A CONJECTURE ON THE UNSTABLE ADAMS SPECTRAL SEQUENCES FOR SO AND U KATHRYN LESH Abstract.In this paper we give a systematic account of a conjecture sug- gested by Mark Mahowald on the unstable Adams spectral sequences for the groups SO and U. The conjecture is related to a conjecture of Bousfield o* *n a splitting of the E2-term and to an algebraic spectral sequence constructe* *d by Bousfield and Davis. In this paper, we construct and realize topologicall* *y a chain complex which is conjectured to contain in its differential the str* *ucture of the unstable Adams spectral sequence for SO. A filtration of this chain complex gives rise to a spectral sequence that is conjectured to be the u* *n- stable Adams spectral sequence for SO. If the conjecture is correct, then* * it means that the entire unstable Adams spectral sequence for SO is available from a primary level calculation. We predict the unstable Adams filtratio* *n of the homotopy elements of SO based on the conjecture, and we give an exam- ple of how the chain complex predicts the differentials of the unstable A* *dams spectral sequence. Our results are also applicable to the analogous situa* *tion for the group U. 1.Introduction In this paper, we consider the unstable Adams spectral sequence (UASS) of the group SO at the prime 2. In particular, we give a systematic account of a conje* *cture suggested by Mark Mahowald concerning the calculation of the differentials in t* *his spectral sequence. We give a geometric realization of the conjecture in the for* *m of a tower with the 2-completion of SO as inverse limit. Our tower comes equipped wi* *th a map from the destabilization of the stable Adams tower for the infinite deloo* *ping of SO. We use this map and theorems of Bousfield on h0-towers in unstable Ext to predict the Adams filtrations of the unstable homotopy of SO. Our results are equally valid for the group U, and thus differentials and unstable filtrations * *can be predicted for this group as well. We note that, of course, the homotopy of SO a* *nd U is well known by Bott periodicity, and that what is of interest is the workin* *gs of the UASS, not the end result. Before we describe our results and conjectures, we establish some notation. We work entirely at the prime 2, all cohomology will be taken with mod 2 coefficie* *nts, and all spaces will be taken to be completed at 2 as appropriate. Let A be the * *mod 2 Steenrod algebra, let U_be the category of unstable A-modules, and let K_be the category of unstable A-algebras. There is a functor U : U_! K_, described by Massey and Peterson [M-P ], which takes the free unstable A-algebra on an unsta* *ble A-module. This functor is left adjoint to the forgetful functor from unstable A- algebras to unstable A-modules. ___________ Date: July 22, 2001. 1991 Mathematics Subject Classification. 55T15, 55Q52, 55U99. 1 2 KATHRYN LESH In general, the unstable Adams spectral sequence for a space X has the form Es,t2= ExtsK_(H*X , H*St) ) ßt-sX, where Ext is a derived functor in the nonabelian category K_. However, for a sp* *ace X with the property that H*X ~=U(N) for some N 2 U_, the unstable Adams spectral sequence has the form Es,t2= ExtsU_(N, tF2) ) ßt-sX. We will follow the stable notation and write Exts,tU_(N, F2) for ExtsU_(N, tF2* *). We will be discussing the unstable Adams spectral sequence for the special or- thogonal group SO and indicating_modifications*to be made to the discussion for the unitary group U. Let M1 = H RP1 , with nonzero elements xiin dimension i and A-action Sqjxi= ijxi+j; then H*SO ~=U(M1 ). Hence the unstable Adams spectral sequence for SO takes the form Exts,tU_(M1 , F2) ) ßt-sSO. Let ff(i) be the number of ones in the dyadic expansion of i, and filter M1 by Mn = {xi|ff(i) n}. This filtration leads to a spectral sequence converging to* * the E2-term of the UASS: Ext*,*U_(Mn=Mn-1 , F2) ) Ext*,*U_(M1 , F2). It is a conjecture of Bousfield from the 1970s that this spectral sequence coll* *apses, giving Ext*,*U_(M1 , F2) ~= n Ext*,*U_(Mn=Mn-1 , F2). A similar conjecture for the_E2-term*of the UASS for the group U arises from the fact that if we take M1 = H CP1+, then H*U ~=U(M1 ), and in this case also, M1 can be filtered by dyadic expansion of the dimension of the elements. In this paper, we use the destabilization of the stable Adams resolution of t* *he connective so spectrum to construct a chain complex whose constituent parts are minimal resolutions of the filtration quotients Mn=Mn-1. When realized topo- logically using the Massey-Peterson theorem [M-P ], this chain complex gives a tower of spaces whose inverse limit is SO (2-completed), and whose homotopy spectral sequence collapses at E2. The E1-term of the homotopy spectral sequence is Ext*,*U_(Mn=Mn-1 , F2), a very large vector space, while E2 = E1 is the as* *soci- ated graded to ß*SO, a rather small vector space (ßiSO ~=Z for i 3 mod 4, and Z=2 for i 0 or 1 mod 8). Hence the spectral sequence has a very complicated d* *1, which is, however, completely calculated by the calculation of the chain comple* *x, a primary level calculation. The conjecture suggested by Mahowald (Conjecture 5.1) is that in a certain precise sense, this d1 differential contains all the diffe* *rentials in the UASS. Because the tower comes equipped with a map from a modified Post- nikov tower for SO, it is possible to use theorems of Bousfield on unstable Ext to predict where the homotopy of SO is represented, and this, in turn, allows a prediction of the unstable Adams filtration of those elements. It is the hope * *of the author that in the future it will be possible to manipulate this tower by an elaboration of methods of [Lesh] to prove Conjecture 5.1. Extensive knowledge of differentials in the UASS for SO would allow the computation of differentials in other unstable Adams spectral sequences by naturality. For example, it should be possible to recover a form of Hopf invariant one from the model's calculation o* *f the UASS for SO. UASS FOR SO AND U 3 The splitting conjecture of Bousfield was discussed and an algebraic model for the UASS for U and SO constructed in [B-D]. However, the model was considered strictly on an algebraic level and was not realized topologically. Although the author believes that the spectral sequence of [B-D] is the same as that of the * *current work, the advantages of the model described here seem to be the following. Firs* *t, the construction of the model is essentially formal, and very similar to a stan* *dard construction of homological algebra to obtain the spectral sequence converging * *to the derived functors of a composite functor. All of the differentials can be ca* *lculated by a primary level calculation that is a strictly mechanical process. Second, t* *he model comes equipped with a topological realization. It seems that in order to prove that the model actually does give the UASS, it will be necessary to have * *such a realization. The rest of this paper is organized as follows. In Section 2, we give some ba* *ck- ground on the stable and destabilized Postnikov towers of so, as well as some algebraic preliminaries. In Section 3, we construct a tower of spaces and an a* *s- sociated chain complex that models the UASS for SO. In Section 4, we study the homotopical properties of the tower. Finally, in Section 5 we use theorems * *of Bousfield to predict the unstable Adams filtration of elements of ß*SO, we give a counterexample to a conjecture of [B-D], we draw some conclusions about what may be necessary to prove Conjecture 5.1, and we give an example of a different* *ial in the UASS that is predicted by our methods. 2.Preliminaries In this section, we review algebraic properties of the category of unstable A- modules, we recall the Massey-Peterson theorem, and we consider the cohomology of the stages of the destabilized Adams tower of so. We begin by reviewing properties of the algebraic looping functor : U_! U_ and its iterates. (See also [M-P ].) The functor : U_! U_is the left adjoint * *to the suspension functor : U_! U_. Given an unstable A-module M, the module M can be calculated as the largest unstable quotient of the desuspension of M: M ( -1M)=( -1Sq0M), where Sq0x = Sq|x|x. The functor is not exact, but it can have at most one nonzero derived functor, which we denote 11. The module 11M can be expressed as a regrading of the kernel of Sq0 on M. In particular, if Sq0 acts freely on * *M, then 11M = 0. We write n for the n-fold iterate of , and we write njfor the* * jth derived functor of n. There is a composite functor spectral sequence (the Sing* *er spectral sequence) si tjM ) s+ti+jM which allows us to calculate derived func* *tors of n inductively. For any unstable module M, njM = 0 for j > k. We will also need the following routine lemma. Lemma 2.1. Let g : N1 ! N2 be a map of unstable A-modules. If im(g) is Sq0- free, then the natural map ker(g) ! ker( g) is a monomorphism. If in addition N2 is Sq0-free, then there is a short exact sequence 0 ! ker(g) ! ker( g) ! 11cok(g) ! 0. Proof.The map g factors as N1 ! im(g) ! N2, and since is right exact, N1 ! im(g) is an epimorphism. Thus there is a short exact sequence (2.1) 0 ! ker[ N1 ! im(g)] ! ker( g) ! ker[ im(g) ! N2] ! 0. 4 KATHRYN LESH To calculate the left-hand term, observe that the short exact sequence 0 ! ker(g) ! N1 ! im(g) ! 0 gives rise to an exact sequence 11im(g) ! ker(g) ! N1 ! im(g) ! 0. Thus if 11im(g) = 0, then ker[ N1 ! im(g)] ~= ker(g), proving that ker(g) injects into ker( g). Consider the right-hand term of equation (2.1). The short exact sequence 0 ! im(g) ! N2 ! cok(g) ! 0 gives rise to a long exact sequence 0 ! 11im(g) ! 11N2 ! 11cok(g) ! im(g) ! N2 ! cok(g) ! 0. If 11N2 = 0, then ker[ im(g) ! N2] ~= 11cok(g), concluding the proof of the lemma. Remark 2.2. Suppose that M is an unstable A-module, that N1and N2are unstable projective A-modules, and that we are given a map M ! ker(N1 ! N2). Then we can consider the composition M ! ker(N1 ! N2) ,! ker( N1 ! N2) ,! N1, and so ker[M ! ker(N1 ! N2)] = ker(M ! N1). We will use this remark repeatedly in Section 3. Going in the opposite direction from looping, we define a "deloopingö n free modules. Given a free unstable A-module P, we write BP for the free unstable A- module whose generators are one dimension higher than those of P. Thus, BF(n) = F(n + 1), and BP ~=P. Note that "delooping" is not a functor on U_, because given a map g : P1 ! P0, there is no canonical choice of map Bg : BP1 ! BP0 with Bg = g. In most cases where we will use this notation, P will itself be an ite* *rated looping, and BP will simply mean one fewer loops: P = iN and BP = i-1N. We remind the reader of the content of the Massey-Peterson theorem, which we will need to use repeatedly. Essentially, this theorem says that under favo* *r- able conditions, the Serre spectral sequence for a fibration behaves much like * *the long exact sequence in cohomology for a stable cofibration. Note that if we wri* *te F(n) for the free unstable A-module on a single generator in dimension n, then H*K(Z=2, n) ~=U(F(n)). Therefore, if P is a free unstable A-module, we write KP for the Eilenberg-MacLane space with H*KP ~=U(P). Definition 2.3. We call a map X ! KP Massey-Peterson if the following hold. (1)There is an unstable A-module M with H*X ~=U(M) (2)There is a map f : P ! M that induces the map on cohomology. That is, H*KP ! H*X is U(f). (3)im(H*KP ! H*X) is contained in a polynomial subalgebra of H*X. (4)X is simple and of finite type. We think of the topological map X ! KP as realizing f, and by abuse of notation we call the topological map f as well. If Y is the homotopy fiber of a Massey-Peterson map f : X ! KP, then the Massey-Peterson theorem says that UASS FOR SO AND U 5 H*Y ~=U(N), where there is a short exact sequence (the fundamental sequence of f) 0 ! cok(f) ! N ! ker(f) ! 0. The short exact sequence does not, in general, split as A-modules, although U(N) is split as an algebra as the tensor product of U(cok(f)) and U( ker(f)). We begin our discussion of SO by describing the stable Postnikov tower of so, which is very_close to its stable Adams resolution.1 We know H*so ~= A =A Sq3, and letting A = A =A Sq1, the stable Postnikov tower of so realizes the acyclic complex of stable A-modules __ 11 9__ 4__ (2.2) . .!. 13A ! A ! A ! A ! A where each_term_is_monogenic and the differentials_run cyclically through the l* *ist Sq2, Sq2, Sq3, Sq5. Only the fact that A is not projective keeps this chain com* *plex from being the Adams resolution. Next we destabilize the stable Postnikov tower for the spectrum so by taking the zero space of the infinite loop spectrum at e* *ach level of the tower. We obtain the unstable Postnikov tower for SO, a tower of spaces {Xn} (Figure 1) with very nice cohomological properties_summarized_in the following lemma. (Recall that Mn is the nth filtration of M H*RP1 by dyadic expansion.) Lemma 2.4. [Long] (1)holimn-Xn ' SO. (2)kn is a Massey-Peterson map. (3)ker(H*Xn ! H*Xn+1) = ker(H*Xn ! H*SO). (4)im(H*Xn ! H*Xn+1) ~=im(H*Xn ! H*SO) ~=U(Mn). However, we will be interested in the destabilization, not of the Postnikov t* *ower for so, but of the Adams tower. The only difference this introduces is that ins* *tead of having only one homotopy group in each dimension, we have to introduce the copies of the integers one Z=2 at a time (building up the completion Z^2). To do this, take a projective resolution of each term in (2.2), take the total comple* *x, and destabilize. The realization of this projective chain complex will have the for* *m of Figure 2. An exercise in homological algebra shows that the tower has the same cohomological properties as those of the Postnikov tower which were summarized in Lemma 2.4: Lemma 2.5. (1)holimn-Yn ' SO^2. (2)There is an unstable A-module Zn with H*Yn ~=U(Zn). (3)ker(H*Yn ! H*Yn+1) = ker(H*Yn ! H*SO). (4)im(H*Yn ! H*Yn+1) ~=im(H*Yn ! H*SO) ~=U(Mn). Remark 2.6. (1)For the reader interested in carrying out this calculation, we note that * *the issues are the same as those laid out in detail in the proofs of Proposit* *ion 4.3 and Proposition 4.1. ___________ 1An appropriate reference for the remainder of the section is [Long]. 6 KATHRYN LESH SO ?? y .. . K(F(8)) ----! X4 ----! K(F(10)) ?? y __ K(F (7))----! X3 ----! K(F(9)) ?? y __ __ K(F (3))----! X2 ----! K(F (8)) ?? y __ K(F(1)) ----! X1 ----! K(F (4)) ?? y * ----! K(F(2)) Figure 1. The Postnikov tower for SO SO^2 ?? y .. . K(F(8) F(7) F(3))-i4---!Y4--k4--!K(F(10) F(8) F(4)) ?? y K(F(7) F(3))--i3--!Y3 --k3--!K(F(9) F(8) F(4)) ?? y K(F(3)) --i2--!Y2 --k2--! K(F(8) F(4)) ?? y K(F(1)) --i1--!Y1 --k1--! K(F(4)) ?? y * ----! K(F(2)) Figure 2. The destabilized Adams tower for SO UASS FOR SO AND U 7 (2)Let Pn be the unstable projective such that KPn is the homotopy fiber of Yn ! Yn-1. Thus P1 = F(1), P2 = F(3), P3 = F(7) F(3), etc. Then it is a consequence of Lemma 2.5(4) that __ker(BPn_!_Pn-1)_~M =M . im(BPn+1 ! Pn) = n n-1 3.A chain complex model for the UASS In this section, we use {Yn}, the destabilized Adams tower of so, to construc* *t a tower {En} that also has SO^2as its inverse limit, but that involves in its k-i* *nvariants the unstable resolutions of the filtration quotients Mn=Mn-1. The tower {En} wi* *ll come equipped with a map {Yn} ! {En}, which will allow us to calculate where the homotopy of SO is represented in the homotopy spectral sequence of {En}. This in turn will allow us in Section 5 to make predictions about unstable Adams filtrations in the homotopy of SO. We need a considerable amount of notation. Choose a minimal projective U_- resolution Dn*of Mn=Mn-1. The tower we are going to build will have the form En+1 # K(Dn0 Dn-11 . . . n-1D1n-1)E!n! KB(Dn+10 Dn1 . . . nD1n). Note that Dn*will make its first appearance at the nth stage of the tower. Beca* *use the module Dniappears in the tower as iDni, we avoid excessive loops in our notation by letting Cni= iDniand BCni= i-1Dni. We write Ln = ni=1Cin-i, and our tower will have the form En+1 # KLn ! En ! KBLn+1. We define the following filtration, along with similar filtrations of BLn and * *Ln: F-jLn = ni=jCin-i. Thus Cn0= F-n F-(n-1) ... F-1Ln = Ln. The tower of spaces {En} that we construct in this section has the following properties. Recall from Lemma 2.5 that Zn is the unstable A-module such that H*Yn ~=U(Zn), and from Remark 2.6 that Pn is the unstable projective such that Yn is the homotopy fiber of a map Yn-1 ! BPn. (1)There exists an unstable A-module Fn-1 such that H*En-1 ~=U(Fn-1), and En is the homotopy fiber of a Massey-Peterson map En-1 ! KBLn. (2)There are commuting diagrams of Massey-Peterson maps KPn-1 ----! Yn-1 ----! KBPn ?? ? ? y ?y ?y KLn-1 ----! En-1 ----! KBLn 8 KATHRYN LESH induced by commuting diagrams of unstable A-modules BLn ----! Fn-1----! Ln-1 ? ? ? hn?y ?y hn-1?y BPn ----! Zn-1----! Pn-1. (3)ker(BLn ! Fn-1) = ker(BLn ! Ln-1). (4)cok(BLn ! Fn-1) ! cok(BPn ! Zn-1) is an isomorphism. (5)Algebraic properties of the map fn described in detail below. Property (3) is analogous to Lemma 2.5(3), and both say that the k-invariants do not kill any cohomology that comes from lower down in the tower. Property (4) is related to Lemma 2.5(4), and arranges for the towers {En} and {Yn} to give the same filtration of H*SO. To describe the last set of properties we recall from the Massey-Peterson the* *orem that if En-1 is the fiber of a Massey-Peterson map En-2 ! KBLn-1, then the fundamental sequence for En-1 is 0 ! cok(BLn-1 ! Fn-2) ! Fn-1 ! ker(BLn-1 ! Fn-2) ! 0, where the righthand term is the contribution of the fiber, KLn-1, to H*En-1. The next space, En, will be the fiber of a Massey-Peterson map En-1 ! KBLn, and our last requirement is on the composition of the k-invariants, KLn-1 ! En-1 ! KBLn. Let fn denote the composite BLn ! Fn-1 ! ker(BLn-1 ! Fn-2). The final requirement on the tower {En} is detailed below. (5)fn has the following algebraic properties: (a)fn is filtration preserving. (b)For 1 j n, on F-j=F-(j+1)the map E0(fn) is the map BCjn-j! ker(BCjn-j-1! Cjn-j-2). that comes from looping down the differential in the resolution Dj*! Mj=Mj-1. (c)E0(ker(fn)) ~=ker(E0(fn)). We will use Remark 2.2 freely throughout this section. In particular, Remark * *2.2 together with requirement (5) tell us that the associated graded of ker(fn) is F-j=F-(j+1)(ker(fn)) ~=ker(BCjn-j! Cjn-j-1). The construction of {En} is inductive. For the first stage we observe that P1* * = L1 = C10, and we define L1 ! P1 to be the identity map. Thus Y1 = KP1 = KL1 = E1, and the theorem is certainly true in this case. Observe that P1 = Z1 = F1 =* * L1. At the next stage, L2 = C20 C11; we want a commuting diagram BL2 ----! L1 = F1 B(C20 C11)----! C10 ? ? ? ? h2?y =?y i.e. h2?y = ?y BP2 ----! P1 = Z1. BP2 ----! P1. We define BC20! C10to be zero, and BC11! C10by the differential for C1*. The composite BC11! C10= P1 ! cok(BP2 ! P1) ~=M1 is zero because BC11! C10! M1 begins a resolution, and so the composite BC11! C10! P1 factors through BP2. We use this factoring to define h2 : BL2 ! BP2 on the factor BC11. To defi* *ne UASS FOR SO AND U 9 h2 on the factor BC20, choose a class x2 2 ker(BP2 ! P1) that, when looped, giv* *es the generator of ker(BP2 ! P1)= im(BP3 ! P2) ~=M2=M1. This gives us the desired commuting diagram above. Looking at the topological realization, Y1 ----! KBP2 ?? ? y ?y E1 ----! KBL2, the properties required for E1 ! KBL2 are easily verified by inspection, and we take homotopy fibers in the diagram to obtain the space E2together a map Y2 ! E2 and maps of fundamental sequences 0 ! cok(BL2 ! L1) ! F2 ! ker(BL2 ! L1) ! 0 # # # 0 ! cok(BP2 ! P1)! Z2 ! ker(BP2 ! P1) ! 0. For an inductive hypothesis, we assume that for i n we have defined spaces Ei and maps fi satisfying the required conditions, and we seek to define En+1. Thus we have maps BPn+1 ! Zn and Fn ! Zn induced by Yn ! KBPn+1 and Yn ! En, respectively. We need to define a commuting diagram BLn+1 ----! Fn ? ? hn+1?y ?y BPn+1 ----! Zn and verify that when we realize it by a diagram of spaces Yn ----! KBPn+1 ?? ? y ?y En ----! KBLn+1, taking horizontal fibers gives rise to a space En+1 and a map Yn+1 ! En+1 that satisfies the inductive hypotheses. Consider the ladder of fundamental sequences for Yn and En: 0 ! cok(BLn ! Fn-1) ! Fn ! ker(BLn ! Fn-1) ! 0 # # # 0 ! cok(BPn ! Zn-1) ! Zn ! ker(BPn ! Zn-1) ! 0. We know by Lemma 2.5 that ker(BPn ! Zn-1) = ker(BPn ! Pn-1), and by the inductive hypothesis, ker(BLn ! Fn-1) = ker(BLn ! Ln-1), with the map between them induced by hn. Our strategy is to construct a commuting diagram BLn+1 ----! ker(BLn ! Ln-1) ? ? (3.1) hn+1?y hn?y BPn+1 ----! ker(BPn ! Pn-1) This will give a map of BLn+1into the right-hand term of the fundamental sequen* *ce above, and then we will lift to Fn using projectivity of BLn+1. We will make the construction in such a way that hn induces an isomorphism between the cokernel* * of 10 KATHRYN LESH BLn+1 ! ker(BLn ! Ln-1) and the cokernel of BPn+1 ! ker(BPn ! Pn-1), which we know to be Mn=Mn-1. This will lead to condition (4) for the tower {En}. To construct diagram (3.1), we compute ker(BLn ! Ln-1). From inductive hypothesis (5), we know the associated graded of ker(BLn ! Ln-1), and since commutes with cokernels, we know that ker(BLn ! Ln-1) has associated graded F-j=F-(j+1)~= ker[BCjn-j! ker(BCjn-j-1! Cjn-j-2)] = ker(BCjn-j! Cjn-j-1). We first define a filtration preserving map gn+1 : BLn+1 ! ker(BLn ! Ln-1) as follows. On the lowest filtration, F-(n+1)= BCn+10, let gn+1 be zero. In filtra* *tion (-j), let gn+1 : BCjn-j+1! ker(BLn ! Ln-1) lift the natural map BCjn-j+1! ker(BCjn-j! Cjn-j-1) = F-j=F-(j+1)( ker(BLn ! Ln-1)) to F-j( ker(BLn ! Ln-1)). Note that F-n( ker(BLn ! Ln-1)) = Cn0splits off from ker(BLn ! Ln-1), and hence we can take gn+1 : n-1j=1BCjn+1-j! Cn0to be zero, and the only factor on which gn+1 : BLn+1 ! Cn0is nonzero is BCn1. Lemma 3.1. gn+1 is filtration preserving and cok(gn+1) ~=Mn=Mn-1. Proof.gn+1 is filtration preserving by its construction. To calculate the coker* *nel, we first consider the cokernel on the level of the associated graded. For j 1* *, in filtration F-j=F-(j+1)we have BCjn-j+1! ker(BCjn-j! Cjn-j-1), that is, n-jDjn-j+1! ker( n-j-1Djn-j! n-j-1Djn-j-1). Because Dj*! Mj=Mj-1is a resolution, for j < n the homology at the middle of the three term sequence n-j-1Djn-j+1! n-j-1Djn-j! n-j-1Djn-j-1calculates n-j-1n-jMj=Mj-1, which we know is zero since n - j > n - j - 1. Hence the map n-j-1Djn-j+1! ker( n-j-1Djn-j! n-j-1Djn-j-1) is a surjection. Looping preserves surjections, and hence BCjn-j+1! ker(BCjn-j! Cjn-j-1) is a surjection. Thus the cokernel of E0(gn+1) is zero on F-j=F-(j+1)for j < n. Consider j = n: on F-n we have defined gn+1 to be the differential BCn1! Cn0, whose cokernel is Mn=Mn-1. Since we have taken gn+1 to be zero from higher filtrations into F-n, we find that cok(gn+1) ~=Mn=Mn-1 as desired. Recall that the cokernel of BPn+1 ! ker(BPn ! Pn-1) is Mn=Mn-1 (Re- mark 2.6). To get diagram (3.1), we must have a map fn+1 : BLn+1 ! ker(BLn ! Ln-1) whose cokernel is Mn=Mn-1and whose composition with hn factors through BPn+1. So far, we have a map gn+1 : BLn+1 ! ker(BLn ! Ln-1) whose coker- nel is Mn=Mn-1, but the composition of gn+1 with hn does not necessarily factor UASS FOR SO AND U 11 through BPn+1. To adjust gn+1, consider the composite n-1j=1BCjn-j+1,! BLn+1gn+1----! ker(BLn ! Ln-1) ---hn-! ker(BP n ! Pn-1) ----! Mn=Mn-1. Choose a lift of the composite across the epimorphism Cn0! Mn=Mn-1. We define fn+1 : BLn+1 ! ker(BLn ! Ln-1) as the sum of gn+1 with the lift n-1j=1BCjn-j+1! Cn0. Observe that fn+1 is the same as gn+1 on the factors BCn+10and BCn1of BLn+1, and further, the adjustment added to gn+1 to obtain fn+1 strictly lowers filtration; thus fn+1 and gn+1 induce the same map on the associated graded. By construction, hn O fn+1 : nj=1BCjn-j+1! ker(BPn ! Pn-1) composes to zero in Mn=Mn-1, and so hn O fn+1 factors through BPn+1. We define hn+1 : BLn+1 ! BPn+1 to be the sum of this factoring with a map BCn+10! BPn+1 that hits a class xn+1 whose looping generates ker(BPn+1 ! Pn)= im(BPn+2 ! Pn+1) ~=Mn+1=Mn. Lemma 3.2. The commuting diagram BLn+1 -fn+1---! ker(BLn ! Ln-1) ? ? hn+1?y hn?y BPn+1 -dn+1---! ker(BPn ! Pn-1) induces an isomorphism cok(fn+1) ~=cok(dn+1). Proof.By the construction of hn : BLn ! BPn at the previous stage, ker(BLn ! Ln-1) ! cok(dn+1) ~=Mn=Mn-1 is an epimorphism. On the other hand, the cokernel of E0(fn+1) is Mn=Mn-1 in filtration -n and zero in higher filtrations, and so hn induces an isomorphism cok(fn+1) ~=cok(dn+1). Corollary 3.3. E0(kerfn+1) ~=ker(E0(fn+1)). Proof.The result follows from the proof of the preceding lemma, since we establ* *ish that E0(cokfn+1) ~=cok(E0(fn+1)). Now we are ready to define the k-invariant that takes us from En to En+1. Let kn+1 be a lift of fn+1 across the epimorphism Fn ! ker(BLn ! Ln-1) that comes from the fundamental sequence for En. Lemma 3.4. kn+1 can be chosen to give a commuting diagram BLn+1 -kn+1---!Fn ? ? hn+1?y ?y BPn+1 ----! Zn Proof.The choice of the lift kn+1can be adjusted if necessary by a routine diag* *ram chase. Use the ladder of fundamental sequences 0 ! cok(BLn ! Fn-1) ! Fn ! ker(BLn ! Fn-1) ! 0 # # # 0 ! cok(BPn ! Zn-1) ! Zn ! ker(BPn ! Zn-1) ! 0. 12 KATHRYN LESH in which the left vertical arrow is an isomorphism by induction, and the commut* *ing diagram BLn+1 -fn+1---! ker(BLn ! Ln-1)= ker(BLn ! Fn-1) ? ? hn+1?y hn?y BPn+1 ----! ker(BPn ! Pn-1)= ker(BPn ! Zn-1). We now begin verification of the inductive hypotheses. Let Yn ----! KBPn+1 ? ? (3.2) ?y ?y En ----! KBLn+1 be a homotopy commutative diagram of spaces that realizes the commutative di- agram of Lemma 3.4, let En+1 be the homotopy fiber of En ! KBLn+1, and let Yn+1 ! En+1 be the map between the homotopy fibers. By construction, En ! KBLn+1 is a Massey-Peterson map, because the image of BLn+1 ! Fn injects to ker(BLn ! Ln-1) Ln, and thus is Sq0-free. The commuting square (3.2)is a map between Massey-Peterson maps by construction, and thus we get the first two inductive hypotheses immediately. Lemma 3.5. ker(kn+1) = ker(fn+1). Proof.fn+1 is the top composite in the commuting diagram BLn+1 -kn+1---!Fn----! ker(BLn ! Ln-1) ? ? ? hn+1?y ?y hn+1?y BPn+1 ----! Zn ----! ker(BPn ! Pn-1). Certainly ker(kn+1) ker(fn+1). Suppose x 2 ker(fn+1); then hn+1(x)2 ker[BPn+1 ! ker(BPn ! Pn-1)] = ker[BPn+1 ! Zn] by Lemma 2.5. Thus kn+1(x) 2 ker(Fn ! Zn). However, by inductive hypothesis (4) and the ladder of fundamental sequences for Yn and En, ker[Fn ! ker(BLn ! Ln-1)] ~=ker[Zn ! ker(BPn ! Pn-1)]. Thus kn+1(x) = 0, which establishes the lemma. Lemma 3.6. cok(BLn+1 ! Fn) ~=cok(BPn+1 ! Zn). Proof.Apply the Snake Lemma to the ladder of short exact sequences 0 ! 0 ! BPn+1 ! BPn+1 ! 0 # # # 0 ! cok(BPn ! Zn-1) ! Zn ! ker(BPn ! Pn-1)! 0. Because ker(BPn+1 ! Zn) ~=ker[BPn+1 ! ker(BPn ! Pn-1)] by Lemma 2.5, the cokernels of the vertical maps form a short exact sequence . Apply the same UASS FOR SO AND U 13 reasoning with BLn+1 and the fundamental sequence for En to get a commuting ladder of short exact sequences 0 ! cok(BLn ! Fn-1) ! cok(BLn+1 ! Fn) ! cok(fn+1) ! 0 # # # 0 ! cok(BPn ! Zn-1) ! cok(BPn+1 ! Zn) ! cok(dn+1) ! 0. The leftmost column is an isomorphism by the inductive hypothesis and the right- hand column is an isomorphism by Lemma 3.2, which implies the desired conclu- sion. Corollary 3.7. The natural map lim-!nFn ! lim-!nZn is an isomorphism. Proof.Consider Fn ----! Zn ?? ? y ?y Fn+1 ----! Zn+1 ?? ? y ?y lim-!nFn----!lim-!nZn By the preceding lemma, im(Fn ! Fn+1) ~=im(Zn ! Zn+1), and by Lemma 2.5, im(Zn ! Zn+1) ~=im(Zn ! Zn+j) for j > 1. The corollary follows. 4.Homotopical properties of {En} In this section we give the homotopical and homological properties of the tow* *er {En}. We prove that it has inverse limit SO^2and that its homotopy spectral sequence collapses at the E2-term. Notation is continued from Section 3. Proposition 4.1. The map of towers {Yn} ! {En} induces a homotopy equiva- lence on the homotopy inverse limits. Proof.We already know from Corollary 3.7 that the map of towers induces an isomorphism lim-!nH*En ! lim-!nH*Yn. Although cohomology is not in general well-related to inverse limits, an application of [Lannes, Lemme 3.2.3] tells u* *s that in our situation, H* holimY-n~=lim-!H*Yn n n and H* holimE-n~=lim-!H*En. n n The essential ingredients that allow the use of Lannes's lemma are: (1)For all n, the spaces Yn and En are connected and have mod 2 cohomology that is finite in each dimension. (2)The towers of groups {ß1Yn} and {ß1En} are constant. (3)The towers of groups {H1Yn} and {H1En} are constant. The proposition then follows by observing that holimn-Yn and holimn-En are 2- complete (each is built from mod 2 Eilenberg-MacLane spaces by fibrations) and that the map between them is a mod 2 cohomology isomorphism. 14 KATHRYN LESH Corollary 4.2. holimn-En ' SO^2. Our next goal is Corollary 4.5, in which we prove that the homotopy spectral sequence of {En} collapses at the E2-term. This follows by using a homological argument to show that the map {Yn} ! {En} induces an isomorphism at E2 of the homotopy spectral sequences, and then observing that the homotopy spectral sequence of {Yn} does in fact collapse at E2. The following proposition performs the main technical calculation. Proposition 4.3. The following ladder gives a homology isomorphism at the mid- dle term: BLn+1 ----! Ln ----! Ln-1 ?? ? ? y ?y ?y BPn+1 ----! Pn ----! Pn-1. Proof.The proof is inductive. For n = 1, we take P0 = L0 = 0 and the result is easily established by direct calculation. Suppose that the lemma is true for BLn ----! Ln-1 ----! Ln-2 ?? ? ? y ?y ?y BPn ----! Pn-1 ----! Pn-2 and consider the next stage. By Lemma 3.2, we already know that __ker(BLn_!_Ln-1)_~__ker(BPn_!_Pn-1)_. im(BLn+1 ! Ln) = im(BPn+1 ! Pn) Let iL : ker(BLn_! Ln-1) ! ker(Ln ! Ln-1) be the natural map ker(fn) ! ker( fn), let iL be the induced map on cokernels, and consider the diagram of exact sequences BLn+1 -fn+1---! ker(BLn ! Ln-1)----!_ker(BLn!Ln-1)_im(BLn+1!Ln)----!0 ? ? ? =?y iL?y __iL?y BLn+1 ----! ker(Ln ! Ln-1) ----! ker(Ln!_Ln-1)_im(BLn+1!Ln)----!0. __ __ By Lemma 2.1 and the Snake Lemma, iL and iLare monomorphisms and cok(iL) ~= cok(iL) ~= 11cok(BLn ! Ln-1). The same argument with iP : ker(BPn_! Pn-1)_!_ker(Pn ! Pn-1) and the corresponding_map of cokernels, iP, shows that iP is a monomorphism and cok(iP) ~= 11cok(BPn ! Pn-1). Consider the diagram _ker(BLn!Ln-1)_~= _ker(BPn!Pn-1)_ im(BLn+1!Ln)----!? im(BPn+1!Pn)? __iL?y __iP?y ker(Ln!_Ln-1)_ ker(Pn!_Pn-1)_ im(BLn+1!Ln)----! im(BPn+1!Pn). __ __ We already know that the top row is an isomorphism. Since iLand iPare monomor- phisms, the lemma will be established_by the Five_Lemma if we prove that the diagram induces an isomorphism cok(iL) ! cok(iP). Thus we must show that 11cok(BLn ! Ln-1) ~= 11cok(BPn ! Pn-1). UASS FOR SO AND U 15 The three term sequence BLn ! Ln-1 ! Ln-2 gives us a short exact sequence 0 ! ker(Ln-1_!__Ln-2)_im(BL! _____Ln-1_____! ______Ln-1______! 0. n !iLn-1)m(BLn ! Ln-1)ker(Ln-1 ! Ln-2) The middle term is cok(BLn ! Ln-1), and the right hand term is Sq0-free, because it injects into Ln-2, which is itself Sq0-free. This argument and a similar o* *ne applied to BPn ! Pn-1 ! Pn-2 give us ~ ~ 11cok(BLn ! Ln-1)~= 11ker(Ln-1_!__Ln-2)_im(BL ~ ~ n ! Ln-1) 11cok(BPn ! Pn-1)~= 11ker(Pn-1_!__Pn-2)_im(BP, n ! Pn-1) and these are isomorphic by the inductive hypothesis, completing the proof of t* *he lemma. Corollary 4.4. The commuting ladder BLn+1 ----! Ln ----! Ln-1 ?? ? ? y ?y ?y BPn+1 ----! Pn ----! Pn-1 induces an isomorphism on H* HomU_(-, tF2) for all t at the middle term. Proof.It is sufficient to prove that in the commuting ladder BLn+1 ----! Ln ----! Ln-1 ----! 2Ln-2 . . .----! n-1L1 ?? ? ? ? ? y ?y ?y ?y ?y BPn+1 ----! Pn ----! Pn-1 ----! 2Pn-2 . . .----! n-1P1, the map between the upper and lower rows is an isomorphism on homology up to and including Ln ! Pn. The proof is by induction, beginning with BL2 ----! L1 ----! 0 ?? ? ? y =?y ?y BP2 ----! P1 ----! 0 In the case of SO, BL2 ! BP2 is an equality. In the case of U, we observe BL2 = BC11 BC20= BP2 BC20where the BP2 summand maps to BP2 by the identity and BC20maps to L1 by the zero map. Thus we have a base for the induction in the case of U also. Suppose that BLn ----! Ln-1 ----! 1Ln-2...----! n-2L1 ?? ? ? ? y ?y ?y ?y BPn ----! Pn-1 ----! 1Pn-2...----! n-2P1 16 KATHRYN LESH induces an isomorphism on homology up to Ln-1 ! Pn-1. Applying to both complexes, we find that Ln ----! Ln-1 ----! 2Ln-2...----! n-1L1 ?? ? ? ? y ?y ?y ?y Pn ----! Pn-1 ----! 2Pn-2...----! n-1P1 is an isomorphism on homology up to Ln-1 ! Pn-1, and joining this with the result of Lemma 4.3, we find that BLn+1 ----! Ln ----! Ln-1 ----! 2Ln-2 . . .----! n-1L1 ?? ? ? ? ? y ?y ?y ?y ?y BPn+1 ----! Pn ----! Pn-1 ----! 2Pn-2 . . .----! n-1P1 is an isomorphism on homology up to and including Ln ! Pn. The lemma follows. Corollary 4.5. The homotopy spectral sequence of {En} collapses at E2. Proof.By Corollary 4.4, the map {Yn} ! {En} induces a map of homotopy spec- tral sequences which is an isomorphism on the E2-term. Since the homotopy spec- tral sequence of {Yn} has no further differentials (in fact, it collapses at E1* *), the homotopy spectral sequence of {En} collapses at E2. 5.A Model for the UASS, and Some Predictions and Reflections In the preceding sections, we used the resolutions of the filtration quotients Mn=Mn-1 to construct a complicated tower {En} that involves those resolutions, converges to SO^2, and has a homotopy spectral sequence that collapses at E2. T* *he tower {En} realizes the chain complex L*, where the notation L*is to be interpr* *eted as BLn+1 ! Ln ! Ln-1at the nth level. The differential of the chain complex L* gives rise to the only nonzero differential in the homotopy spectral sequence o* *f {En}, since the E1-term is Hom U_(Ln, *F2) at level n, and En,t2~=En,t1(Corollary 4.* *5). In this section, we describe how the complex L* gives a model for the unstable Adams spectral sequences of SO and U, we make some predictions based on the model, and we discuss some related work of Bousfield and Davis [B-D]. 5.1. A Model for the UASS. The conjecture suggested by Mahowald is, loosely, that the differential of the chain complex L* contains all the information of the unstable Adams spectral sequence, including all of its many nonzero differentials. We already know that H*[Hom U_(L*, t tF2)] is the associated graded for the filtration of ß*SO^2by * *the destabilized Adams tower (Corollary 4.4 and Corollary 4.5). The assertion is th* *at it is possible to produce the UASS from the complex Hom U_(L*, *F2) by a com- bination of filtering and regrading. To describe the proposed model, let L* be the cochain complex of graded vector spaces defined by (Ln)j= HomU_(Ln, jF2), UASS FOR SO AND U 17 and use the differential BLn+1 ! Ln and adjointness to define d : (Ln)j ! (Ln+1)j-1by Hom U_(Ln, jF2)H!omU_(BLn+1, jF2) ~=Hom U_( BLn+1, j-1F2) ~=Hom U_(Ln+1, j-1F2). We filter Ln by (FsLn)j= HomU_( ni=sCn-ii, jF2). We have F0 F1 F2 . .,.and comparing to the construction of BLn+1 ! Ln in Section 3, it is easy to check that the differential on L* is filtration-pre* *serving. Thus the filtration gives rise to a spectral sequence that converges to H*L*, a* *nd we grade it as Es,t1= n HomU_(Cns, t-sF2). Recall that the abutment, H*L*, is the associated graded to ß*SO^2. Also, Cns= sDns, and hence by the adjointness of and , we have Es,t1= n HomU_(Dns, tF2). The d1-differential is induced by differential in the resolution Dn*! Mn=Mn-1, * *and thus the spectral sequence becomes Es,t2= n ExtsU_(Mn=Mn-1 , tF2) ) ß*SO^2. Conjecture 5.1. The spectral sequence Es,trdefined above is the UASS for SO. If Conjecture 5.1 is correct, then it has the consequence that all of the dif* *ferentials in the unstable Adams spectral sequence can be computed from the primary level calculation of the complex L*. In principle, this could be done indefinitely fa* *r out by computer. Corollary to Conjecture 5.1 . Exts,tU_(M1 , F2) ~= n Exts,tU_(Mn=Mn-1 , F2). Proof.The E2-term of the UASS for SO is given by Es,t2~=ExtsU_(M1 , tF2), and hence if Conjecture 5.1 is correct, these two must be isomorphic. In fact, there is a general spectral sequence that is very close to the spect* *ral se- quence of Conjecture 5.1, namely the Grothendieck spectral sequence for the cal* *cu- lation of the derived functors ExtsA( A=Sq3 , tF2). Let D be the destabilizati* *on functor from the category of (stable) A-modules to U_, the category of unstable A-modules. (This functor is often denoted 1 .) Because tF2 is an unstable A-module, any map to tF2 from a stable A-module factors through the desta- bilization. Hence the functor Hom A(-, tF2) can be written as the composition Hom U_(-, tF2) O D(-), giving rise to a composite functor spectral sequence Exts-rU_(Dr- , tF2) =) ExtsA(- , tF2). In the case of A=Sq3, ExtsA( A=Sq3 , tF2) actually gives the associated graded to the stable homotopy, because there are no differentials in the stable Adams spectral sequence for infinite delooping of SO. Thus the Grothendieck spectral sequence gives a spectral sequence starting from an unstable Ext and converging to ß*SO. 18 KATHRYN LESH The Grothendieck spectral sequence is very closely related to the spectral se- quence we have constructed, but it is not quite the same. In particular, let X = A=Sq3, so that we are considering the case of SO. Then it can be shown that Mn+1=Mn ~=Dn -nX, the ingredients being found in Lemma 2.5, Lemma 2.1, and the proof of Proposition 4.3, and our construction gives a spectral sequence Exts-rU_(Dr -rX , tF2) =) ExtsA(X, tF2). However, the situation for the group U is a little different, the difference_be* *ing* caused by the fact that while H*SO is the free unstable A-algebra_on*H RP1 , which is Sq0-free, H*U is the free unstable A-algebra on H CP1+, which_is not. In fact, contrary to the assertion of [B-D, Proposition 4.1], if X ~= A = 1, wh* *ere 1 is the subalgebra of A generated by the Milnor primitives Q0 and Q1, then Dn -nX is not Mn+1=Mn Z=2 but a much larger module. The problem lies not in the spectral sequence constructed in the proof of the proposition, but i* *n its assumption that the homology being converged to is Mn+1=Mn. However, a small variation can repair the problem. Let X be an A-module, and let C* be a stable resolution of X. For n 1, define D0rX = __ker(D_Cr_!_D_Cr-1)_im(DC. r+1! DCr) Using methods similar to those of Proposition 4.3, one can show that the defini* *tion of D0rX is independent of the resolution used, and that the modules D0rX and DrX are different exactly_when Dr-1 X is not Sq0-free. If we let X = A=Sq3 (in the case of SO) or X = A =Sq3 (in the case of U), then for both SO and U, D0n -nX ~=Mn+1=Mn, __* where the modules_Mn=Mn-1 are the filtration quotients of H RP1 (in the case of SO) or H *CP1+ (in the case of U). The construction of the previous section gives, for a general A-module X, two spectral sequences, depending on whether we use D0ror Dr: (5.1) Exts-rU_(D0r -rX, tF2)=) ExtsA(X, tF2) (5.2) Exts-rU_(Dr -rX , tF2)=) ExtsA(X, tF2). (The spectral sequence of Conjecture 5.1 is (5.1).) These spectral sequences ca* *n be given a construction almost exactly like that of the Grothendieck spectral sequ* *ence. Conjecture 5.1 observes that because the stable Adams spectral sequences for SO and U collapse, the target of spectral sequence (5.1) is actually the associate* *d graded to the homotopy of the space. Since the E2-term is closely related to the homol* *ogy of the space, because D0r -rX is the associated graded for the cohomology of SO (or U), this variation of the Grothedieck spectral sequence could actually be t* *he unstable Adams spectral sequence. 5.2. Predictions. Next we discuss some predictions that arise from Conjecture 5.1 and some empiri* *cal data that support the conjecture. The main tool in making these predictions is a vanishing theorem of Bousfield [B , Theorem 2.6] that describes the location of* * h0- towers in unstable Ext by giving values of t - s where towers occur, though not the value of s in which they begin. Application of Bousfield's theorem gives us* * the UASS FOR SO AND U 19 following proposition. Recall that ff(n) denotes the number of ones in the dyad* *ic expansion of n. Proposition 5.2. __* (1)For M = H RP1 : (a)The h0-towers of ExtsU_(M, tF2) are found in stem degrees satisfying (t - s) 3 mod 4, and there is exactly one h0-tower in each such dimension. (b)The h0-towers of ExtsU_(Mn=Mn-1 , tF2) are found in stem degrees satisfying (t - s) 3 mod 4 and ff(t - s) = n, and there is exactly * *one h0-tower_in*each such dimension. (2)For M = H CP1+, (a)The h0-towers of ExtsU_(M, tF2) are found in stem degrees satisfying (t - s) 1 mod 2, and there is exactly one h0-tower in each such dimension. (b)The h0-towers of ExtsU_(Mn=Mn-1 , tF2) are found in stem degrees satisfying (t - s) 1 mod 2 and ff(t - s) = n, and there is exactly * *one h0-tower in each such dimension. Proof.An easy calculation with [B , Theorem 2.6]. Remark 5.3. Proposition 5.2 says that Corollary to Conjecture 5.1 is correct at* * least at the level of h0-towers, since ExtsU_(M, tF2) and n ExtsU_(Mn=Mn-1 , tF2) * *have exactly the same towers. Bousfield's theorem also gives a vanishing line above which Ext is zero except for h0-towers. To describe his theorem as it applies to our situation, we defin* *e a function OE(m) for positive integers m as follows. Suppose that m = 8k + i where i < 8. Then (1)OE(m) = 4k + i for i = 0, 1, 2, 3; (2)OE(m) = 4k + 3 for i = 4, 5, 6; (3)OE(m) = 4k + 4 for i = 7. We specialize Bousfield's theorem to our situation as follows. Theorem 5.4 ([B , Theorem 2.6]). Let N be an unstable A-module such that Ni= 0 for i < c, where c 5. Then ExtsU_(N, tF2) is free over F2[h0] for s > OE(t-s* *-c). This gives a vanishing line of slope 1=2 in the UASS. We are going to use Theorem 5.4 to predict the unstable Adams filtrations of the elements of ß*SO and ß*U. From the map of towers {Yn} ! {En}, the maps KPn+1 ! KLn+1 induce on homotopy a map (5.3) ExtnA( A=Sq3 , tF2) ! nr=1Extn-r+1U_(Mr=Mr-1 , t-r+1F2), and this map commutes with the action of h0. All of the elements on the left represent homotopy, and since the right-hand side is the E2-term for the spectr* *al sequence of Conjecture 5.1, the map tells us where the homotopy is represented * *in this spectral sequence, which predicts the unstable Adams filtration of ß*SO. Consider first the case of SO. Suppose k 3 mod 4; if k 3 mod 8, define n = (k - 1)=2, and if k 7 mod 8, define n = (k - 3)=2. Then ßkSO ~=Z, represented by an h0-tower in Ext*A( A=Sq3 , *+kF2) beginning in filtration s * *= n. On the right side of (5.3), the only term with an h0-tower in dimension k is r * *= ff(k) 20 KATHRYN LESH (Proposition 5.2), and so the part of (5.3)that carries the bottom element of t* *he h0-tower is ExtnA( A=Sq3 , tF2) ! Extn-ff(k)+1U_(Mff(k)=Mff(k)-1, t-ff(k)+1F2). Thus we obtain the following prediction. Conjecture 5.5. If ßkSO is torsion free, then the unstable Adams filtration of ßkSO is ff(k) - 1 less than the stable Adams filtration of the corresponding st* *em. By exactly the same reasoning we obtain the same prediction for the case of U, where all the homotopy is torsion free. Conjecture 5.6. The unstable Adams filtration of ßkU is ff(k) - 1 less than the stable Adams filtration of the corresponding stem. Next, we predict the unstable Adams filtration of the torsion elements of ß*S* *O, namely ßkSO ~=Z=2 for k 0 or 1 mod 8. Consider first the case k 0 mod 8, and let n = (1=2)k - 1. Then ßkSO is represented in ExtnA( A=Sq3 , n+kF2). As before, we predict the unstable Adams filtration by considering the image of th* *is element under the map of (5.3): ExtnA( A=Sq3 , n+kF2) ! nr=1Extn-r+1U_(Mr=Mr-1 , n+k-r+1F2). Using Theorem 5.4, we will prove that only the r = 3 summand has h0-torsion elements in high enough filtration to be in the image of this map. We already k* *now that M1 has exactly one torsion element in Ext for s = 0 and nothing else, and M2=M1 has exactly one h0-tower in Ext for k = 3, and nothing else. Suppose that r 4, and note that Mr=Mr-1begins in dimension 2r- 1. To use Theorem 5.4 to show that Extn-r+1U_(Mr=Mr-1 , n+k-r+1F2) has no h0-torsion elements, we must show that (n - r - 1) > OE[(n + k - r - 1) - (n - r - 1) - (2r- 1)], a task that is easily accomplished using k 0 mod 8 and n = (1=2)k - 1. An almost identical calculation leads to the same conclusion if k 1 mod 8. This leaves the r = 3 summand as the only one where the torsion elements can go, and since r = 3 causes a filtration drop of 2 from the stable Ext, we arrive at* * the following prediction. Conjecture 5.7. If ßkSO ~=Z=2 is represented in filtration n in the stable Adams spectral sequence, then it has filtration n-2 in the unstable Adams spectral se* *quence. Remark 5.8. The author has verified the preceding conjectures as to filtration * *for ß*SO up to approximately ß50, using charts of unstable Extprovided by R. Bruner* *'s computer calculations. Likewise the author has verified the Corollary to Conjec- ture 5.1 for SO in the same range. We close this discussion by giving an example of the calculation of a differe* *ntial in the spectral sequence modelling the UASS for SO In the Figure 3, we exhibit part of the UASS for SO. We will show how to use the spectral sequence of Conjecture 5.1 to predict the first differential in the UASS for SO, which goes from (s, t - s) = (0, 15) to (s, t - s) = (2, 14). (This differential propagate* *s to give differentials connecting the two lightning flashes, but we will deal only with * *the first differential.) UASS FOR SO AND U 21 | | 10 |_ b r r rd | | | |_ b| r| r| rd| | | | |_ b| r| r| rd| | | | |_ b| r| r| rd| | | | |_ b| r| r| rd| r | | | |_ b| r| r| rd|r r 5 | | | | | | | | | |_ b r r| r rd r r | | | | | BB | |_ b r| r| r| rrd |r rd | | | | |_ b| r| r r| rd rdrd| | | AK | |_ b| r| r A rd|rd rd| | | ______________________________________________________|||||||||||||||||||b* *b|r|rd|A 5 10 15 Figure 3. The E2-term of the UASS for SO Elements represented by open circles arise from M1 and M2=M1. El- ements represented by black dots arise from M3=M2. Elements repre- sented by circled dots arise from M4=M3. In order to do this, we will have to calculate the first few stages of the co* *mplex L*. In particular, we will be looking at the commuting three term sequences BL5 ----! L4 ----! L3 ? ? ? (5.4) h5?y h4?y 2h3?y BP5 ----! P4 ----! P3, which is detailed in Table 1. We need the result that Mn=Mn-1 ~= F(2n - 1)=Sq1, Sq2, ..., Sq2n-2[Massey], and we remind the reader that in the diagram above, the top row involves resolutions of Mn=Mn-1 for n = 1, 2, 3, 4, and 5, where the resolution of Mn=Mn-1 is looped down 4 - n times. When n = 1, M1 ~=F(1) is a projective,_and has a resolution of length 1. Hence C1i= 0 for i > 0. Further, M2=M1 ~=F(3) is almost projective. Its projective resolution is . .!.F(5) ! F(4) ! F(3) (each map given by Sq1), and so all the elements contributed lie in t - s = 3. It turns out that this resolution does not intera* *ct with any of the other parts of L*, corresponding to the fact that no differentials i* *n the UASS for SO involve t - s = 3. In Table 1, we provide all the summands of each of the terms in (5.4)and show the horizontal maps between them. In the commuting square L4----! L3 ? ? h4?y 2h3?y P4----! P3, 2h3 is the identity, and h4 is the map is the identity on the summands F(3), F(7), and F(8). To describe h4 on the summand F(15) of L4, we recall that 22 KATHRYN LESH ________BL5_________________-!__________________L4_____-!_______L3__ C2*: F(4) Sq1'3 F(3) Sq1'2 F(2) 8 1 < F(8) Sq '7 F(7) Sq1' C3*: : F(10) Sq2'8+ Sq3'7 ____F(8) Sq2'6 F(6) F(15) Sq7'8+ Sq4,2,1'8+ Sq6,2'7+ |_'15|_ 6 8 < F(16) Sq1'15 C4*: : F(17) Sq2'15 F(15) F(19) Sq4'15 C5*: F(32) _____________________________________________________________________ ________BP5__________________-!_________________P4______-!______P3__ F(4) Sq1'3 F(3) Sq1'2 F(2) F(8) Sq1'7 F(7) Sq1'6 F(6) ________F(10)____________Sq2'8+_Sq3'7__________F(8)____Sq2'6________ Table 1. The chain complexes of Section 3 '15 2 L4 must hit an element of P4 that represents an A -module generator of the homology of the three term sequence BP5 ! P4 ! P3, and the element in question is Sq7'8+ Sq4,2,1'8+ Sq6,2'7 2 P4. Now for the differential, which is predicted by the construction of the map B* *L5 ! L4. It comes about because the map BL5 ! L4 must be defined in such a way that the composite BL5 ! L4 ! P4 lifts across BP5 ! P4. Since there are no interactions between the filtrations in the map L4 ! L3, the map BL5 ! L4 can be constructed simply by using the differentials within the resolutions Cn** *, and then making adjustments as needed to ensure the required lifting. In terms of t* *he construction of Section 3, this is saying that the map g5 is just the sum of the differentials in the individual resolutions. No corrections need to be made until we reach F(15) BL5. At this point, if * *no adjustments were made, the composite BL5 ! L4 ! P4 would take the generator '152 BL5 to Sq7'8 + Sq4,2,1'8 + Sq6,2'7 2 P4. Since this element generates the homology at P4, it certainly does not lift to BP5. Thus we add '152 L4to the im* *age of '152 BL5 (boxed for emphasis in the table). This gives a differential between adjoining filtrations in L*, which translates to the prediction of the nonzero * *d2 differential taking (s, t - s) = (0, 15) to (s, t - s) = (2, 14) in the UASS of* * SO. UASS FOR SO AND U 23 5.3. Relation to [B-D]. Bousfield and Davis make a much more general conjecture than our Conjecture 5.1 in [B-D]. Suppose given a diagram of unstable A-modules F1 F2 F3 ?? ? ? yf0 ?yf1 ?yf2 X0 --p0--!X1 --p1--!X2 --p2--!. .-.---!X ? ? i1?y i2?y F1 F2 satisfying the following conditions. (1)Fn ! Xn-1 ! Xn ! Fn ! Xn-1 is exact. (2)Fn is a direct sum of F(m)'s and/or F0(m)'s (where F(m) is a free unstable A -module on a generator of dimension m and F0(m) = F(m)=Sq1). (3)(infn)* : ExtsU_( Fn , tF2) ! ExtsU_(Fn+1, tF2) is the zero map. (4)ker(Xn ! X) = ker(Xn ! Xn+1). (5)X ~=lim-!n(Xn). Let Mn = im(Xn ! X). Conjecture 5.9 ([B-D, Conjecture 5.1]). ExtsU_(X, tF2) ~= n ExtsU_(Mn=Mn-1 , tF2). However, this conjecture is false, as shown by the counterexample that follow* *s. Consider the following tower, whose k-invariants are described below. K(Z=2, 10)--i4--!Y4 ?? y K(Z=2, 8)--i3--!Y3--k3--!K(Z=2, 10) ?? y K(Z=2, 8)--i2--!Y2--k2--!K(Z=2, 9) ?? y K(Z, 7)--i1--!Y1--k1--!K(Z=2, 9) ?? y * ----! K(Z, 8) Let H*Yi= U(Zi). The first k-invariant is k1 = Sq2'7 and the second is k2 = 0. For the_third,_let x10 be a class in Z2 with (i2)*(x10) = Sq2'9 2 ker(Sq2 : F(9) ! F(7)), and let x010denote its image in Z3. Let x8 be a class in Z3 with (i3)*(x8) = '8, the fundamental class. Then the third k-invariant is defined by k3 = x010+ Sq2x8. 24 KATHRYN LESH We consider Bousfield and Davis's conjecture for this situation, where the di* *a- gram is given by __ F(8) F(9) F(9) F(10) ?? ? ? ? y Sq2'7?y 0?y x010+Sq2x8?y __ p1 p2 p3 0 ----! F(7)----! Z2 ----! Z3 ----! Z4 = X ? ? ? ? i1?y i2?y ?y ?y __ F(7) F(8) F(8) F(9). In particular, we consider Ext0, so that we are really looking at A-module gene* *ra- tors. We find that Ext0has nonzero groups only in the following dimensions. (1)Ext0U_(M1 , tF2) = Z=2 if t = 7. (2)Ext0U_(M2=M1 , tF2) = Z=2 if t = 10 or 15. (3)Ext0U_(M3=M2 , tF2) = Z=2 if t = 8. (4)Ext0U_(M4=M3 , tF2) = Z=2 if t = 12 or 31. (5)Ext0U_(X, tF2) = Z=2 if t = 7, 8, 12, 15 and 31. In particular, Ext0U_(X, tF2) has no nonzero class for t = 10. In fact, M3=M2 * *~= F(8)=Sq2, and in the spectral sequence for Ext*U_(X, tF2) arising from the fil* *tration of X, there is a nonzero differential Ext0U_(M2=M1 , 10F2) ! Ext1U_(M3=M2 , 10F2). In effect, what we have done in this example is to introduce a generator in M2 (namely x10, corresponding to Sq2'8) and then to equate it with a Steenrod oper* *a- tion on another class at a later stage, thus eliminating it from the list of ge* *nerators. However, it is possible to revise Conjecture 5.9 to deal with this problem. T* *he salient feature that distinguishes the situation for SO and U from the example above is that there is a stable resolution in the background. In other words, * *in the case of the_tower_{Yn} defined in Section 2, the tower realizes a destabili* *zed resolution of A =Sq3, whereas in the counterexample above, it realizes the uns* *table complex __ Sq2 0 Sq2 F(7)---- F(9)---- F(10) ---- F(12), which is certainly not the destabilization of a resolution. To reflect this, we* * refine Bousfield and Davis's conjecture as follows. Conjecture_5.10._Conjecture_5.9_is_true if we add the hypothesis that there exi* *st A-modules_Fn and maps dn: Fn+1! Fn satisfying the following_conditions: (1)F nis_the sum of copies of A and A=Sq1, and nDF n~=Fn. (2) nD(dn)_=_in O fn. (3)(F *, d*) is a chain complex whose only nonzero homology group occurs in the lowest homological dimension. References [B] A. K. Bousfield, A vanishing theorem for the unstable Adams spectral sequ* *ence, Topology 9 (1970) 337 - 344. [B-D] A. K. Bousfield and D. M. Davis, On the unstable Adams spectral sequence * *for SO and U, and splittings of unstable Ext groups, Bol. Soc. Mat. Mex. 37 (1992) 4* *1-53. UASS FOR SO AND U 25 [H-M] J. R. Harper and H. R. Miller, Looping Massey-Peterson towers, in: S. M. * *Salamon, B. Steer, W. A. Sutherland, eds., Advances in Homotopy Theory (London Math. * *Soc. Lec. Notes 139, Cambridge University Press, Cambridge, 1989) 69-86. [Lannes]J. Lannes, Sur les espaces fonctionels dont la source est le classifian* *t d'un p-groupe ab`elien 'elementaire, Publ. Math. Inst. Hautes Etudes Sci. 75 (1992), 13* *5 - 244. [Lesh]K. Lesh, The unstable Adams spectral sequence for two-stage towers, Topol* *ogy Appl. 101 (2000), no. 2, 161-180. [Long]J. Long, Two contributions to the homotopy theory of H-spaces, Princeton * *University thesis (1979). [Massey]W. S. Massey, unpublished manuscript, 1978. [M-P] W. S. Massey and F. P. Peterson, The mod 2 cohomology structure of certai* *n fiber spaces, Mem. Amer. Math. Soc. Number 74, Providence, 1967. [S-E] N. E. Steenrod and D. B. A. Epstein, Cohomology Operations, (Ann. Math. S* *tudies Number 50, Princeton University Press, Princeton, NJ, 1962). [S] R. Stong, Determination of H*(BO(k, 1)) and H*(BU(k, 1)), Trans. Amer. Ma* *th. Soc. 107 (1963) 526-544. Department of Mathematics, Union College, Schenectady, NY 12308 E-mail address: klesh@member.ams.org