The Unstable Adams Spectral Sequence for Two-Stage Towers

Transcription

1 The Unstable dams Spectal Sequence fo Two-Stage Towes Kathyn Lesh 1 Depatment of Mathematics, Univesity of Toledo, Toledo, Ohio US bstact Let KP denote a genealized mod 2 Eilenbeg-MacLane space and let Y be the fibe of a map X KP to which the Massey-Peteson theoem applies. We study the elationship of the mod 2 unstable dams spectal sequence (USS) fo X and fo Y. Given conditions on X, we split the E 2 -tem fo Y, and we use a pimay level calculation to compute d 2 fo Y up to an eo tem. If the USS fo X collapses at E 2 (fo example, if X is an Eilenbeg-MacLane space), the USS fo Y collapses at E 3, and we have the entie USS fo Y. We also give examples and addess a conjectue of Bousfield on the USS fo the Lie goup SO. Key wods: unstable dams spectal sequence, unstable Ext goups, SO 1991 MSC: 55T15, 55Q52, 55R45 1 Intoduction Let KP denote a mod 2 genealized Eilenbeg-MacLane space. In this pape we study the mod 2 unstable dams spectal sequence (USS) of a space Y with polynomial cohomology which is obtained as the fibe of a map between simply connected spaces X KP to which the Massey-Peteson theoem applies. By using specially constucted dams esolutions, we study the elationship between the USS fo X and fo Y. Given cetain conditions on X, wegive a splitting of the E 2 -tem of the USS fo Y, and we show how to use a pimay level calculation to compute almost complete infomation about the d 2 diffeentials in the USS fo Y. In the case that the USS fo X collapses at E 2, the USS fo Y collapses at E 3, and we have almost complete infomation about it. 1 klesh@uoft02.utoledo.edu Pepint submitted to Topology and Its pplications 15 May 1998

2 To state ou esults moe pecisely, we make some definitions. Thoughout the pape, we take p = 2 and cohomology with mod 2 coefficients. Let U denote the categoy of unstable modules ove the mod 2 Steenod algeba. If M is an object in U, let U(M) denote the fee unstable -algeba geneated by M, as descibed by [5]. If P is a fee unstable -module, we wite KP fo the Eilenbeg-MacLane space with H KP = U(P ). We call a map X KP Massey-Peteson if H X = U(M), H X is polynomial of finite type, X and KP ae simply connected, and H KP H X is induced by a map f : P M. We think of the topological map as ealizing f. IfX KP is Massey-Peteson, with fibe Y, then the Massey-Peteson theoem [4] tells us that H Y = U(N), whee thee is a shot exact sequence 0 cok(f) N Ωke(f) 0. In geneal, this fundamental sequence does not split as -modules. Howeve, the following theoem states conditions unde which it does split at the level of Ext. Theoem 1.1 Let X be a simply connected space with polynomial mod 2 cohomology H X = U(M), and let X KP be Massey-Peteson, inducing f : P M in cohomology. Let Y be the fibe of X KP, and suppose H Y = U(N) is polynomial. If X has E 2 = E 3 in its unstable dams spectal sequence, then thee is a splitting Ext U (N,Σ t F 2 ) = Ext U (cok(f), Σ t F 2 ) Ext U (Ω ke(f), Σ t F 2 ). In geneal, the E 2 -tem of an USS is an Ext goup taken ove the categoy K of unstable algebas ove the Steenod algeba. Because H Y = U(N), Ext K (H Y,H S t ) = Ext U (N,Σt F 2 ), and hence Theoem 1.1 descibes the E 2 -tem of the USS fo Y. Many diffeentials in the USS fo Y can be descibed in this case; we make a pecise statement in Theoem 4.5. Roughly speaking, the d 2 diffeentials that go between the two summands of Ext U (N,Σ t F 2 ) can be computed by a pimay level calculation involving an algebaic mapping cone. In fact, if X has E 2 = E in its USS, then the USS fo Y can be computed almost completely by this calculation (Coollay 4.7). The motivation fo studying this situation was povided by a conjectue of Bousfield fom the 1970s about the E 2 -tem of the USS fo the Lie goup SO. We descibe this conjectue in Section 5, along with examples of the application of Theoem 1.1 and Theoem 4.5. Ou stategy fo studying the fibation Y X KP whee H X = U(M) 2

3 and the fibation is induced by f : P M is to build a esolution of M using P and esolutions of cok(f) and ke(f). When ealized, this esolution gives an dams towe fo X which can be manipulated to give an dams towe fo Y. Compaison of the two towes gives infomation about the diffeentials in the USS fo Y. The oganization of the pape is as follows. In Section 2 we constuct a special chain complex esolving M and use it to constuct an dams towe fo X and a towe with invese limit Y which ae closely elated. In Section 3, we constuct an dams towe fo Y by eaanging the towe fo X. In Section 4, we study the homotopy spectal sequences of X and Y to pove Theoem 1.1, and we state and pove futhe esults on diffeentials in the USS fo Y (Theoem 4.5 and Coollay 4.7). In Section 5, we give examples of the application of Theoem 1.1 and Theoem 4.5. The notation set up in the intoduction will be caied though the pape. 2 dams towe fo X In this section we constuct an dams towe fo X and use it to obtain a towe whose invese limit is the 2-completion of Y.ThetowefoY will not be an dams towe, but we show in Section 3 that its k-invaiants can be eaanged in a contollable way to give an dams towe fo Y. We will etain enough infomation though the eaangement that we can elate the homotopy spectal sequences of the two towes, thus obtaining infomation about the USS fo Y fom that of X. We will need to make fequent use of the algebaic looping functo on U, and so we eview the basic facts hee. (See also [4].) The functo Ω : U U is ight adjoint to the suspension functo Σ : U U. The module ΩM is the lagest unstable quotient of the desuspension of M: ΩM (Σ 1 M)/(Σ 1 Sq 0 M), whee Sq 0 x =Sq x x. We wite Ω k fo the k-fold iteate of Ω. The functo Ω is not exact, but only its fist deived functo, which we denote Ω 1 1,canbe nonzeo. The module Ω 1 1M can be expessed as a egading of the kenel of Sq 0 on M. In paticula, if Sq 0 acts feely on M, thenω 1 1 M = 0. We wite Ωk j fo the jth deived functo of Ω k and note that thee is a composite functo spectal sequence (the Singe spectal sequence) Ω s i Ω t jm Ω s+t i+j M which allows us to calculate deived functos of Ω k inductively. Fo any unstable module M, Ω k j M =0foj>k. Now we tun to the constuction of an dams towe fo X. Recall that H X = 3

4 U(M) and that Y is the fibe of a Massey-Peteson map X KP which ealizes f : P M. We know that H Y = U(N) wheen is given by an extension 0 cok(f) N Ωke(f) 0. We will constuct an dams towe fo X that can be elated to a towe fo Y by using a paticula esolution of M. Thee is a shot exact sequence 0 P/ke(f) M cok(f) 0, and so we note that both M and N ae given by extensions which involve cok(f) andke(f). Let C and D be minimal pojective esolutions of cok(f) and ke(f), espectively. Late in the pape we will need the following lemma. Lemma 2.1 ΩD is a pojective esolution of Ωke(f). Poof. The kth homology goup of the complex ΩD is Ω 1 k ke(f). If k>1, then the functo Ω 1 k is identically zeo. On the othe hand, Sq 0 acts feely on ke(f) because ke(f) P, which is itself Sq 0 -fee. Hence Ω 1 1 ke(f) =0. We constuct a pojective esolution of M as follows. Since D is a pojective esolution of ke(f) andp is pojective, we can make a esolution of P/ke(f) by shifting D i to homological dimension i + 1 and putting P into homological dimension 0. Let B be this chain complex, D 1 D 0 P, which esolves P/ke(f). ugment B by f : P M to obtain H i B =0fo i 0andH 1 B = cok(f). Let Shift(C ) be the augmented chain complex obtained by shifting C down one homological degee, that is, Shift(C ) i = C i+1, and let the augmentation be the diffeential C 1 C 0.ThenH i Shift(C )=0 fo i 0andH 1 Shift(C )=cok(f). It is easy to constuct a chain map g : Shift(C ) B, ɛ C 3 C 2 C 1 C 0 D 1 D 0 P ɛ M, which is an isomophism on all homology goups. In dimension 1, we define g by a lift C 0 M of C 0 cok(f). Then we note that C 1 C 0 M lifts to P,sinceC 1 C 0 M cok(f) is zeo. The est of the map g follows fom pojectivity of C and acyclicity of D. 4

5 Let X be the algebaic mapping cone. Hence X 0 = C 0 P, X i = C i D i 1 fo i 1, X has an augmentation to M, and the diffeential is defined by d =(d C,g d D ): D D D P M C 3 C 2 C 1 The augmented complex X is acyclic because g is a homology isomophism. Hence X M is a pojective esolution, and it can be ealized to give an dams towe {X k } fo X. (See, fo example, [2, Section 3].) We note the following facts about the towe {X k }, which follow fom its being a ealization of the pojective esolution X M whee H X = U(M): (1) lim k X k Xˆ2. (2) The k-invaiants X k KΩ k (C k+1 D k ) ae Massey-Peteson maps. (3) The composite of the inclusion of a fibe with a k-invaiant C 0 KΩ k (C k D k 1 ) X k KΩ k (C k+1 D k ) ealizes the looped down diffeential of X. (4) X 0 KC 0 KP. To constuct a chain complex modeling Y, we define E as the quotient of X by P in dimension 0: E i = X i fo i 1andE 0 = X 0 /P. If we egad P as a chain complex concentated in degee 0, thee is a shot exact sequence of chain complexes 0 P X E 0 which can be egaded as modeling the fibation Y X KP. We ealize this situation by defining a new towe as follows. Define a map of {X k } to the constant towe {KP} by X k X 0 KP, whee the second map is pojection, and let {E k } be the towe of fibes. The following lemma gives the essential facts about {E k }. Lemma 2.2 (1) lim k E k Y ˆ2. (2) The k-invaiants E k KΩ k (C k+1 D k ) ae Massey-Peteson maps. (3) The composites KΩ k (C k D k 1 ) E k KΩ k (C k+1 D k ) ealize the looped down diffeential of E. 5

6 (4) The commutative diagam of k-invaiants poduced by the map of towes {E k } {X k } ealizes the quotient map of chain complexes X E appopiately looped down. Poof. Fo the fist item, we note that since E k is the fibe of the map X k KP,wehavethatlimk E k is the fibe of the invese limit map limk X k KP. In view of the fact that KP is its own 2-completion, since it is a mod 2 Eilenbeg-MacLane space, the map between invese limits is the 2-completion of X KP. Since all the spaces ae simply connected, the fibe lemma fo p-completion tells us that the fibe of the map between completions is the completion of the fibe, that is, lim k E k Fibe(lim X k KP) k Fibe(Xˆ2 KP) [Fibe(X KP)]ˆ2 Y ˆ2 To elate the towe {X k } and the towe {E k }, we study the following commutative diagam: E k+1 E k KΩ k (C k+1 D k ) = X k+1 X k KΩ k (C k+1 D k ) KP = KP The uppe ight hand squae gives the k-invaiants, and hence the thid pat of the lemma follows immediately fom the coesponding fact fo {X k } and the following commuting diagam of k-invaiants and inclusions of fibes: KΩ k (C k D k 1 ) E k KΩ k (C k+1 D k ) = = KΩ k (C k D k 1 ) X k KΩ k (C k+1 D k ). Likewise the fouth pat of the lemma follows fom noting that at the bottom 6

7 of the towes we have E 0 KC 0 K(C 1 D 0 ) = X 0 K(P C 0 ) K(C 1 D 0 ) π KP. To pove the second pat of the lemma, we poceed inductively. Conside the map of fibations aising fom the fist k-invaiant: E 1 E 0 KC 0 K(C 1 D 0 ) X 1 X 0 K(P C 0 ) K(C 1 D 0 ). Each of the k-invaiants is clealy a Massey-Peteson map, and so the cohomologies of E 1 and X 1 ae vey nice in the sense of Massey-Peteson, say H E 1 = U(N1 )andh X 1 = U(M1 ). The fundamental sequence is natual, giving us a ladde of shot exact sequences: 0 cok(f) N 1 Ωke(C 1 D 0 C 0 ) 0 0 M M 1 Ωke(C 1 D 0 C 0 P ) 0. In ode to show that E 1 KΩ(C 2 D 1 ) is Massey-Peteson, we need to know that H E 1 is polynomial. It suffices to check that Sq 0 acts feely on the two ends of the shot exact sequence fo N 1. Cetainly Sq 0 acts feely on Ωke(C 1 D 0 C 0 ), since this is a submodule of the pojective Ω(C 1 D 0 ). Futhe, by the assumption that H Y = U(N) is polynomial and the fact that cok(f) N,Sq 0 acts feely on cok(f). It follows easily that E 1 KΩ(C 2 D 1 ) is Massey-Peteson. We need to examine what happens in cohomology fo the fibation E 2 E 1 KΩ(C 2 D 1 ), and then the est of the induction will follow. In paticula, we need to know cok(ω(c 2 D 1 ) N 1 ) in ode to undestand the cokenel side of the fundamental sequence fo H E 2. Conside the following commuting diagam, whee the ows ae Massey-Peteson fibations: Y X KP E 1 E 0 KC 0 K(C 1 D 0 ). (Commutativity of the ight hand squae follows fom the fact that the com- 7

8 posite C 1 D 0 C 0 P M is zeo because it is the squae of the diffeential in the chain complex X.) We obtain a commuting ladde of fundamental sequences fo H Y and H E 1 : 0 cok(f) N Ωke(f) 0 0 cok(f) N 1 Ωke(C 1 D 0 C 0 ) 0. Now the map E 1 KΩ(C 2 D 1 ) induces a map Ω(C 2 D 1 ) N 1 which goes into the Ω ke(c 1 D 0 C 0 ) pat of N 1 ; specifically, it maps to Ω ke(c 1 D 0 C 0 P ). Theefoe cok(ω(c 2 D 1 ) N 1 ) is an extension of cok(f) by cok[ω ke(c 1 D 0 C 0 P ) Ωke(C 1 D 0 C 0 )], which is Ω ke(f), and the ladde of exact sequences above shows that this extension is N. It is now easy to see that im(h E k H E k+1 ) = U(N) fok > 1, and since the pat of H E k+1 coming fom the fibe is also polynomial, we have that H E k+1 is polynomial, as equied. The induction now follows without difficulty. 3 dams towe fo Y In this section, we take the towe {E k } constucted in Section 2 and eaange it into an dams towe fo Y. We do this by taking the k-invaiants, which ae poducts, and sepaating them in such a way as to give a towe {E k } which is inteleaved with the towe {E k}. In Section 4, we will study the homotopy spectal sequence of the towe {E k} and show that this towe is an dams towe fo Y. Recall that E k+1 is the fibe of a Massey-Peteson map E k KΩ k (C k+1 D k ). The towe {E k} is defined as follows. Definition 3.1 Let E k be the homotopy fibe of E k KΩ k D k. Define a map E k+1 E k by the commutative ladde E k+1 E k KΩ k (C k+1 D k ) = π E k E k KΩ k D k The map E k+1 E k is defined as the composite E k+1 E k+1 E k. 8

9 Coollay 3.2 lim k E k lim k E k. Ou fist task is to undestand the k-invaiants fo the towe {E k}, sothat we will late be able to analyze the homotopy spectal sequence. We need the following lemma, which allows us to combine k-invaiants of successive fibations. Lemma 3.3 Suppose given a towe of pincipal fibations W 3 g W 2 K 2 p h W 1 f K 1 and suppose that g factos up to homotopy as g = hp. Then the homotopy fibe of (f,h) :W 1 K 1 K 2 is homotopy equivalent to W 3. Poof. The commutative squae W 2 p W 1 f K 1 is a homotopy pullback squae, hence emains so when we coss the bottom ow with K 2 to obtain the squae W 2 g K 2 p W 1 (f,h) (,id) K 1 K 2. Since the fibes of the vetical maps ae homotopy equivalent and the fibe of g is W 3, the lemma follows. We want to apply this lemma to find a k-invaiant that will get us fom E k to E k+1, and as a peliminay, we find a k-invaiant to go fom E k to E k+1. Lemma 3.4 The homotopy fibe of the composite E k E k KΩ k C k+1 is E k+1. 9

10 Poof. pply the same easoning as in the pevious lemma, cossing the bottom ow of the homotopy pullback squae E k E k KΩ k D k, with KΩ k C k+1. Then the fibe of the new vetical maps will be E k+1. To show thee is a k-invaiant to go fom E k to E k+1 look at the following towe of thee spaces: and to compute it, we E k+1 E k+1 KΩ k+1 D k+1 E k KΩ k C k+1. We next show that the second k-invaiant factos though E k Lemma 3.3. and apply Lemma 3.5 The map E k+1 KΩ k+1 D k+1 factos though E k+1 E k homotopy. up to Poof. Since we ae mapping into an Eilenbeg-MacLane space, the statement amounts to a factoing on the level of cohomology. To explain the algebaic fact that makes this wok, conside a pat of the towe {E k }: KΩ k+1 (C k+1 D k ) E k+1 KΩ k+1 (C k+2 D k+1 ) E k KΩ k (C k+1 D k ). The composite KΩ k+1 (C k+1 D k ) E k+1 KΩ k+1 (C k+2 D k+1 )isa ealization of the looped down chain complex diffeential of the complex E. The citical fact we need fo ou poof is that the composite D k+1 C k+2 D k+1 C k+1 D k C k+1 is zeo. Heuistically, this means that putting in KΩ k+1 C k+1 is not necessay in ode to put in KΩ k+1 D k+1 ; once we have put in KΩ k+1 D k (to obtain E k ), we can then put in KΩ k+1 D k+1. 10

11 We compae the fundamental sequences fo the cohomology of E k+1 and E k aising fom the fibations E k+1 E k KΩ k (C k+1 D k ) = π E k E k KΩ k D k. Let H E k = U(Nk ), let H E k = U(N k ), and note that the cokenel of Ω k (C k+1 D k ) N k is N. The ladde of fundamental sequences coesponding to the above diagam is 0 cok(ω k D k N k ) N k Ωke(Ω k D k N k ) 0 0 N N k+1 Ωke(Ω k (C k+1 D k ) N k ) 0. The left vetical aow is an epimophism because N = cok[ω k (C k+1 D k ) N k ], and we ae mapping to it fom the cokenel of Ω k D k N k. Now we want to conside the map Ω k+1 D k+1 N k+1 and show that it lifts to N k. The composite Ω k+1 D k+1 N k+1 Ωke(Ω k (C k+1 D k ) N k ) is zeo into the facto Ω k (C k+1 ), because of the citical algebaic fact mentioned above. Hence Ω k+1 D k+1 lifts to Ωke(Ω k D k N k ). Then a standad diagam chase using the pojectivity of Ω k+1 D k+1 and the fact that cok(ω k D k N k ) N is an epimophism shows that Ω k+1 D k+1 N k+1 lifts to N k. Coollay 3.6 E k+1 is the fibe of a Massey-Peteson map E k KΩ k C k+1 KΩ k+1 D k+1. Note that, in paticula, E 0 KC 0 KΩD 0. 4 Homotopy Spectal Sequences In this section we study the homotopy spectal sequence fo the towe {E k} constucted in Section 3. By examining d 1 we show that {E k } is an dams towe fo Y. Because the homotopy spectal sequence of {E k } is almost the same as that of {X k }, that is, the USS fo X, we can compae the USS 11

12 fo X and fo Y using the two towes {E k } and {E k }. We deduce elationships between the diffeentials to establish Theoem 1.1 and the elated esults Theoem 4.5 and Coollay 4.7. To study d 1 of the homotopy spectal sequence of {E k }. we need to examine the composite KΩ k C k KΩ k+1 D k E k KΩ k C k+1 KΩ k+1 D k+1. The following lemma elates these maps to the coesponding k-invaiants and inclusions of fibes fo {E k }. Lemma 4.1 The following diagam of k-invaiants and inclusions of fibes commutes: KΩ k+1 C k+1 KΩ k+1 D k E k+1 KΩ k+1 C k+2 KΩ k+1 D k+1 (,π) (,π) KΩ k C k KΩ k+1 D k E k KΩ k C k+1 KΩ k+1 D k+1 (π, ) (π, ) KΩ k C k KΩ k D k 1 E k KΩ k C k+1 KΩ k D k Poof. The uppe left squae commutes by the commutative ladde which defines the map E k+1 E k. (See Definition 3.1.) The uppe ight squae commutes by Lemma 3.5 and the fact that E k+1 E k KΩk C k+1 is a fibation. The lowe ight squae commutes because E k KΩ k C k+1 is defined as the composite E k E k KΩ k C k+1 and because E k E k KΩ k D k is a fibation. Fo the lowe left squae, we expand the 2 2 squae E k E k 1 KΩ k D k to a 3 3squae: KΩ k C k KΩ k+1 D k E k E k 1 = KΩ k C k E k E k 1 KΩ k D k = KΩ k D k. 12

13 Because the composite KΩ k C k E k KΩ k D k is known to be null, the map KΩ k C k KΩ k+1 D k KΩ k C k is homotopic to pojection. Poposition 4.2 {E k } is an dams towe fo Y. Poof. We aleady know that lim k E k lim k E k Y ˆ2. We must show that (1) Y E k gives an epimophism H E k H Y ; (2) ke(h E k H E k+1 ) = ke(h E k H Y ). Conside the commutative ladde of fundamental sequences at the end of the poof of Lemma 3.5. The map E k+1 E k induces a map N k N k+1 which is onto N. SinceY E k+1 induces an isomophism fom N N k+1 to N H Y = U(N), the desied epimophism H E k H Y follows. To pove theseconditem,wenotethaty E k factos as Y E k+1 E k+1 E k. Since ke(h E k H E k+1 ) = ke(h E k H Y ) the desied isomophism follows. Now we wish to calculate the homotopy spectal sequence of {E k}, whichis the USS fo Y.Thed 1 diffeential comes fom the composite KΩ k C k KΩ k+1 D k E k KΩ k C k+1 KΩ k+1 D k+1. The components π KΩ k C k π KΩ k C k+1 and π KΩ k+1 D k π KΩ k+1 D k+1 ae zeo because C and D ae minimal esolutions. Futhe, by Lemma 4.1 KΩ k+1 D k E k KΩk C k+1 is actually null because it factos though E k E k and KΩ k+1 D k E k E k is a fibation. Hence the only possible nonzeo pat of d 1 takes π KΩ k C k π KΩ k+1 D k+1. Remak 4.3 Note the contast between the d 1 diffeentials in the homotopy spectal sequences fo {E k } and {E k }. In the homotopy spectal sequence fo {E k }, d 1 comes fom the composite KΩ k C k KΩ k D k 1 E k KΩ k C k+1 KΩ k D k. The only possible nonzeo component of d 1 is π KΩ k D k 1 π KΩ k C k+1, coesponding to the pat of the chain complex diffeential of E taking C k+1 D k 1. Befoe giving the poof of Theoem 1.1, we need the following peliminay lemma. Lemma 4.4 Let x π KΩ k C k be a bounday in the homotopy spectal sequence of {E k }. Then x is in the kenel of d 1 of the homotopy spectal sequence of {E k}. 13

14 Poof. We use the diagam of Lemma 4.1. Let y be the image of x in π E k. If x is a bounday in the homotopy spectal sequence of {E k },thenx goes to0inπ E k,andsoy does also. The long exact sequence fo the fibation KΩ k+1 D k E k E k gives a peimage fo y, callitz, inπ KΩ k+1 D k. Howeve, d 1 (z) is cetainly zeo, and theefoe since x and z have the same image in π E k,wehaved 1(x) =0also. We ae now eady to pove ou Ext splitting esult, which we epoduce hee fo the convenience of the eade. Theoem 1.1 Let X be a simply connected space with polynomial mod 2 cohomology H X = U(M), and let X KP be Massey-Peteson, inducing f : P M in cohomology. Let Y be the fibe of X KP, and suppose H Y = U(N) is polynomial. If X has E 2 = E 3 in its unstable dams spectal sequence, then thee is a splitting Ext U (N,Σt F 2 ) = Ext U (cok(f), Σt F 2 ) Ext U (Ω ke(f), Σt F 2 ). Poof. In view of the emaks peceding the statement of the theoem in the intoduction, and the fact that ΩD Ωke(f) andc ke(f) ae esolutions, we have only to show that d 1 in the homotopy spectal sequence of {E k} is zeo. ssuming that C and D ae chosen to be minimal esolutions, this will follow fom showing that π KΩ k C k π KΩ k+1 D k+1 is zeo. We will elate this pat of d 1 to the USS fo X using the diagam of Lemma 4.1. To do this, we must discuss the elationship of the homotopy spectal sequences fo {E k } and {X k }, the latte being the USS fo X. We claim that they ae almost the same. To be moe pecise, note the map of towes {E k } {X k } is the identity on the k-invaiants except at the vey bottom, whee E 0 X 0 is the inclusion KC 0 KC 0 KP. Hence any nonzeo diffeential in {E k } gives ise to a coesponding nonzeo diffeential in {X k }, and any zeo diffeential in {E k } gives ise to a coesponding zeo diffeential in {X k }. In fact, thee ae only two diffeences between the homotopy spectal sequences fo {E k } and {X k }. The fist is that classes in the homotopy spectal sequence of {X k } which ae boundaies of π KP will suvive to E in the homotopy spectal sequence of {E k }. These coespond to the kenel of π Y π X. The second is that infinite cycles in the homotopy spectal sequence of {X k } which ae in π KP will not appea in the homotopy spectal sequence of {E k }. These coespond to the cokenel of π Y π X. To complete the poof of the theoem, we claim that the diagam of Lemma 4.1 shows that a nonzeo map π KΩ k C k π KΩ k+1 D k+1 would poduce a nonzeo d 2 in the homotopy spectal sequence of {E k }, and hence in the USS 14

15 of X. To pove this, we use the inteplay between the homotopy spectal sequences fo the towes {E k } and {E k }.Ifx π KΩ k C k has a nonzeo image in π KΩ k+1 D k+1,thenx is not in the kenel of d 1 fo the homotopy spectal sequence of {E k }. Hence by Lemma 4.4, we know that x is not a bounday in the homotopy spectal sequence fo {E k }. Howeve, this means that x suvives to E 2 in the homotopy spectal sequence fo {E k }, because all elements of π KΩ k C k ae in the kenel of d 1. On the othe hand, since E k+1 is the fibe of E k+1 KΩ k+1 D k+1, x does not lift even to E k+1, let alone to E k+2,and so x must suppot a d 2 diffeential. Theefoe if X has E 2 = E 3 in its unstable dams spectal sequence, that is, thee ae no nonzeo d 2 s, it must be that π KΩ k C k π KΩ k+1 D k+1 is zeo, completing the poof of the theoem. Lastly, we pove a theoem about d 2 in the USS fo Y. Theoem 4.5 Given the hypotheses of Theoem 1.1, let x π KΩ k D k 1 and let δ : π KΩ k D k 1 π KΩ k C k+1 be given by d 1 in the homotopy spectal sequence of {E k }. In the unstable dams spectal sequence of Y, d 2 (x) = δ(x) y whee y π KΩ k+1 D k+1.ifx ke(δ), then y =0. Remak 4.6 If we wite Hom t (, B) fo mophisms Σ t B in the categoy of unstable -modules, then δ is the only nonzeo pat of the diffeential in the cochain complex Hom (E, F 2 ), whee E is the chain complex constucted in Section 2. Hence this theoem says that d 2 s in the unstable dams spectal sequence fo Y can be detemined up to an eo tem y by the pimay level algebaic calculation of E. Futhemoe, if an element x looks like a cycle unde d 2 (δ(x) = 0), then it is a cycle (the eo tem vanishes). Poof of Theoem 4.5 gain we examine the diagam in Lemma 4.1, this time extending it downwad by a ow: KΩ k C k KΩ k+1 D k E k KΩ k C k+1 KΩ k+1 D k+1 (π, ) (π, ) KΩ k C k KΩ k D k 1 E k KΩ k C k+1 KΩ k D k (,π) (,π) KΩ k 1 C k 1 KΩ k D k 1 E k 1 KΩ k 1 C k KΩ k D k. The fist pat of the theoem follows fom a diagam chase stating in the lowe left hand cone. The second pat follows once we note that if x ke(δ), then x lifts to E k+1, and theefoe also to E k+2 (thee ae no nonzeo d 2 diffeentials 15

16 in the homotopy spectal sequence fo {E k }). Theefoe x must go to zeo in π KΩ k+1 D k+1 o thee would be an obstuction to lifting. Coollay 4.7 If E 2 = E fo X, then E 3 = E fo Y. Essentially, the coollay says that if E 2 = E fo X, then the USS fo Y is almost completely detemined by the pimay level calculation of the chain complex E. The only piece of infomation missing is the potentially nonzeo tem y of Theoem 4.5. Poof. If E 2 = E fo X, then once a class in homotopy lifts fom E k to E k+1, it lifts all the way up the towe to Y ˆ2. If a class lifts fom E k 1 to E k+1, it has lifted fom E k to E k+1 in the pocess, hence lifts all the way to Y ˆ2. 5 Examples In this section, we discuss two examples of Theoem 1.1 and Theoem 4.5. In the fist example, we show how the theoy woks itself out in a vey easy case, by consideing the unstable dams spectal sequence fo Y = K(Z/4,n). In the second example, which motivated looking fo a theoem along the lines of Theoem 1.1, we discuss a conjectue of Bousfield on the unstable dams spectal sequence fo the Lie goup SO, and we give a esult elated to the fist stage of the conjectue. 5.1 The USS fo K(Z/4,n) s an example whee Theoems 1.1 and 4.5 can be easily and completely woked out, we let X = K(Z/2,n), KP = K(Z/2,n+ 1), and f : X KP be Sq 1. Hence the space Y whose USS is being descibed by the theoems is Y = K(Z/4,n). Of couse, we aleady know eveything thee is to know about the USS in this case, so it is simply an execise in following the definitions of the pevious sections and seeing whee they lead, paticulaly egading the diffeentials. We wite F (n) fo the fee unstable -module on a geneato of dimension n, and we wite F (n) F (n)/sq 1.ThenM = F (n), P = F (n +1), cok(f) = F (n), and ke(f) = F (n + 2). The esolution C cok(f) hasc i = F (n + i), and the esolution D ke(f) hasd i = F (n + i + 2), while in both cases all the diffeentials ae Sq 1. 16

17 If we lift C 0 cok(f) tom, wegettheidentitymapf (n) F (n). Hence the chain map g : Shift(C ) B, becomes ɛ C 3 C 2 C 1 C 0 D 1 D 0 P Sq 1 Sq 1 ɛ M, F (n +3) F (n +2) F (n +1) F (n) = = = = F (n +3) Sq 1 F (n +2) Sq 1 F (n +1) ɛ ɛ F (n). The complex E, which we ealize to obtain Y, is the mapping cone, without augmentation and factoed out by P : D D D C 1 C 0. C 3 C 2 In ou example, the complex E becomes F (n +4) F (n +3) F (n +2) F (n +3) F (n +2) F (n +1) F (n), whee the slant maps ae the identities, since they come fom the vetical maps in g. To obtain the dams esolution fo Y, we pull D down one homological dimension (looping as we do so since we ae beginning in a lowe homological dimension and the topological dimension must emain the same): ΩD 3 ΩD 2 ΩD 1 ΩD 0 C 1 C 0. C 3 C 2 The dashed slant aows indicate that we ae emembeing whee d 1 used to go, because it will become d 2 in the USS. In ou example, we get ΩF (n +5) ΩF (n +4) ΩF (n +3) ΩF (n +2) F (n +3) F (n +2) F (n +1) F (n). 17

18 Finally, to get the E 2 -tem of the USS fo Y, we compute Hom (, F 2 ): Σ n+4 F 2 Σ n+3 F 2 Σ n+2 F 2 Σ n+1 F 2 Σ n+3 F 2 Σ n+2 F 2 Σ n+1 F 2 Σ n F 2. Rewiting this in the usual s vesus (t s) fomat, and including the d 2 s indicated by the slant aows, we obtain the USS fo Y = K(Z/4,n): 1 d K d K d K d K d d n n +1 The USS fo K(Z/4,n). 5.2 The USS fo SO Theoem 1.1 and Theoem 4.5 came up as pat of an effot to pove a conjectue of Bousfield about the E 2 -tem of the unstable dams spectal sequence fo the Lie goup SO. Let P = H RP, and ecall that H SO = U(P ). Hence in the USS fo SO, we have E s,t 2 = Ext s (P U, Σ t F 2 ). Witing the elements of P as x i,wefiltep by letting P n be the elements x i with at most n nonzeo digits in the dyadic expansion of i. Thee is an associated spectal sequence conveging to Ext s U (P, Σ t F 2 ), and Bousfield s conjectue is that this spectal sequence collapses, giving Ext s U (P, Σ t F 2 ) = n Ext s U (P n/p n 1, Σ t F 2 ). Fo an algebaic exploation of the conjectue and of the conjectued wokings of the USS of SO, see [1]. It is a suggestion of Mahowald that one ty to elate Bousfield s conjectue to the Postnikov towe fo SO, which consists of spaces with polynomial cohomology of the fom U( ) and k-invaiants which ae Massey-Peteson maps. In Figue 1 we display the beginning of the towe. Note that the composites k n i n un cyclically though the list Sq 3, Sq 5,Sq 2,Sq 2.IfweletX n be the nth space in the Postnikov towe, then im(h X n H SO) = U(P n ). That is, the filtation of H SO given by the Postnikov towe is the same as that 18

19 . K(Z/2, 8) K(Z, 7) K(Z, 3) K(Z/2, 1) i 4 k 4 X4 K(Z/2, 10) i 3 k 3 X3 K(Z/2, 9) i 2 k 2 X2 K(Z, 8) i 1 k 1 X1 K(Z, 4) Fig. 1. The Postnikov towe fo SO given by dyadic expansion. Hence the Postnikov towe gives some geometic significance to the dyadic filtation of P. The theoems in this pape ae a fist attempt to study the conjectue of Bousfield by compaing neighboing tems in the Postnikov towe of SO. It is an easy matte to split off the fist two filtations. Poposition 5.1 Ext s U (P, Σ t F 2 ) is isomophic to the diect sum Ext s U (P 1, Σ t F 2 ) Ext s U (P 2/P 1, Σ t F 2 ) Ext s U (P /P 2, Σ t F 2 ). Poof. Fist, we claim that the bounday map in the long exact sequence fo Ext aising fom the shot exact sequence 0 P 1 P P /P 1 0 is zeo. To pove this, it is sufficient to show that the map Ext s U (P, Σ t F 2 ) Ext s U (P 1, Σ t F 2 ) is an epimophism. Since P 1 = F (1), a fee module in the categoy of unstable -modules, we have Ext s U (P 1, Σ t F 2 )=0whens>0, so sujectivity is clea fo s>0. Fo s = 0, we need only note that P 1 P takes the geneato of P 1 to a geneato of P. To finish the poof of the poposition, we will show that the bounday map in 19

20 Ext associated to the shot exact sequence 0 P 2 /P 1 P /P 1 P /P 2 0 is zeo. gain, it is sufficient to show that Ext s U (P /P 1, Σ t F 2 ) Ext s U (P 2/P 1, Σ t F 2 ) is an epimophism. Since P 2 /P 1 = F (3), we know that Ext s (P U 2/P 1, Σ t F 2 )= 0 unless t s = 3, so we cetainly have an epimophism fo t s 3.Onthe othe hand, P /P 2 has its fist class in dimension 7, so Ext s U (P /P 2, Σ t F 2 )= 0fot s<7. Theefoe we have the equied epimophism fo t s =3 also. Because of the peceding poposition, to pove Bousfield s conjectue it is sufficient to pove a splitting fo Ext s U (P /P 2, Σ t F 2 ). Note that U(P /P 2 ) = H SO 7, whee SO 7 denotes the 6-connected cove of SO. The fist stage of a splitting of Ext s U (P /P 2, Σ t F 2 )wouldbe Ext s U (P 4/P 2, Σ t F 2 ) = Ext s U (P 4/P 3, Σ t F 2 ) Ext s U (P 3/P 2, Σ t F 2 ). We can use Theoems 1.1 and 4.5 to offe a step in this diection. The fist two stages of the Postnikov towe fo SO 7 ae K(Z/2, 8) SO[7, 8] K(Z, 7) Sq 2 K(Z/2, 9), and this is a situation to which we can apply Theoems 1.1 and 4.5. We note the following facts, which follow diectly fom [3], [6], and the Massey- Peteson Theoem: (1) Let f : F (9) F (7) be Sq 2. Then cok(f) = P 3 /P 2,andsoH SO[7, 8] = U(N) whee thee is a shot exact sequence (2) Thee is a shot exact sequence (3) Thee is a shot exact sequence 0 P 3 /P 2 N Ωke(f) 0. 0 Sq 2 ι 8 Ωke(f) P 4 /P Sq 2 ι 8 N P 4 /P

21 s an immediate coollay of these facts and Theoem 1.1, we have the following. Poposition 5.2 Ext s U (N,Σt F 2 ) = Ext s U (P 3/P 2, Σ t F 2 ) Ext s U (Ω ke(f), Σt F 2 ). It futhe follows fom the facts enumeated above that thee is a filtation with the popety that 0=N 0 N 1 N 2 N 3 = N N 1 /N 0 = P3 /P 2 N 2 /N 1 = Sq 2 ι 8 N 3 /N 2 = P4 /P 3. Thee is an associated spectal sequence conveging to Ext s U (N,Σt F 2 ), and the peceding poposition tells us that all elements of Ext s U (P 3 /P 2, Σ t F 2 )ae infinite cycles. nothe esult necessay fo the fist level of a splitting esult would be the following. Conjectue 5.3 d 1 :Ext s U ( Sq 2 ι 8, Σ t F 2 ) Ext s+1 U (P 4 /P 3, Σ t F 2 ) is zeo. Coollay to Conjectue 5.3 Ext s U (P 4/P 2, Σ t F 2 ) = Ext s U (P 4/P 3, Σ t F 2 ) Ext s U (P 3/P 2, Σ t F 2 ). Poof. Conjectue 5.3 would tell us that Ext s U (N,Σt F 2 ) = Ext s (P U 3/P 2, Σ t F 2 ) Ext s U ( Sq2 ι 8, Σ t F 2 ) Ext s U (P 4 /P 3, Σ t F 2 ). Howeve, it is also possible to filte N by 0=N 0 N 1 N 2 N 3 = N with the popety that N 1/N 0 = Sq 2 ι 8 N 2/N 1 = P 3 /P 2 N 3/N 2 = P 4 /P 3 21

22 and the spectal sequence fo Ext aising fom this filtation would also collapse. Theefoe the inclusion Sq 2 ι 8 N would give an epimophism and hence the shot exact sequence Ext s U (N,Σt F 2 ) Ext s U ( Sq2 ι 8, Σ t F 2 ), 0 Sq 2 ι 8 N P 4 /P 2 0 would split on Ext. The coollay would then follow. Even in the absence of Conjectue 5.3, howeve, we can apply Theoem 4.5 and Coollay 4.7 to get the following chat fo the USS of SO[7, 8]. The plain dots come fom the summand Ext s U (P 3 /P 2, Σ t F 2 ). The cicled dots come fom the summand Ext s U (Ω ke(f), Σt F 2 ). No cicled dot can suvive to be an element of homotopy, which is found only in dimensions 7 and 8, so since E 3 = E, all cicled dots suppot d 2 diffeentials. Futhe, the eo tem y of Theoem 4.5 is always zeo, because y must be a d 2 cycle and in this situation thee ae no such cycles, as we have just discussed. 10 d d d d 5,,,,, K d d,,,,,,, d% % K d d d K,,, K d d d d,,, d d d The USS fo SO[7, 8] d d d d cknowledgement I am gateful to Mak Mahowald fo many convesations about SO and to Don Davis fo intoducing me to Bousfield s conjectue. 22

23 Refeences [1]. K. Bousfield and D. M. Davis, On the unstable dams spectal sequence fo SO and U, and splittings of unstable Ext goups, Bol. Soc. Mat. Mex. 37 (1992) [2] J. R. Hape and H. R. Mille, Looping Massey-Peteson towes, in: S. M. Salamon, B. Stee, W.. Sutheland, eds., dvances in Homotopy Theoy (London Math. Soc. Lec. Notes 139, Cambidge Univesity Pess, Cambidge, 1989) [3] J. Long, Two contibutions to the homotopy theoy of H-spaces, Pinceton Univesity thesis (1979). [4] W. S. Massey and F. P. Peteson, The mod 2 cohomology stuctue of cetain fibe spaces, Mem. me. Math. Soc. Numbe 74, Povidence, [5] N. E. Steenod and D. B.. Epstein, Cohomology Opeations, (nn. Math. Studies Numbe 50, Pinceton Univesity Pess, Pinceton, NJ, 1962). [6]R.Stong,DeteminationofH (BO(k, )) and H (BU(k, )), Tans. me. Math. Soc. 107 (1963)