *100
stems to compute the homotopy root invariants of some low dimensional Greek
letter elements ffi=jat p = 3. These results are summarized in the following pr*
 *opo-
sition.

Proposition 12.1. We have the following root invariants at p = 3.

                             R(ff1)= fi1

                             R(ff2).=fi21ff1

                           R(ff3=2)= -fi3=2

                             R(ff3)= fi3

                             R(ff4).=fi51

                             R(ff5)= fi5

                           R(ff6=2)= fi6=2

                             R(ff6)= -fi6

  It is interesting to note that although fi2 exists, it fails to be the root i*
 *nvariant
of ff2. The element fi4 does not exist, so it cannot be the root invariant of f*
 *f4. In

Section 15 we will prove that fii2 R(ffi) for i   0, 1, 5 (mod  9). The remaind*
 *er of
this section is devoted to proving Proposition 12.1.
  Figure 5 shows the Adams-Novikov E2 term. These charts were created from the
computations in [33].  Solid lines represent multiplication by ff1 and dotted l*
 *ines
represent the Massey product <-, ff1, ff1>. Dashed lines represent hidden exten*
 *sions.
  In the Section 10, we proved in Corollary 10.3 that we have filtered root inv*
 *ariants

                        (-1)i-jfii=j2 R[2](ffi=j).

We supplement those results with two higher filtered root invariants particular*
 * to
p = 3.

Proposition 12.2. We have the following higher filtered root invariant of ff2.
                                     [5]
                           fi21ff1 2RBP (ff2)

Proof.We know by Corollary 10.3 that

                            -fi2 2 R[2]BP(ff2).
                                                                           __
The element fi2 is a permanent cycle in the ANSS, which detects an element fi22
ssS*.  We shall apply Theorem 5.4 to determine the higher filtered root invaria*
 *nt
from the following hidden Toda bracket
                                           __
                            fi31.=<ff1, 3, fi2>

(see, for example, [33]). Considering the attaching map structure of P-1  we ha*
 *ve
the following equalities of Toda brackets (the first is an equality of homotopy*
 * Toda
brackets whereas the second is an equality of Toda brackets in the ANSS).
                                __   .          __   .   3
                      <P--2025>(fi2) = <ff1, 3, fi2> = fi1

                   <P--1725>(fi21ff1) .=<ff1, ff1, fi21ff1> .=fi31




46                          MARK BEHRENS



          Figure 5. The Adams-Novikov spectral sequence at p = 3




            ROOT INVARIANTS IN THE ADAMS SPECTRAL SEQUENCE           47


We conclude from Theorem 5.4(3) that
                                     [5]
                           fi21ff1 2RBP (ff2).



Proposition 12.3. We have the following higher filtered root invariants of ff4.
                                      [6]
(12.1)                     fi3=3fi212RBP (ff4)
                                    [10]
(12.2)                        fi512RBP (ff4)

Proof.We know from Proposition 10.3 that

                            -fi4 2 R[2]BP(ff4)

but fi4 is not a permanent cycle. There are Adams-Novikov differentials

                             d5(fi4).=fi3=3fi21ff1

                         d5(fi3=3fi21).=fi51ff1.

The higher filtered root invariants of ff4 are derived from two applications of*
 * The-
orem 5.3. The relevant Toda brackets are computed below.

                      <P--3640>(fi3=3fi21) .=ff1 . fi3=3fi21

                          <P--4044>(fi51) .=ff1 . fi51



  We now apply Procedure 9.1 to our filtered root invariants.  The results of
executing Steps 1-3 in Procedure 9.1 are summarized in Table 2. The meanings of
the contents of the columns are described below.

    Element:  contains the element ff which we wish to calculate the root invar*
 *i-
       ant of.
    R[k]:indicates the kth BP -filtered root invariant fi of ff. We would like *
 *fi to
       detect an element of R(__ff).
    Cell: indicates the cell of P-1  that carries the filtered root invariant.
    {fli[ni]}:contains the list of elements of the ANSS which could survive to
       the difference between the filtered root invariant and the homotopy root
       invariant. This is the list described in Step 1 of Procedure 9.1.
    Diffs:indicates the differentials in the AAHSS which kill the elements fli[*
 *ni],
       as described in Step 2 of Procedure 9.1, or the differentials supported *
 *by the
       fli[ni], as described in Step 3 of Procedure 9.1. If the element flisupp*
 *orts a
       non-trivial differential in the ANSS, we shall place a "(1)" to indicate*
 * that
       it is not a permanent cycle, and so cannot be a homotopy root invariant.*
 * If
       the fli[ni] satisfies the conditions of Proposition 8.4, then Steps 2 an*
 *d 3 of
       Procedure 9.1 are automatically satisfied, as explained in Remark 9.2. We
       shall indicate this with a "(2)".




48                          MARK BEHRENS


               Table 2. Steps 1-3 of Procedure 9.1 at p = 3
           ________________________________________________
           _Element_|_R[k]|__Cell___|____{fli[ni]}____Di|ffs__

           ____ff1___|fi1__|_-2q___|________________|______
           ____ff2___|fi21ff1-|4q_-_1_|_____________|______
           ___ff3=2__|fi3=2_|-7q___|________________|______
           ____ff3___|fi3__|_-8q___|_fi2fi1ff1[-7q_-_1](2|)_
           ____ff4___|fi51_|_-9q___|________________|______
               ff5   |fi5  |-14q  |  fi4fi1ff1[-13q - 1](1|)

                    |     |        |  j1fi1[-12q - 1]  (|1)
                    |     |        |  j1fi1ff1[-13q]2  (|1)
                    |     |        |   fi2fi1[-11q]3   (|2)
                    |     |        |fi3=3fi1ff1[-12q3- 1](|1)
                    |     |        | fi2fi1ff1[-10q4- 1](2|)
                    |     |        |   fi2fi1[-12q]6   (|2)
                    |     |        |  fi1ff1[-11q7- 1] (2|)
           _________|_____|________|____fi1[-13q]______|(2)_
              ff6=2  |fi6=2 |-16q  |  fi2j1[-15q - 1]  (|1)
                                       3
                    |     |        | fi2=ff1[-13q - 1] (|2)
                    |     |        | fi4fi1ff1[-12q3- 1](1|)
                    |     |        |    fi2[-14q]      |(2)
                    |     |        |  j1fi1ff1[-12q]22 (|1)
                    |     |        | fi2fi1ff1[-13q2-31](2|)
                    |     |        |   fi2fi1[-15q]4   (|2)
                    |     |        |   fi3=3fi1[-13q]5 (2|)
                    |     |        | fi2fi1ff1[-14q7- 1](2|)
                    |     |        |    fi1[-12q]8     |(2)
           _________|_____|________|__fi1ff1[-15q_-_1]_(2|)_
               ff6   |fi6  |-17q  |   j1fi2[-15q - 1]  (|1)
                                       3
                    |     |        | fi2=ff1[-13q - 1] (|2)
                    |     |        | fi4fi1ff1[-12q - 1](1|)
                    |     |        |  j1fi2ff1[-16q]3  (|1)
                    |     |        |    fi2[-14q]      |(2)
                    |     |        |  j1fi1ff1[-12q]3  (|1)
                    |     |        |  j1fi1[-16q2-21]  (|1)
                    |     |        |     fi2fi1ff123   (|2)
                    |     |        |   fi2fi1[-15q]4   (|2)
                    |     |        |   fi3=3fi1[-13q]5 (1|)
                    |     |        |fi3=3fi1ff1[-16q5- 1](|2)
                    |     |        | fi2fi1ff1[-14q6- 1](2|)
                    |     |        |   fi2fi1[-16q]8   (|2)
           _________|_____|________|__fi1ff1[-15q_-_1]_(2|)_
  Some care must be taken in the case of ff3. If <ff1, 3, fi3> is nonzero (i.e.*
 * contains
 fi2fi21), then the AAHSS differential



                   d4(fi2fi1ff1[-7q - 1]) .=fi2fi21[-9q - 1]




            ROOT INVARIANTS IN THE ADAMS SPECTRAL SEQUENCE           49


may be trivial in the AHSS with an intervening differential supported by fi3[-8*
 *q]
(see Remark 9.2). This does not happen, though, as the Toda bracket <ff1, 3, fi*
 *3>
is actually zero. It is easily seen to have zero indeterminacy. If the Toda bra*
 *cket
were non-trivial, it would have to contain  fi2fi21, and thus ff1. <ff1, 3, fi3*
 *> would be
nonzero. This cannot be the case, since we have

               ff1 . <ff1, 3, fi3> = -<ff1, ff1, 3> . fi3 = ff2 . fi3 = 0.

  Now we must complete Step 4 of Procedure 9.1. We must determine that the
filtered root invariants survive to non-trivial elements in the AHSS, and subse-
quently in the ANSS. For this we must determine which elements in the AAHSS
(or subsequently in the ANSS) can support differentials which kill the filtered*
 * root
invariants.
  Proposition 8.7 implies that for all of the ffi=jabove whose filtered root in*
 *variants
were fii=j, the image of fii=jin the appropriate stunted projective space is no*
 *n-trivial.
Thus we conclude that
                             fii=j2 R(ffi=j)

for i   6 and i 6= 2, 4.
  Step 4 of Procedure 9.1 is completed for ff2 and ff4 in the following lemmas.
In both of these lemmas, we denote by __xthe element in homotopy detected by
the ANSS element x. There is no ambiguity arising from higher Adams-Novikov
filtration for the elements that we will be considering.
                        _____
Lemma 12.4. The element fi21ff1[-4q - 1] is not the target of a differential in*
 * the
AHSS for P-4q-1.
                                     _____
Proof.Differentials which could kill fi21ff1[-4q - 1] must have their source in*
 * the 7-
stem. Proposition 8.5 demonstrates that in the the targets of any AHSS differen*
 *tials
supported by ImJ elements in the 7-stem are also ImJ elements. In the range we
are considering, there_is no room for_shorter_differentials.  The only element *
 *left
which could kill fi21ff1[-4q - 1] is fi21[-3q - 1].  However, the complex P--3q*
 *-14q-1is
reducible, so there can be no such differential.
                        ___
Lemma 12.5. The element fi51[-9q] is not the target of a differential in the AH*
 *SS
for P-9q.

Proof.Exactly as in the proof of Lemma 12.4, the differentials supported by ImJ
elements in the 15-stem can only hit other ImJ elements. The only elements left
which can support differentials are given below.
                ___             _____           ____
                fi21[-q - 1]    fi21ff1[-2q]    fi2fi1[-5q - 1]
                      ___              ________
                      fi41[-6q - 1]    fi3=3fi1ff1[-8q]
        ________
All but fi3=3fi1ff1[-8q] are the target or source of the AHSS d2 differentials *
 *displayed
below.
                              ___      O       _____
                              fi21[-q -_1]____//fi21ff1[-2q - 1]
             ___   O          _____
             fi21[-q]________//_fi21ff1[-2q]
                            ____         O    _______
                            fi2fi1[-5q - 1]__//_fi2fi1ff1[-6q - 1]
         ______        O     ___
         fi3=3ff1[-5q -_1]___//fi41[-6q - 1]




50                          MARK BEHRENS


The last of these_is_the result of a hidden ff1-extension in the ANSS. The rema*
 *in-_
ing element fi3=3fi1ff1[-8q] cannot support a differential whose target is fi51*
 *[-9q],

because P--8q9qis coreducible.

               13. Algebraic filtered root invariants


  In this section we will describe the Mahowald spectral sequence (MSS), which
is a spectral sequence that computes Ext(H*X) by applying Ext(H*-) to an E-
Adams resolution of X. This spectral sequence is described in [28]. We will bri*
 *efly
recall its construction for the reader's convenience. We will then define algeb*
 *raic
E-filtered root invariants, and indicate how some of the results of Section 5 c*
 *arry
over to the algebraic setting to compute algebraic root invariants from the alg*
 *ebraic
filtered root invariants in the MSS.
  Let E be a ring spectrum and suppose that H is an E-ring spectrum in the sense
that there is a map of ring spectra E ! H. Suppose furthermore that H*H is a fl*
 *at
H* module. Let X be a (finite) spectrum. By applying H* to the cofiber sequences
that make up the E-Adams resolution, we obtain short exact sequences

(13.1)     0 ! H*(Ws(X)) ! H*(Wss(X)) ! H*( Ws+1(X)) ! 0

which are split by the E-action map
                            __(s)         __(s)
       H ^ Wss(X) = H ^ E ^ E   ^ X ! H ^ E   ^ X = H ^ Ws(X).

Thus the short exact sequences 13.1 give rise to long exact sequences when we a*
 *pply
ExtH*H(-).  Therefore, upon applying Ext to the E-Adams resolution, we get a
spectral sequence

             Es,t,k1(H*X) = Exts-k,t(H*Wkk(X)) ) Exts,t(H*X)

called the Mahowald spectral sequence (MSS). For the remainder of this section *
 *we
shall assume H = HFp.
  Many of the notions that we defined for the homotopy root invariant carry over
to the algebraic context. We may define Tate comodules (over the dual Steenrod
algebra)
               t(H*X) = lim-(H*(X ^  P-N )) = (tH ^ X)*.
                        N
Here, it is essential that the limit is taken after taking homology. We may use*
 * this
to define algebraic E-root invariants.

Definition 13.1 (Algebraic E-root invariant ). Let ff be an element of the Ext
group Exts,t(H*X).  We have the following diagram of Ext groups which defines
the algebraic E-root invariant RE,alg(ff).

                           RE,alg(-)    s,t
              Exts,t(H*X) //o//o///o///oooooExt(H*E ^  -N+1 X)

                  f||                        'N||
                   fflffl|                     fflffl|
          Exts,t(t(H*(E ^ X)))__N__//_Exts,t(H*(E ^  P-N ^ X))

Here f is induced by the inclusion of the 0-cell of tS0,  N is the projection o*
 *nto
the -N-coskeleton, 'N is inclusion of the -N-cell, and N is minimal with respect
to the property that  N O f(ff) is non-zero.  Then the algebraic E-root invaria*
 *nt
RE,alg(ff) is defined to be the coset of lifts fl 2 Exts,t+N-1(H*E ^X) of the e*
 *lement




            ROOT INVARIANTS IN THE ADAMS SPECTRAL SEQUENCE           51


 N O f(ff). It could be the the case that f(ff) = 0, in which case we say that *
 *the
algebraic E-root invariant is trivial.

  For simplicity, we_now_restrict our attention to the case E = BP , and we wor*
 *k_at
an odd prime p. Let   be the periodic lambda algebra [14]. The_cohomology of
is Ext(H*). The_MSS has a concrete description in terms of  . Define a decreasi*
 *ng
filtration on   by    __    __    __    __
        __              = F0   _F1_   F2    . . .

where Fk  is the subcomplex of   generated by monomials containing_k_or more
~i's. The spectral sequence associated to the filtered complex {Fk  } is isomor*
 *phic
to the MSS starting with the E2-term. The spectral sequence of the filtered com*
 *plex
would agree with the MSS on the level of E1-terms if one were to apply Ext(H*-)*
 * to
the correct BP -resolution (which differs from the canonical BP -resolution). T*
 *here-
fore,_we shall also refer to the spectral sequence associated to the filtered c*
 *omplex
{Fk  } as a MSS.                                           __
  In analogy with the spaces Ws(X), define subcomplexes of     H*(X) (the
complex which computes Ext(H*X)) by
                                   __
                       Wk(H*X) = Fk    H*(X).

We define quotients Wkl(H*X) by

                   Wkl(H*X) = Wk(H*X)=Wl+1(H*X).

The cohomology H*(Wkl(H*X)) is computed by restricting the MSS by setting the
E*,*,i1-term equal to zero for i < k or i > l. We define complexes Wkl(H*P N) t*
 *o be
the inverse limit
                     Wkl(H*P N) = lim-Wkl(H*P-NM).
                                  M
  For sequences

                       I = {k1 < k2 < . .<.kl}

                       J = {N1 < N2 < . .<.Nl}

we can define filtered Tate complexes
                                 X
                  WI(H*P J^ X) =    Wki(H*P Ni^ X).
                                  i

Given another pair of sequences (I0, J0)   (I, J), we define complexes
             0    J              J               J0-1
          WII(H*PJ0^ X) = WI(H*P  ^ X)=WI0+1(H*P     ^ X)

where I0+1 (respectively J0-1) is the sequence obtained by increasing (decreasi*
 *ng)
every element of the sequence by 1.
  We shall now define algebraic filtered root invariants in analogy with the de*
 *f-
inition of filtered root invariants given in Section 3.  Let ff be an element of
Exts,t(H*X). We shall describe a pair of sequences

                     I = {k1 < k2 < . .<.kl}

                     J = {-N1 < -N2 < . .<.-Nl}

associated to ff, which we define inductively. Let k1   0 be maximal such that *
 *the
image of ff under the composite

            Ext s,t-1(H* -1X) ! Hs,t-1(W0k1-1(H*P ^ X)-1 )




52                          MARK BEHRENS


is trivial.  Next, choose N1 to be maximal such that the image of ff under the
composite

        Exts,t-1(H* -1X) ! Hs,t-1(W0(k1-1,k1)(H*P(-N1+1,1)^ X))

is trivial. Inductively, given

                        I0= (k1, k2, . .,.ki)

                       J0 = (-N1, -N2, . .,.-Ni)

let ki+1be maximal so that the image of ff under the composite
                                     0-1,ki+1-1)
       Exts,t-1(H* -1X) ! Hs,t-1(W0(I        (H*P(J0+1,1)^ X))

is trivial. If there is no such maximal ki+1, we declare that ki+1= 1 and we are
finished. Otherwise, choose Ni+1to be maximal such that the composite
                                0-1,ki+1-1,ki+1)
  Exts,t-1(H* -1X) ! Hs,t-1(W0(I           (H*P(J0+1,-Ni+1+1,1)^ X))

is trivial, and continue the inductive procedure. We shall refer to the pair (I*
 *, J) as
the BP -bifiltration of ff.
  Observe that there is an exact sequence

Hs,t-1(WI(H*P J^ X)) ! Exts,t-1(t(H* -1X)) ! Hs,t-1(W I-1(H*PJ+1 ^ X)).

Our choice of (I, J) ensures that the image of ff in Hs,t-1(W I-1(H*PJ+1 ^ X)) *
 *is
trivial. Thus ff lifts to an element fff2 Hs,t-1(WI(H*P J^ X)).

Definition 13.2 (Algebraic filtered root invariants ). Let X be a finite complex
and let ff be an element of Exts,t(H*X) of BP -bifiltration (I, J).  Given a li*
 *ft
fff2 Hs,t-1(WI(H*P J^ X)), the kthalgebraic filtered root invariant is said to *
 *be
trivial if k 6= ki for any ki 2 I. Otherwise, if k = ki for some i, we say that*
 * the
image fi of fffunder the quotient map

            Hs,t-1(WI(H*P J^ X)) ! Hs,t-1(Wkkii(H* -NiX))

is an element of the kth algebraic filtered root invariant of ff.  The kth alge*
 *braic
filtered root invariant is the coset R[k]E,alg(ff) of the MSS E1-term Es,t+Ni-1*
 *,k1(H*X)
of all such fi as we vary the lift fff.

  We wish to indicate how our filtered root invariant theorems carry over to the
algebraic context.  In order to do this we must produce algebraic versions of K-
Toda brackets. Suppose that M is a finite A*-comodule concentrated in degrees 0
through n with a single Fp generator in degrees 0 and n. In what follows, we let
Mj be the sub-comodule of M consisting of elements of degree less than or equal
to j, and let Mjibe the quotient Mj=Mi-1. We shall omit the top index for the
quotient Mi= M=Mi-1.

Definition 13.3 (Algebraic M-Toda bracket ). Let

                      f : Exts,t(M1) ! Exts+1,t(Fp)

be the connecting homomorphism associated to the short exact sequence

                        0 ! Fp ! M ! M1 ! 0.

Let   : M1 !  nFp be the projection onto the top generator.  Suppose ff is an
element of Exts,t(H*X). We have

         Exts,t(H*X) - *Exts,t+n(H*X   M1) f-!Exts+1,t+n(H*X).




            ROOT INVARIANTS IN THE ADAMS SPECTRAL SEQUENCE           53


We say the algebraic M-Toda bracket

                       <M>(ff)   Exts+1,t+n(H*X)

is defined if ff is in the image of  *.  Then the algebraic M-Toda bracket is t*
 *he
collection of all f(fl) 2 Exts+1,t+n(H*X) where fl 2 Exts,t+n(H*X   M1) is any
element satisfying  *(fl) = ff.

  We leave it to the reader to construct the dually defined algebraic M-Toda
bracket analogous to the dually defined K-Toda bracket, and MSS version of the
algebraic M-Toda bracket analogous to the E-ASS version of the K-Toda bracket.
The definitions of these Toda brackets given in Section 4 carry over verbatim to
the algebraic context.
  With these definitions, our filtered root invariant results given in Section *
 *5 have
analogous algebraic statements.  The first algebraic filtered root invariant is*
 * the
algebraic BP -root invariant. If the kthalgebraic filtered root invariant is a *
 *perma-
nent cycle in the MSS, then it is the algebraic root invariant modulo BP -filtr*
 *ation
greater than k.  One may deduce higher algebraic filtered root invariants from
differentials and compositions in the MSS. In other words, there are algebraic *
 *ver-
sions of Theorem 5.1, Corollary 5.2, Theorem 5.3, Theorem 5.4, Proposition 5.7,
Proposition 5.8, and Corollary 5.9, where one makes the following replacements:

     o Replace ss* with Ext.
     o Replace root invariants with algebraic root invariants.
     o Replace E-root invariants with algebraic E-root invariants.
     o Replace filtered root invariants with algebraic filtered root invariants.
     o Replace K-Toda brackets with algebraic H*(K)-Toda brackets.
     o Specialize from E to BP .
     o Replace the E-ASS with the MSS.


                14.Modified filtered root invariants

  In this section we will recall the modified root invariant that Mahowald and
Ravenel define in [25].  We will then explain how to define modified versions of
filtered root invariants, and how to adapt our theorems to compute modified root
invariants. We also discuss algebraic modified filtered root invariants, which *
 *is the
tool we will be using in Section 15.
  Let X be a finite complex.  We shall begin be recalling the definition of the
modified root invariant R0(ff) for an element ff 2 sst(X).  Let V (0) be the mod
p Moore spectrum.  We define modified versions of the stunted Tate spectra PMN
which will have the same underlying space, but which we shall view as being bui*
 *lt
out of V (0) instead of the sphere spectrum. To this end, define spectra
                      (  N+1

                P 0N=  P      if N is of the form kq - 1
                       P N    otherwise.

Likewise, for M   N, define quotient spectra

                          P 0NM= P 0N=P 0M-1.

Thus the spectrum P 0NMhas a V (0)-cell in every dimension from M to N congruent
to -1 modulo q. There are cofiber sequences

                    P 0(l-1)q-1kq-1! P 0lq-1kq-1!  lq-1V (0).




54                          MARK BEHRENS


Definition 14.1 (Modified root invariant ). Let X be a finite complex, and supp*
 *ose
we are given ff 2 sst(X). The modified root invariant of ff is the coset of all*
 * dotted
arrows making the following diagram commute.

                        St_____//_________ -N+1 V (0) ^ X

                         ff||          ||
                         fflffl|       ||

                        X              |
                         |             |
                         |             |
                         fflffl|       fflffl|
                       tX _______// P-0N^ X

This coset is denoted R0(ff). Here N is chosen to be minimal such that the comp*
 *osite
St !  P-0N^ X is non-trivial.

  Thus the definition of the modified root invariant differs from the definitio*
 *n of
the root invariant in that it takes values in ss*(V (0) ^ X) instead of ss*(X),*
 * and
that P has been replaced with P 0.  By making these replacements elsewhere, we
may produce modified versions of our other definitions, as summarized below. In
the_case_of the algebraic filtered root invariants,_we choose to use the quotie*
 *nt
 (0)=  =(v0) described in [14]. The cohomology of  (0)is given by
                           __
                        H*(  (0)) = Ext(H*V (0)).

One has modified versions of the subcomplexes Wk(H*X) given by
                                  __          __
         Wk0(H*X) = Im(Wk(H*X) ,!     H*(X) !  (0)  H*(X)).

By modifying our definitions, we produce:

     o Modified E-root invariants

                     R0E: ss*(X) _ ss*(E ^ V (0) ^ X)

     o Modified algebraic root invariants

                  R0alg: Ext(H*X) _ Ext(H*V (0) ^ X)

     o Modified filtered Tate spectra

                               WI(P 0J)

     o Modified filtered root invariants
                       0                  k
                    R[k]E: ss*(X) _ ss*(Wk(V (0) ^ X))

     o Modified algebraic E-root invariants

                R0E,alg: Ext(H*X) _ Ext(H*E ^ V (0) ^ X)

     o Modified algebraic filtered root invariants
                      0                  *   0k
                  R[k]E,alg: Ext(H*X) _ H (W  k(H*X))

  We would like to reformulate the results of Section 5 in terms of modified ro*
 *ot
invariants. We need a modified version of the Toda brackets which appear in the
statements of the main theorems. Suppose that K is a finite complex built out of
V (0) with a single bottom V (0)-cell in dimension 0 and a single top V (0)-cel*
 *l in
dimension n. Let Kj be the jth V (0)-skeleton of K, and let Kjibe the quotient
Kj=Ki-1. We shall omit the top index for the ithV (0)-coskeleton Ki= K=Ki-1.




            ROOT INVARIANTS IN THE ADAMS SPECTRAL SEQUENCE           55


Definition 14.2 (Modified K-Toda bracket ). Let

                           f :  -1K1 ! V (0)

be the attaching map of the first V (0)-coskeleton of K to the zeroth V (0)-cel*
 *l, so
that the cofiber of f is K. Let   : K1 !  nV (0) be the projection onto the top
V (0)-cell. Suppose ff is an element of sst(X ^ V (0)). We have


           sst(X ^ V (0)) - *sst+n(X ^ K1) f*-!sst+n-1(X ^ V (0)).

We say the modified K-Toda bracket

                      <K>0(ff)   sst+n-1(X ^ V (0))

is defined if ff is in the image of  *.  Then the modified K-Toda bracket is the
collection of all f*(fl) 2 sst+n-1(X ^ V (0)) where fl 2 sst+n(X ^ K1) is any e*
 *lement
satisfying  *(fl) = ff.

  Similarly we may provide a modified version of the dually defined and E-ASS
K-Toda brackets given in Section 4. Furthermore, for an A*-comodule M which
is cofree over the coalgebra E[o0], it is straightforward to define the modifie*
 *d al-
gebraic M-Toda bracket by varying the definition of the M-Toda bracket given in
Section 13.
  Modified versions of all of the results in Section 5 hold, with proofs that go
through with only superficial changes.  Specifically, one must make the followi*
 *ng
adjustments.

     o Replace P with P 0
     o Replace root invariants with modified root invariants.
     o Replace E-root invariants with modified E-root invariants.
     o Replace filtered root invariants with modified filtered root invariants.
     o Replace K-Toda brackets with modified K-Toda brackets.
     o Replace the E-ASS for X with the E-ASS for X ^ V (0).

Furthermore, there are algebraic modified versions of Theorems 5.1, 5.3 and 5.4,
Propositions 5.7 and 5.8, and Corollaries 5.2 and 5.9, where one makes the foll*
 *owing
replacements:

     o Replace P with P 0.
     o Replace ss* with Ext.
     o Replace root invariants with modified algebraic root invariants.
     o Replace E-root invariants with modified algebraic E-root invariants.
     o Replace filtered root invariants with modified algebraic filtered root i*
 *nvari-
       ants.
     o Replace K-Toda brackets with modified algebraic H*(K)-Toda brackets.
     o Specialize from E to BP .
     o Replace the E-ASS for X with the MSS for H*X ^ V (0)

  We now modify our definitions even further, to give variations of the second
modified filtered root invariant R00(ff) given in [25]. In what follows we shal*
 *l always
be working at a prime p   3.
  It is observed in [25] that there is a splitting


            Pk(l+1)qq-1^ V (0) ' P 00lq-1kq-1_  kqV (0) _  (l+1)q-1V (0)




56                          MARK BEHRENS


In the notation of [25], we have

                                          __lq
                            P 00lq-1kq-1= Pkq-1


The spectra P 00lq-1kq-1are built out of the Smith-Toda complex V (1).  There i*
 *s a
V (1)-cell every in dimension from kq - 1 to lq - 1 congruent to -1  mod q. The
decomposition of P-001into V (1)-cells is described on the level of cohomology *
 *in
the following lemma, which is a straightforward computation.


Lemma 14.3. Let e*nbe the generator of H*(P )-1  in dimension n, where n
0, -1 (mod  q). The cohomology of the Moore spectrum is given by


                           H*(V (0)) = E[Q0]


as a module over the Steenrod algebra.  The cohomology H*(P 00)-1  = H*(P ^
V (0))-1  is free over the subalgebra E[Q0, Q1] of the Steenrod algebra on the *
 *gen-
erators e*kq-1 1. The action E[Q0, Q1] on the free E[Q0, Q1]-submodule generated
by e*kq-1  1 is then given by the following formulas.


                       Q0(e*kq-1  1)= e*kq  1 - e*kq-1  Q0

                       Q1(e*kq-1  1)= e*(k+1)q  1

              Q1(e*kq  1 - e*kq-1  Q0)= -e*(k+1)q  Q0

                      Q0(e*(k+1)q  1)= e*(k+1)q  Q0


  It is convenient to allow arbitrary subscripts and superscripts, so we define*
 * for
integers M   N

                            P 00NM= P 00lq-1kq-1


where k (respectively l) is minimal (maximal) such that M   kq - 1   N (M
lq - 1   N). If there is no such k and l, then we have P 00NM= *.
  The following lemma is Lemma 3.7(e) of [25].


Lemma 14.4. If k and l are congruent to 0 (mod  p), then H*(P 00lq-1kq-1) is fr*
 *ee over
the subalgebra A(1) of the Steenrod algebra.


  Since there is no difference between the spectra P-001and P-1 ^ V (0), and si*
 *nce
V (0) is p-complete, there is an analog of Lin's theorem [25, Lemma 3.7(b)].


Lemma 14.5. The map

                            -1V (0) ! P-001


is an equivalence.


  In light of Lemma 14.5, Mahowald and Ravenel [25] define a second modified
root invariant.


Definition 14.6 (Second modified root invariant ). Let X be a finite complex, a*
 *nd
suppose we are given ff 2 sst(X ^ V (0)). The second modified root invariant of*
 * ff is




            ROOT INVARIANTS IN THE ADAMS SPECTRAL SEQUENCE           57


the coset of all dotted arrows making the following diagram commute.


                        St________//______________ -N+1 V (1) ^ X
                                          |
                         ff||             |
                         fflffl|          ||
                     X ^ V (0)            |
                                          |
                         |                |
                         |                |
                         fflffl|          fflffl|
                    tX ^ V (0)_______// P-00N^ X


This coset is denoted R00(ff). Here N is chosen to be minimal such that the com-
posite St !  P-00N^ X is non-trivial.

  The definition of the second modified root invariant may be obtained from the
definition of the (first) modified root invariant by smashing X with V (0), rep*
 *lac-
ing P 0with P 00, and replacing V (0) with V (1). Similarly, one may obtain sec*
 *ond
modified versions of all of the other definitions we have been working with, as*
 * sum-
marized below._The_second modified_lambda_complex Wk00(H*X) is the appropriate
subcomplex of  (1)  H*(X), where  (1)=  =(v0, v1).

     o Second modified E-root invariants

                  R00E: ss*(X ^ V (0)) _ ss*(E ^ V (1) ^ X)

     o Second modified algebraic root invariants

               R00alg: Ext(H*X ^ V (0)) _ Ext(H*V (1) ^ X)

     o Second modified filtered Tate spectra


                               WI(P 00J)

     o Second modified filtered root invariants
                    00                        k
                R[k]E: ss*(X ^ V (0)) _ ss*(Wk(V (1) ^ X))

     o Second modified algebraic E-root invariants

             R00E,alg: Ext(H*X ^ V (0)) _ Ext(H*E ^ V (1) ^ X)

     o Second modified algebraic filtered root invariants

                  00                          *   00k
               R[k]E,alg: Ext(H*X ^ V (0)) _ H (W  k(H*X))


If K is a finite complex built from V (1) with single bottom and top dimensional
V (1)-cells, then we may define the second modified K-Toda bracket

                  <K>00: ss*(V (1) ^ X) _ ss*(V (1) ^ X).

Likewise, if M is an A*-comodule which is cofree over E[o0, o1], one can define*
 * a
second modified algebraic M-Toda bracket on Ext(H*V (1) ^ X). Just as outlined
for the case of modified root invariants in the first half of this section, sec*
 *ond
modified and second modified algebraic versions of the results of Section 5 hol*
 *d.




58                          MARK BEHRENS


15. Computation of some infinite families of root invariants at p = 3

  In this section we will extend the computation

                            (-v2)i2 R00(vi1)

of [25] for p   5 to the prime 3 for i   0, 1, 5 (mod  9). We will use our modi*
 *fied
root invariant computations to deduce that, at the prime 3, the root invariant *
 *of
the element ffi2 ssS*is given by the element fii for i   0, 1, 5 (mod  9).
  Throughout this section we work at the prime 3. Low dimensional computations
indicate that there is a map
                            v2 : S16 ! V (1).

Oka [31] demonstrates that there is a map

                            v52: S80 ! V (1).

Composing these maps with iterates of the map

                         v92:  144V (1) ! V (1)

given in [4], we have maps
                            vi2: S16i! V (1)

for i ~=0, 1, 5 (mod  9). The computation of ss*(L2V (1)) given in [12] indicat*
 *es that
these are the only i for which vi2can exist.

Theorem 15.1. At the prime 3, for i   0, 1, 5 (mod  9), the second modified root
invariant of vi12 ss*(V (0)) is given by

                           (-v2)i2 R00(vi1).

  We shall prove Theorem 15.1 with a sequence of lemmas.

Lemma 15.2. For all i, we have the second modified BP -root invariant

                           (-v2)i2 R00BP(vi1).

Proof.Modulo I1, the 3-series of the formal group associated to BP is given by

                   [3](x)   v1x3 + v2x9 + . . .(mod  I1).

Thus in tBP ^ V (0)* (see Lemma 10.4), we have

                           v1 = -v2x6 + . . .

and upon taking the ithpower, we have

                         (v1)i= (-v2)ix6i+ . . .

Using the second modified version of Corollary 10.5, we may deduce the result.

Lemma 15.3. For all i, we have the second modified algebraic BP -root invariant

                          (-v2)i2 R00BP,alg(vi1).

Here, v1 and v2 are viewed as elements in the cohomology of the periodic lambda
algebra.

Proof.Apply Ext(H*-) to the diagram which realizes the second modified BP -root
invariant given by Lemma 15.2.

  In order to eliminate obstructions in higher Adams filtration, we prove the f*
 *ol-
lowing lemma. This lemma is essentially contained in the proof of Lemma 3.10 of
[25].




            ROOT INVARIANTS IN THE ADAMS SPECTRAL SEQUENCE           59


Lemma 15.4. We have the following Ext calculation.
                                           (

                Exts,iq-1+s(H*P-003iq-1) =  0       s > i
                                            Fp{vi2} s = i


Proof.In Lemma 14.4, we saw that H*(P-003iq-1) is free over the subalgebra A(1)
of the Steenrod algebra on generators in dimensions congruent to -1 (mod  3q).
Therefore ExtA*(H*P-003iq-1) is built out of ExtA*( 3kq-1A(1)*) for k   -i. Now
the May spectral sequence E2-term for ExtA*(A(1)*) is given by

E2 = E[hi,j: i   1, j   0, i+j   2] P [bi,j: i   1, j   0, i+j   2] P [vi : i  *
 * 2].

Analyzing the degrees that these elements live in establishes that the only ele*
 *ments
in ExtA*(A(1)*) which lie on or above the line of slope 1=|v2| (in the (t-s, s)*
 *-plane)
are those of the form hffl11,1hffl22,0vk2for fflj = 0, 1. Therefore, one easily*
 * deduces that
there are no elements which lie above the line of slope 1=|v2| passing through *
 *the
point (0, -3iq - 1) which lie in the dimensions we are considering, and the only
element which lies in the iq - 1 stem with Adams filtration equal to i is vi2.


Lemma 15.5. For all i, we have the second modified algebraic root invariant

                           (-v2)i2 R00alg(vi1)

Proof.Applying the second modified algebraic version of Proposition 5.7, we may
deduce that the zeroth second modified algebraic filtered root invariant is giv*
 *en by

                                       00      i
                          (-vi2) 2 R[0]BP,alg(v1).


The element vi2is a permanent cycle in the MSS for Ext(H*V (1)), so we may use
the second modified algebraic version of Theorem 5.1 to deduce that the differe*
 *nce
of the images of (-v2)i and vi1in

                         Exti,iq-1+i(H*P-003iq-1)

is of BP -filtration greater than 0.  Lemma 15.4 says that there are no non-zero
elements of this Ext group which could represent the difference of these images*
 *, so
we actually have

                           (-v2)i2 R00alg(vi1).




  We shall need a slightly different result for i   5 (mod  9).

Lemma 15.6. Modulo indeterminacy, the second modified algebraic root invariant
of v9t+51is also given by

                   (-v2)9t+5  v9t2v3g0b0 2 R00alg(v9t+51)


Proof.By Lemma 15.4, the element v9t2v3g0b0 is in the kernel of the map

           Exti,iq-1+i(H* -3iq-1V (1)) ! Exti,iq-1+i(H*P-003iq-1)

where i = 9t + 5.  Therefore it is in the indeterminacy of the second modified
algebraic root invariant.




60                          MARK BEHRENS


Proof of Theorem 15.1.The element vi2is a permanent cycle in the ASS for V (1)
for i = 0, 1. The existence of the self map v92of the complex V (1) implies tha*
 *t vi2is
a permanent cycle in the ASS for i   0, 1 (mod  9). However, in the ASS for V (*
 *1),
the element v52is not a permanent cycle; the element



                              v52  v3g0b0



is the permanent cycle that detects the Oka element v522 ss80(V (1)) [4, Remark
8.2]. Thus the element v9t+522 ss*(V (1)) is detected in the ASS by the permane*
 *nt

cycle v9t+52  v9t2v3g0b0. For uniformity of notation, let evi2denote the ASS el*
 *ement
that detects vi22 ss*(V (1)) for i   0, 1, 5 (mod  9).
  Lemmas 15.5 and 15.6 imply that we have second modified algebraic root invari-
ants


                           (-1)ievi22 R00alg(vi1)


where i   0, 1, 5 (mod  9).  By the second modified version of Theorem 5.10, we
have second modified filtered root invariants given by


                                        00  i
                          (-1)ievi22 R[i]H(v1).



The elements evi2are permanent cycles in the ASS, so by the second modified ver-
sion of Theorem 5.1, we may conclude that the difference between the images of
(-v2)i and vi1in ssiq-1(P-003iq-1) has Adams filtration greater than i.  Accord*
 *ing
to Lemma 15.4, there are no such elements, so the images of (-v2)i and vi1in
ssiq-1(P-003iq-1) are actually equal, and we have



                           (-v2)i2 R00(vi1).







  We will now use our second modified root invariant computations to deduce the
following.


Theorem 15.7. We have root invariants



                           (-1)i+1fii2 R(ffi)



for i   0, 1, 5 (mod  9) at p = 3.


Proof.Let   denote the map which is given by projection onto the top cell of V *
 *(0).
Since Pj00q-1is a summand of Pjq-1^ V (0), we have an induced map



                          0: Pj00q-1!  P(j+1)q.




            ROOT INVARIANTS IN THE ADAMS SPECTRAL SEQUENCE           61


We have the following diagram for i   0, 1, 5 (mod  9).

                               (-1)ifii
                          _____________________________________________________*
 *_____________________________________________________________________________*
 *_____________________________________________________________________________*
 *_____________________________________________________________________________*
 *_______________________________@
                    ___________________________________________________________*
 *_____________________________________________________________________________*
 *_____________________________________________________________________________*
 *_____________________________________________________________
               ____________________________________))__________________________*
 *_____________________________________________________________________________*
 *_________________________________
          Sqi-1K____________i____//_ -3iq-1V (1)00//_S(-3i+1)q+1
               KKK i   (-v2)           |               ||
          ffi||  Kv1KKKK               |               |
            fflffl|  K%%               ||              |
           S0 oo_____-1V (0)   (*)     |     (-1)      ||
                                       |               |
            |           |              |               |
            |           |              |              '|
            ||          fflffl|        fflffl|         |
            |         P-001________//_P-003iq-1        |
            |       tt                    OOO          |
            |     ttt                        OOO0      |
            |   ttt                             OOOO   |
            fflffl|yytt                            O'' fflffl|
          P-1  ___________________________________// P(-3i+1)q


The commutivity of the inner portion (*) is our modified root invariant computa-
tions of vi1. The portion of the diagram marked with a "(-1)" commutes with the
introduction of a minus sign. This is due to the fact that the elements fiiare *
 *given
by  00*(vi2) where

                             00: V (1) ! S6

is the projection onto the cell corresponding to the element Q1Q0 in the cohomo*
 *logy
group

                         H*(V (1)) = E[Q0, Q1].

According to the computations of Lemma 14.3, this is the negative of the cell in
P-00i3q-1which corresponds to the cohomology element

                   e*(-3i+1)q  Q0 = Q0Q1(e*-3iq-1  1).

Following the outer portion of the diagram, and introducing the minus sign, we *
 *see
that we have

                           (-1)i+1fii2 R(ffi)

unless '*(fii) = 0. That is not possible by Proposition 8.7.


Remark 15.8. If one believes that the root invariant takes v91multiplication to
v92multiplication at p = 3, analogous to the results of Sadofsky [35] and Johns*
 *on

[18], then it is seems likely that fii2 R(ffi) for i   0, 1, 3, 5, 6 (mod  9). *
 *The present
methods will not extend because vi2is not a permanent cycle in the ANSS for V (*
 *1)
for i 6  0, 1, 5 (mod  9).  In fact, at the time of writing we still do not kno*
 *w that
the elements fi9t+3exist. The elements fi9t+1fi1ff1 and fi9t+1fi41do exist, and*
 * are
non-trivial by calculations of Shimomura and Wang [34].  We conjecture that we
have root invariants

                         fi9t+1fi1ff12R(ff9t+2)

                           fi9t+1fi412R(ff9t+4)

The case t = 0 was already handled in Section 12.




62                          MARK BEHRENS


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  Department of Mathematics, Massachusetts Institute of Technology, Cambridge,
MA 02139, U.S.A.