AN APPLICATION OF THE THOMIFIED EILENBERG-MOORE
SPECTRAL SEQUENCE TO THE SPECTRA T (n)
DOUGLAS C. RAVENEL
December 7, 1999
Theorem 1. The first nontrivial differential in the Adams-Novikov spectral se-
quence for the spectrum T (1) at an odd prime p is
d2p-1(b3;0) = h2;0bp2;0
4-2p
where b3;02 E2;2p2 .
This element is in dimension 2p4 - 2p - 2, which is 154 for p = 3 and 1238 for
p = 5. The notation will be explained below.
T (1) is the p-local spectrum with
BP*(T (1)) = BP*[t1]:
It is the minimal summand of the p-local Thom spectrum for the stable complex
vector bundle induced by the map
SU(p) ! SU = BU:
It is also the p-local Thom spectrum for the map
S2p-1 ! BU
obtained by extending the generator of
ss2p-2(BU) = Z
from S2p-2 to S2p-1.
More generally for each n 0 there is a p-local ring spectrum T (n) with
BP*(T (n)) = BP*[t1; : :;:tn]:
It is a minimal summand of the p-local Thom spectrum associated with the map
SU(m) ! SU = BU
for any m with pn m < pn+1. This was proved in [Rav86 ] as Theorem 6.5.1.
We will now consider the Adams-Novikov spectral sequence for T (n). For a
spectrum let, let
Ext(BP*(X)) = ExtBP*(BP)(BP*; BP*(X))
Then the following was proved in [Rav86 ] as Proposition 6.5.9 and Theorem 6.5.*
*11.
Theorem 2. (i)Let A(n) = Z(p)[v1; : :;:vn] BP*. Then the map
Ext0(BP*(T (n))) ! Ext0(BP*(BP )) = BP*
is monomorphic with image A(n).
1
2 DOUGLAS C. RAVENEL
(ii)Unless n = 0 and p = 2 (which is the subject of [Rav86 , 5.2.6]), Ext1(BP**
*(T (n)))
is the A(n)-module generated by the images of the elements
vn+1_2 Ext0(BP (T (n)) N1)
pi *
where N1 = BP* Q=Z(p)and we are mapping into Ext1with the connecting
homomorphism for the short exact sequence
0 ! BP* ! BP* Q ! N1 ! 0:
For the 2-line and above, we have the following, essentially proved as Theorem
7.1.13 in [Rav86 ].
Theorem 3. For n > 0, Ext2;t(BP*(T (n))) for t 2(pn+3 - p) is the A(n + 1)-
module generated by
ae i p oe( p pn+1 )
vn+2_ vn+2_ vn+3_ v2vn+2_ v2___vn+1_
ipv1; pvi1:1 i p[ pv1 - pv1+p1+ p2v1
and each of these supports a copy of
E(hn+1;0) P (bn+1;0) where hn+1;0= vn+1_pand bn+1;0= vn+2_pv:
1
We also let
vpn+1
bn+1;1 = ____pv
1
n+1
v2vpn+2 vp2 vn+1
and bn+2;0 = vn+3_pv- ______1+p+ _________2:
1 pv1 p v1
Remark: Theorem 7.1.13 of [Rav86 ] omits the element b3;0and is only correct
for t < |vpn+2|. The extra generator in that dimension follows easily from oth*
*er
results of [Rav86 , Chapter 6], and we will extend the methods and resilts of C*
*hapter
7 in a future paper.
This Ext group is illustrated for n = 1 and p = 3 in Figure .
To prove Theorem 1 we use cohomology operations in T (1)-theory, i.e., maps
T (1) ----ri-----!2i(p-1)T (1)
derived from the splitting of T (1) ^ T (1). They have properties similar to St*
*eenrod
and Quillen operations, but they commute with each other. In E2 we have
r1(b3;0)= 0;
rp(b3;0)= -b2;1;
and r1(b2;1) = rp(b2;1)= 0:
Lemma 4. In ss*(T (1)),
r1(b2;1) = bp2;0 and rp(b2;1) = 0:
This means that in ss*(T (1)),
r1rp(b3;0)= bp2;0
but rpr1(b3;0)= 0;
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|| THE THOMIFIED EILENBERG-MOORE SPECTRAL SEQUENCE 3 ||
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| |_ s| |
| 7 | | |
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| 6 |_ b32;s0 | |
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| |_ s | |
| 5 | | |
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| | 2 s v3b2;0s | |
| 4 |_ b2;0 _____ | |
| |6 | 3v1 | |
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| | 3 |_ h2;0b2;0s s | |
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|s | 2 | |
| |_ s v3_ s s_sss_s| |
| 2 | b2;0 b2;1 s| b3;0 |
| | 3v1 |
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| | h2;0 |
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| 1 |_ s s ss| s s ss| s s ss| s |
| | s| |
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| 0_|_________________________________________________________|||||_||_|
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| 0 40 80 120 160 |
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| t - s ___________- |
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Figure 1. The Adams-Novikov E2-term for T (1) at p = 3 in di-
mensions 154, showing the first nontrivial differential. Elements
on the 0- and 1-lines divisible by v1 are not shown. Elements on
the 2-line and above divisible by v2 are not shown.
which is a contradicts the commutativity of the ri. This means that b3;0cannot
survive to represent a homotopy element. The indicated differential
d2p-1(b3;0) = h2;0bp2;0
is the only one it can support.
To prove Lemma 4 we need the Thomified Eilenberg-Moore spectral sequence of
[MRS ]. Given a fibration of spaces
X ! E ! B
with a stable vector bundle over E, we get a spectral sequence converging to the
homotopy of Y , the Thom spectrum for the induced bundle over X.
If the fibration is
SU(p2 - 1) ! SU ! B
4 DOUGLAS C. RAVENEL
with the evident vector bundle over SU = BU, we get the usual Adams-Novikov
spectral sequence for X(p2- 1), the Thom spectrum associated with SU(p2- 1),
which has T (1) as a summand.
If we take the Cartesian product of this fibration with
2-1 2p2-1
2S2p ! pt:! S ;
the E2-term is a subquotient of the tensor product of the one above with H*(2S2*
*p2-1)
equipped with the Eilenberg-Moore filtration.
The map
2-1 2
2S2p ! SU(p - 1)
sends this homology to
E(h2;0; h2;1; : :): P (b2;0; b2;1; : :):
in the Adams-Novikov spectral sequence for T (1). The lemma then follows from
the fact that in H*(2S2p2-1),
P*1(b2;1) = bp2;0 and P*p(b2;1) = 0:
Next we will show that there are no differentials in a similar range for T (n*
*) for
n > 1. Figure illustrates this for p = 3 and n = 2. There is no generator on t*
*he
2-line in the right position to kill h3;0b33;0, and there is no target for a di*
*fferential
supported by b4;0.
References
[MRS] M. E. Mahowald, D. C. Ravenel, and P. Shick. The Thomified Eilenberg-Moor*
*e spectral
sequence. http://www.math.rochester.edu:8080/u/drav/temss.dvi.
[Rav86]D. C. Ravenel. Complex Cobordism and Stable Homotopy Groups of Spheres. *
*Academic
Press, New York, 1986.
University of Rochester, Rochester, NY 14627
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|| THE THOMIFIED EILENBERG-MOORE SPECTRAL SEQUENCE 5 ||
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| 7 |_ q |
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| 6 |_ b3;q0 |
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| |_ q q |
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| | v4b3;0 |
| 4 |_ b23;0q _____ q |
| | 3v1 |
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| |6 | |
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| | 3 |_ q q |
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| | 2 |
| |_ b q v4_ q b qqqq__qb |
| s 2 | 3;0 3;1 q| 4;0 |
| | 3v1 |
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| | h3;0 |
| 1 |_ q q qq| q q qq| q q qq| q |
| | q| |
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| 0 ||_ |
| 0 100 200 300 400 500 |
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| t - s__________- |
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Figure 2. The Adams-Novikov E2-term for T (2) at p = 3 in
dimensions 520. Elements on the 0- and 1-lines divisible by v1
or v2 are not shown. Elements on the 2-line and above divisible by
v2 or v3 are not shown.