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
(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 other
results of [Rav86 , Chapter 6], and we will extend the methods and resilts of Chapter 7 in a future paper.