On the unstable Adams spectral sequence for SO and U, and splittings of unstable Ext groups
A. K. Bousfield Donald M. Davis
University of Illinois at Chicago LehighUniversity
Chicago, IL 60680 Bethlehem, PA 18015 y
This paper isdedicated to the memory of Jose Adem.
1 Introduction
Let U denote the category of unstable modules over the mod 2 Steenrod algebra
A,and let Exts;t() = ExtsU(;tZ2). Let M1 =fH (CP1+) denote the unstable
A-module withnonzero classes xisuch that i is odd and positive, and
ikj
Sq2jx2k+1= j x2(j+k)+1:
ThenH!(U)!ss U (M1), where the left side is the mod 2 cohomology of the infinite
!
unitarygroup,!and!the right side the free unstable A-algebra generated by M1 .T*
*hus
there!isan!unstable Adams spectral sequence (UASS), defined as in [5], convergi*
*ng
to!ss(U!)with E2s;tssExts;t(M1 ). We shall construct an algebraic spectral sequ*
*ence
!
(SS)which!we!conjecture agrees with this UASS. In Section 3, we perform the min*
*or
modificationsrequired!to!yield the analogous results for the infinite special o*
*rthogonal
!
groupSO.!
!
!! Part of our conjecture is a splitting result for Ext(M1). Let Mn denote the
subspaceof!M1! spanned by those xisuch that ff(i) n, whereff(i) denotes the
!
AMSSubject Classification 55T15, 55R45. Key words: unstable Adams spectral
sequence, splittings of Ext groups, looping resolutions
yBoth authors were supported by National Science Foundation research grants.
number of 1's in the binary expansion of i. This provides a filtration of M1 *
*by
A-submodules,associated to which is a trigraded SS with
En;s;t1= Exts;t(Mn=Mn1 );
dr: En;s;tr! En+r;s+1;tr, and Exts;t(M1 ) filtered withnth subquotient En;s;t1.
Conjecture 1.1 This SS collapses to an isomorphism
M
Ext s;t(M1 ) ss Exts;t(Mn=Mn1 ):
n1
Conjecture 1.1, which should be of much interest in its own right, is implied b*
*y our
L
main conjecture, because the algebraic SSwhich we construct has Ext(Mn=Mn1 )
as its E2-term. The following analogue for SOmay be of even more interest.
Conjecture 1.2 Let Qndenote the subquotient of fHRP1 spanned by classes xiwith
ff(i) = n. Then
M
Exts;t(fH RP1 ) ss Exts;t(Qn):
n1
TheUASS on which we focus is somewhat unusual,since we know the homotopy
groups which it is computing, by Bott periodicity. We do not, however, know the
Adams filtrations of their classes. One thing that is known in this direction i*
*s the
completeAdams spectral sequence (ASS) converging to the homotopy groups of the
connective unitary spectrum u localized at 2. The 2nth space of this spectrum *
*is
U[2n + 1;1], the space obtained from U by killing ssi() for i < 2n + 1. Usingre*
*sults
of [6],we have H (u) ss A==A1, where 1 is the exterior subalgebra of A generated
by the Milnor primitives of degree 1 and 3. Thus this SS has
Es;t1ss E2s;t= ExtA(H u) ss Ext1 (Z2): (1:3)
In the usual (t s; s) depiction of ASS, this SS consists of, for each i 0,*
* an
infiniteh0-tower in stem t s = 2i + 1 beginning in filtration s =i. The filtra*
*tions
of the generators of the towers in E1 of the UASS for U are certainly not this *
*nice.
But we use an algebraic SS converging to the nice E1-term of (1.3) as an aid to
constructing our conjectural UASS(U).