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
* 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.=
(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
*