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 Lehigh University
Chicago, IL 60680 Bethlehem, PA 18015 y
This paper is dedicated 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*(CP+1) denote the unstable
Amodule with nonzero classes xisuch that i is odd and positive, and
i j
Sq2jx2k+1= kjx2(j+k)+1:
Then H*(U) U(M1 ), where the left side is the mod 2 cohomology of the infinite
unitary group, and the right side the free unstable Aalgebra generated by M1 .*
* Thus
there is an unstable Adams spectral sequence (UASS), defined as in [5], converg*
*ing
to ss*(U) with Es;t2 Exts;t(M1 ). We shall construct an algebraic spectral sequ*
*ence
(SS) which we conjecture agrees with this UASS. In Section 3, we perform the mi*
*nor
modifications required to yield the analogous results for the infinite special *
*orthogonal
group SO.
Part of our conjecture is a splitting result for Ext(M1 ). Let Mn denote the
subspace of M1 spanned by those xi such that ff(i) n, where ff(i) denotes the
___________________________
*AMS Subject Classification 55T15, 55R45. Key words: unstable Adams spectr*
*al
sequence, splittings of Ext groups, looping resolutions
yBoth authors were supported by National Science Foundation research grants.
1
number of 1's in the binary expansion of i. This provides a filtration of M1 *
*by
Asubmodules, 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 with nth subquotient En;s;t*
*1.
Conjecture 1.1 This SS collapses to an isomorphism
M s;t
Exts;t(M1 ) Ext (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 SS which we construct has Ext(Mn=Mn1)
as its E2term. The following analogue for SO may be of even more interest.
Conjecture 1.2 Let Qn denote the subquotient of fH*RP 1 spanned by classes xiw*
*ith
ff(i) = n. Then
M s;t
Exts;t(fH*RP 1) Ext (Qn):
n1
The UASS 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 *
*is the
complete Adams 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. Using*
* results
of [6], we have H*(u) 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;t1 Es;t2= ExtA(H*u) Ext1 (Z2): (1:3)
In the usual (t  s; s) depiction of ASS, this SS consists of, for each i 0*
*, an
infinite h0tower in stem t  s = 2i + 1 beginning in filtration s = i. The fil*
*trations
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).
2
Let F (n) denote the free unstable Amodule on a generator n of degree n, F *
*0(n) =
F (n)=ASq1, and Jn F 0(2n  1) the Asubmodule generated by Sq32n1. Define
ae
Ds;t= Z2 if t  s = 2i + 1 and 0 i s
0 otherwise
and, for any integer n,
(
Ds;t if t  s 2n + 1
Ds;tn= sn1;tn
Ext (Jn+1) otherwise.
If n < 0, then Ds;tn= 0. Note that Ds;tagrees with the ASS for u described in (*
*1.3).
We will prove the following key result in Section 2.
Proposition 1.4 For n 0, there are exact sequences
js1;tn sn1;tn ffis1;tns;tis;tns;tjs;tnsn;tn ffis;tn
! Ext (Mn+1=Mn) ! Dn1 ! Dn ! Ext (Mn+1=Mn) ! :
These can be spliced together to give an exact couple, and hence a SS. ([4]) Wr*
*itten
in tableau form, the SS begins as follows.
j s1;t ffi
! Ext (M1) ! 0?
?yi
j s2;t1 ffi s;t j s;t ffi
! Ext (M2=M1) ! D0? ! Ext (M1) ! 0?
?yi ?yi
j s3;t2 ffi s;t j s1;t1 ffi s+1;t j
! Ext (M3=M2) ! D1? ! Ext (M2=M1) ! D0? !
?yi ?yi
j s4;t3 ffi s;t j s2;t2 ffi s+1;t j
! Ext (M4=M3) ! D2? ! Ext (M3=M2) ! D1? !
?yi ?yi
If t  s < 2n+1 2, then is;tnis bijective, since both groups surrounding it ar*
*e 0. Thus
limits are attained in this SS, and so the following corollary is immediate.
Corollary 1.5 There is a SS with
En;s;t2= Extsn;tn(Mn+1=Mn);
with dn;s;tr: En;s;tr! Enr+1;s+1;trand En;s;t1the nth subquotient of a filtrat*
*ion of Ds;t.
3
Thus this algebraic SS, which has a very complicated E2term, converges to the *
*nice
ASS for u. But its E2term is not the E2term of the UASS converging to ss*(U),
for it effectively pushes the chart of Ext*;*(Mn+1=Mn) up by n units, by letting
Extsn;tn(Mn+1=Mn) contribute to the limit (s; t)group. In order to have a ch*
*ance
of obtaining the correct E2term, we regrade.
Define a new SS with
M n;s+n;t+n M s;t
Es;t2= E2 = Ext (Mn+1=Mn); (1:6)
n n
with dr on the nth summand of Es;trequal to dn;s+n;t+nrof the SS of 1.5. The E1*
* term
of this SS is a regraded version of E1 of the SS of 1.5. When an element of Ds*
*;tis
pulled back to Ds;tnfor smallest possible n, it will now be seen in filtration *
*sn rather
than s, as it was in E. These n's are a nonincreasing function of s as we move *
*up a
tower (fixed t  s) of Ds;t, and will eventually stabilize. But changes in this*
* n cause
what look like filtration jumps in the SS of (1.6).
We illustrate with the situation through t  s = 9, where this jump first ha*
*ppens.
The left SS is that of 1.5, and the right that of (1.6). Classes with n = 0 are*
* indicated
by x, with n=1 by o, and with n = 2 by O. As usual, coordinates are (t  s; s).
    
         
         
         
 r r b r  r r b r
x    x    
        
x r r rb b x r r b r
   l %    
x r r r blb% x r r rb
   l     T
x r r lb x r r r bT b
     T  %
x r x r r bTb%
   T 
________________________x ________________________xrbT
1 3 5 7 9 1 3 5 7 9
Note that when the SS of 1.5 is pictured as above, all differentials look like *
*d1's, for
they all go from a group contributing to Ds;tto a group contributing to Ds+1;t.*
* But
the subscripts of differentials in the SS of 1.5 are related to changes of summ*
*and.
Our second main conjecture for U is as follows.
Conjecture 1.7 The SS of (1.6) agrees with the UASS of U.
4
As evidence, we observe that if Conjecture 1.1 is true, then the SS of (1.6) ha*
*s the
correct E2term, and its E1 could be correct, for it has 0 in even stems and Z*
*2's
of strictly increasing filtrations in odd stems. Further evidence is given by t*
*he fact
that our proof of Proposition 1.4 will involve the cohomology of the spaces in *
*the
Postnikov system for U.
As pointed out by Mark Mahowald, this algebraic SS for ss*(U) can be demysti*
*fied
somewhat by thinking of it as arising from the destabilization of an Adams reso*
*lution
of u. This is discussed in Section 4, where we also explain how the unstable A*
*modules
Mn+1=Mn can be obtained as derived functors of the destabilization functor appl*
*ied
to a shifted version of H*u. In Section 5, we present a generalization of our r*
*esults
and conjectures. It is not clear whether the situation for SO and U contains es*
*sential
ingredients not present in the much more general context of Section 5.
The reader attempting to prove Conjectures 1.1 and 1.2 should keep in mind
that they are not true if an arbitrary module is allowed in the second variable*
*. For
example, Ext0;0(fH*RP 1; Q1) = 0, while Ext0;0(Q1; Q1) 6= 0.
2 Construction of the spectral sequence
In this section, we prove Proposition 1.4, which we have already seen implies t*
*he
existence of the SS's of 1.5 and (1.6).
Define Fn = F 0(2n), and define Kn so that there is a short exact sequence (*
*SES)
in U
.Sq3
0 ! Kn ! Fn ! Jn1 ! 0: (2:1)
Since Jm = 0 for m < 2, and
ae
Z2 t  s = 2n, s 0
Exts;t(Fn) =
0 otherwise,
the Ext sequence of (2.1) yields
ae
Z2 t  s = 2n, s 0
Proposition 2.2 i) if n = 1 or 2, then Exts;t(Kn) =
0 otherwise;
ii) if n 3, then Exts;t(Jn1) Exts1;t(Kn) unless t  s = 2n and s 0,
in which case Exts;t(Jn1) = Z2 and Exts1;t(Kn) = 0.
5
We will need the following elementary result, where is the loop functor in *
*U as
defined in [5].
Proposition 2.3 Exts;t(Kn) Exts;t+1(Kn).
Proof. Letting 1 denote the first derived functor of , [5, pp 4344] and (2.1) *
*imply
that 1Kn = 0, since it injects to 1Fn = 0. Then the exact sequence
! Exts;t(Kn) ! Exts;t+1(Kn) ! Exts1;t(1Kn) ! Exts+1;t(Kn) !
(see, e.g., [2, 3.7]) implies the desired result. 
The Postnikov tower for U is a tower
U
#
..
. (2.2)
#
K(Z; 5)!i U[1; 5] k!K(Z; 8)
??
yp
K(Z; 3)!i U[1; 3] k!K(Z; 6)
??
yp
K(Z; 1)  k!K(Z; 4);
p k
in which each i! !  ! is a fiber sequence, and the map U ! U[1; 2n+1] induc*
*es
an isomorphism in ss() for i 2n + 1, while ssi(U[1; 2n + 1]) = 0 for i > 2n +*
* 1.
There are isomorphisms of Aalgebras H*(K(Z; 2n  1)) U(Fn) and, by [6]
or [3], H*(U[1; 2n  1])) U(Xn) for a certain unstable Amodule Xn, which can
be thought of as 1QH*(BU[2; 2n]), where Q denotes the indecomposable quotient.
The morphisms of H*() of (2.2) are induced by morphisms of these unstable A
modules as below.
6
(2:4)
in+1 fn+1
Fn+1  Xn+1  Fn+2
"
fn
Fn in Xn  Fn+1
"
fn1
Xn1  Fn
The following result is culled from [6] (see esp. [6, 8.6] for (2.7) and [6,*
* 7.1] for
(2.8)), with Kn, Jn, and Mn as above.
Theorem 2.5 In the above diagram, infn = .Sq3,
ker(fn) = Kn+1 and im (fn+1) Jn+1; (2:6)
and there are SES's in U
0 ! Mn ! Xn+1 ! ker(fn) ! 0 (2:7)
and
0 ! im(fn+1) ! Xn+1 ! Mn+1 ! 0: (2:8)
Note that (2.7) and (2.8) imply that
Mn im(Xn ! Xn+1): (2:9)
Taking quotients by Mn of the latter modules in (2.8), then using (2.7) to rewr*
*ite the
middle group, and finally using (2.6) yields a SES
0 ! Jn+1 ! Kn+1 ! Mn+1=Mn ! 0: (2:10)
Remark 2.11 Note that this says that when the exact sequence
.Sq3 1 .Sq33 1
3A=ASq1! A=ASq ! A=ASq
is destabilized so that the generator of its middle group has dimension 2n + 1,*
* then
the homology is Mn+1=Mn. We will elaborate on this remark in Section 4.
7
From (2.10) and (2.3), we obtain a LES
! Extsn1;tn+1(Kn+1) ! Extsn1;tn(Jn+1) ! Extsn;tn(Mn+1=Mn)
! Extsn;tn+1(Kn+1) ! : (2.11)
We will deduce the exact sequence of Proposition 1.4 from this.
When n = 0, the sequence of 1.4 reduces to the fact that
ae
Ds;t0 Exts;t(M1) Z2 t  s = 1 and s 0
0 otherwise.
When n = 1, both (2.11) and 1.4 reduce to the same isomorphism for t  s > 3,
using 2.2 and the definition of Ds;tn. The initial towers (t  s = 1) in Ds;t0a*
*nd Ds;t1
correspond under the morphism of 1.4. The tower in Ds;t1in t  s = 3, s 1, maps
in 1.4 to a tower in Ext(M2=M1) which in (2.11) maps to the tower in Ext(K2).
If n 2, exactness of (2.11) is maintained if Ds;tfor t  s 2n + 1 is added*
* to
both Extsn1;tn+1(Kn+1) and Extsn1;tn(Jn+1). The latter clearly becomes Ds*
*;tn,
while 2.2 shows that the former becomes Ds;tn1, as desired.
Two remarks are in order. First, the reader may object that this is not an a*
*lgebraic
SS, as advertised, because it involves the cohomology of the spaces U[1; 2n  1*
*]. To
this, we counter that the SS comes completely from (2.10), which is just algebr*
*aic.
Moreover, a completely algebraic derivation of (2.10) can be given, but we felt*
* the
one presented here is more likely to lead to a proof of our conjectures.
Second, as more evidence for the relationship between our SS and the UASS, we
point out that the Postnikov tower of U resembles an AdamsPostnikov tower, as
defined in [5, p 82], because
ker(H*(U[1; 2n  1]) ! H*(U)) = ker(H*(U[1; 2n  1]) ! H*(U[1; 2n + 1])):
3 Results for SO
The entire argument can be directly adapted to SO, using results of [7] and [3]*
* instead
of [6]. We just list the replacements for the various symbols.
8
o M1 = fH*(RP 1). Thus M1 has classes xi for all positive integers i,
and Mn is spanned by those with ff(i) n.
o For 0 b 3, let ae(4a+b) = 8a+2b. Thus ae(n) is the grading of the nth
nonzero homotopy group of BSO. Then Xn = 1QH*(BSO[2; ae(n)]),
and H*(SO[1; ae(n)  1]) U(Xn).
ae
F (ae(n)) if n 0; 1 mod 4
o Fn = 0
F (ae(n)) if n 2; 3 mod 4
o infn : Fn+1 ! Fn is .Sqd, where
8
< 2 n 0; 3 mod 4
d = ae(n + 1)  ae(n) + 1 = : 3 n 1 mod 4
5 n 2 mod 4
and, as before, Kn+1 = ker(infn) = ker(fn), Jn = im(infn) im(fn),
and Mn = im(Xn ! Xn+1).
ae
1 i 0; 1 mod 4
o Let h(i) = Then
1 i 2; 3 mod 4.
ae
Ds;t= Z2 if t  s = ae(i)  1 and 0 i  1 s < i  1 + h(i)
0 otherwise
and (
Ds;t if t  s ae(n + 1)  1
Ds;tn= sn1;tn
Ext (Jn+1) otherwise
o Proposition 2.2 is replaced by
ae
Z2 t  s = ae(n), 0 s < h(n)
Proposition 3.1 i) if n 3, then Exts;t(Kn) =
0 otherwise;
ii) if n 4, then Exts;t(Jn1) Exts1;t(Kn) unless t  s = ae(n) and
0 s < h(n), in which case Exts;t(Jn1) = Z2 and Exts1;t(Kn) = 0.
With the new meaning of terms, Theorem 2.5, Proposition 1.4, and Corollary 1*
*.5
are valid. Analogous to Remark 2.11 is that the following sequences on the left*
* have
homology as indicated on the right.
sequence homology
9
F (8t + 1) ! F 0(8t  1) ! F 0(8t  6) M4t1=M4t2
F (8t + 2) ! F (8t) ! F 0(8t  2) M4t=M4t1 (3.12)
F 0(8t + 4) ! F (8t + 1) ! F (8t  1) M4t+1=M4t
F 0(8t + 8) ! F 0(8t + 3) ! F (8t) M4t+2=M4t+1
Also, there is a SS (1.6), which we conjecture agrees with the UASS of SO. Conj*
*ecture
1.2 would be a corollary of this agreement, since it relates two ways of expres*
*sing the
E2term.
We remark that a chart for Ds;twith coordinates (t  s; s) is the chart for *
*the
connective so spectrum, which begins as follows.

r6 r6 r6 r6r
r r r rr
r r r r
r r r
r rr
r rr
r r
r
____________________________r
1 3 7 11 15
This can be seen since (see e.g., [3])
H*(so) A=ASq3 = A A1 A1=A1Sq3;
where A1 is the subalgebra of A generated by Sq1and Sq2, and hence in the ASS f*
*or
so
Es;t1= Es;t2= ExtA1(A1=A1Sq3):
4 Topological and algebraic destabilizations
Our construction of the conjectured UASS for U (resp. SO) can be viewed as a
consequence of destabilizing the ASS for u (resp. so). Let Hn = nHZ(2)denote the
10
EilenbergMacLane spectrum. An AdamsPostnikov tower for u has the simple form
u
#
..
.
#
H5 ! u[1; 5]! H8
#
H3 ! u[1; 3]! H6
#
H1 ! H4:
Diagram (2.2) is just the destabilization of this. Whereas the sequence
H*(H2n+4) ! H*(H2n+1) ! H*(H2n2)
is exact, its destabilization
F 0(2n + 4) ! F 0(2n + 1) ! F 0(2n  2)
has homology Mn+1=Mn. Our SS deals with the way in which the groups Ext(Mn+1=Mn)
build ss*(U) ss*(u).
As pointed out by Paul Goerss, there is an algebraic analogue of this. Let 1*
* :
Mod A ! U be the left adjoint to the inclusion functor. Thus, if M is an Amodu*
*le,
then 1 M is the quotient of M mod relations Sqix = 0 if i > x. Let 1s denote
the sth derived functor of 1 . Then we have
Proposition 4.1 (Goerss) If Mi is as in Section 1, then
1n(n+1A==1) Mn+1=Mn Z=2;
while if Mi is as in Section 3, then
1n+1(n2A==A1) Mn=Mn1:
Proof. Define an acyclic chain complex C by
. .!.C1 ! C0 ! A==1 ! 0
11
with Ci = 3iA=ASq1 and boundary morphisms .Sq3. Apply 1 to the modules in
an Aresolution of each n+1Cito obtain a spectral sequence
E1p;q= 1pn+1Cq =) 1p+qn+1A==1; (4:2)
with dr : Erp;q! Erp+r1;qr. Noting that the only homology of the sequence
. .!.F (3) ! F (2) ! F (1) ! F (0) ! 0;
with morphisms .Sq1, is F (0) = Z2 and Z2 in F (1), we deduce
8 m+p
< Z2 if p > 0 and m + p = 0 or 1
1pm A=ASq1 = : F 0(m) if p = 0
0 otherwise.
Thus the only nonzero groups E1p;qin (4.2) satisfying p + q = n + ffl with 1 *
*ffl 1
are E10;n+ffl= F 0(2n + 1 + 3ffl), E1n;0= Z2, and E1n1;0= Z2. By (2.10) and Re*
*mark
(2.11), E20;n= Mn+1=Mn. The only possible remaining differential involving grou*
*ps
with p + q = n is dn : En0;n! Enn1;0, but this must be zero since these differ*
*entials
preserve internal grading.
The SOcase is proved similarly, using the acyclic complex over n2A==A1
with C4t = 12tn2A=ASq1, C4t+1= 12tnA, C4t+2= 12tn+2A, and C4t+3=
12tn+5A=ASq1. For ffl = 0, 1, 2, and 3, let ffi(ffl) = 0, 2, 4, and 7, respect*
*ively. Then
in the SS analogous to (4.2),
( 1
1p12tn2+ffi(ffl)A=ASqffl = 0; 3
E1p;4t+ffl= 1 12tn2+ffi(ffl)
p A ffl = 1; 2
8 0
>>>F (12t  n  2 + ffi(ffl))ffl = 0; 3 and p = 0
>> Z2 ffl = 0; 3 and 12t  n  2 + ffi(ffl) + p =*
* 0
>>>
>:Z2 ffl = 0; 3 and 12t  n  2 + ffi(ffl) + p =*
* 1
0 otherwise
E20;n+1is the homology of E10;n+2! E10;n+1! E10;n, which by (3.12) is Mn=Mn+*
*1.
Higher differentials must be 0 for dimensional reasons as in the previous case.*
* Since
for ffl = 0; 3 we have ffi(ffl) ffl mod 4, the only Z2's or Z2's occur in E1p;*
*qwith
p + q p + 4t + ffl p + 12t + ffi(ffl) n + 2 or 3 mod 4;
and so there are none when p + q = n + 1.
12
5 A generalization
In this section, we propose conditions on an algebraic resolution which are sat*
*isfied
by (2.4) and its SO analogue. We show that the analogue of Theorem 1.5 is true *
*in
this generality, and conjecture that the analogue of Conjecture 1.1 is true.
Suppose the diagram of unstable Amodules
F1? F2? F3?
?yf0 ?yf1 ?yf2
p0 p1 p2
X0 ! X1!? X2?  ! . .!.X
?yi1 ?yi2
F1 F2
satisfies
o Fn ! Xn1 ! Xn ! Fn ! Xn1 is exact.
o Fn is a direct sum of F (m)'s and/or F 0(k)'s
o (infn)* = 0 : Ext(Fn) ! Ext(Fn+1)
o ker(Xn ! X) = ker(Xn ! Xn+1)
o X =dirlim(Xn).
Let Mn = im(Xn ! X).
L
Conjecture 5.1 Ext(X) = Ext(Mn+1=Mn)
The following proposition generalizes Theorem 1.5.
Proposition 5.2 Let
M sn;tn
Ds;t= Ext (Fn):
n
There is a spectral sequence with
En;s;t2= Extsn;tn(Mn+1=Mn)
and En;s;t1the nth subquotient of a filtration of Ds;t.
13
The proof is identical to the proof for X = fH*(CP+1) in Sections 1 and 2. One
derives a SES
0 ! im(fn+1) ! ker(fn) ! Mn+1=Mn ! 0;
and modifies its exact Ext sequence using that Exts;t(imfn) and Exts1;t(kerfn)*
* differ
only by Ext(Fn+1). This allows one to splice the exact sequences, yielding the *
*desired
spectral sequence.
One could modify the spectral sequence of the proposition to obtain a SS with
L s;t
Es;t2= Ext (Mn+1=Mn). This E2term is the same as the E2term of the SS
converging to Ext(X) whose collapsing we would like to prove. Somehow the fact
that the SS of the proposition allowed us to consider the filtrations as being *
*much
higher is supposed to yield a proof of Conjecture 5.1. If the conjecture isn't *
*true in
this generality, then what extra conditions are required to make it true?
References
[1]J. F. Adams, Stable Homotopy and Generalised Homology, University of
Chicago Press, 1974.
[2]A. K. Bousfield and E. B. Curtis, A spectral sequence for the homotopy
of nice spaces, Trans Amer Math Soc 151 (1970) 457479.
[3]J. Long, Two contributions to the homotopy theory of Hspaces, Prince
ton Univ thesis (1979).
[4]W. S. Massey, Exact couples in algebraic topology, Annals of Math, 56
(1952) 364396.
[5]W. S. Massey and F. P. Peterson, The mod 2 cohomology of certain fibre
spaces, Memoirs Amer Math Soc, 74 (1967).
[6]W. M. Singer, Connective fiberings over BU and U, Topology 7 (1968)
271303.
[7]R. Stong, Determination of H*(BO(k; 1)) and H*(BU(k; 1)), Trans
Amer Math Soc 107 (1963) 526544.
14