MULTIPLICATIVE STRUCTURES ON HOMOTOPY SPECTRAL SEQUENCES II DANIEL DUGGER Contents 1. Introduction 1 2. Sign conventions in singular cohomology 1 3. Spectral sequences for filtered spaces 3 4. The Postnikov/Whitehead spectral sequence 8 5. Bockstein spectral sequences 13 6. The homotopy-fixed-point spectral sequence 15 7. Spectral sequences from open coverings 16 References 19 1.Introduction This short paper is a companion to [D1 ]. Here the main results of that paper are used to establish multiplicative structures on a few standard spectral sequ* *ences. The applications consist of (a) applying [D1 , Theorem 6.1] to obtain a pairing* * of spectral sequences, and (b) identifying the pairing on the E1- or E2-term with something familiar, like a pairing of singular cohomology groups. Most of the arguments are straightforward, but there are subtleties that appear from time to time. Originally the aim was just to record a careful treatment of pairings on Post- nikov/Whitehead towers, but in the end other examples have been included because it made sense to do so. These examples are just the ones that I personally have needed to use at some point over the years, and so of course it is a very limit* *ed selection. In this paper all the notation and conventions of [D1 ] remain in force. In p* *articu- lar, the reader is referred to [D1 , Appendix C] for our standing assumptions a* *bout the category of spaces and spectra, and for basic results about signs for bound- ary maps. The symbol ^_ denotes the derived functor of ^, and W? denotes an `augmented tower' as in [D1 , Section 6]. Ho (-, -) denotes maps in the homotopy category of Spectra. If A is a pointed space we will write F(A, X) as an abbrev* *ia- tion for F( 1 A, X). Finally, the phrase `globally isomorphic' is often used in* * the identification of E2-terms of spectral sequences. It is explained in Remark 3.5. 2.Sign conventions in singular cohomology If E is a ring spectrum and X is a space, then the Atiyah-Hirzebruch spectral sequence Ep,q2= Hp(X; Eq) ) Ep+q(X) is multiplicative. The naive guess about 1 2 DANIEL DUGGER what this means is that there is an isomorphism of bi-graded rings p,qEp,q2(X) ~= p,qHp(X; Eq), where the products on the right-hand-side are the usual ones ~: Hp(X; Eq) Hs(X; Et) ! Hp+s(X; Eq+t) induced by the pairings Eq Et ! Eq+t. Unfortunately, this statement is just not true in general_one has to add an appropriate sign into the definition of ~, and these signs cannot be made to go away. In order to keep track of such signs in a simple way, it's useful to re-evaluate the `standard' conventions about si* *ngular cohomology. I'm grateful to Jim McClure for conversations about these sign issu* *es. Let X be a CW-complex and C*(X) be its associated cellular chain complex. In most algebraic topology textbooks the corresponding cellular cochain complex is defined by Cp(X) = Hom (Cp(X), Z) and (ffiff)(c) = ff(@c), for any ff 2 Cp(X). The cup-product of a p-cochain ff and a q-cochain fi is defined by the formula (ff [ fi)(c d) = ff(c) . fi(d) where c is a p-chain and d is a q-chain. (Note that we have written the above formula as if it were an external cup-product, so we technically need to throw * *in a diagonal map somewhere_we omit this to simplify the typography). Both of these formulas obviously violate the Koszul sign rule: we will abandon them and inste* *ad define (2.1) (ffiff)(c) = -(-1)pff(@c) and (ff [ fi)(c d) = (-1)qpff(c) . fi(* *d). The first equation may seem to have an unexpected minus sign, but here is the e* *x- planation. Recall that if A* and B* are chain complexes then there is an associ* *ated chain complex Hom (A, B). Our definition of ffi corresponds to the differential* * on the chain complex Hom (C*(X), Z[0]), where Z[0] is the complex with Z concentrated in dimension 0. The sign conventions from (2.1) appear in [Do ]. We'll of course use these sa* *me conventions for cohomology with coefficients, external cup products, and any si* *milar construction we encounter. Exercise 2.1. Check that ffi is a derivation with respect to the cup-product, a* *nd that the dga C*(X; Z) defined via our new formulas is isomorphic to the dga C*classical(X; Z) defined via the old formulas. In particular, our singular coh* *omology ring H*(X; Z) is isomorphic to the classical one. 2.2. Cohomology with graded coefficients. If A* is a graded ring we next want to define the singular cohomology ring with graded coefficients H*grd(X; A* *), making use of the natural sign conventions. It would be nice to just use the i* *n- ternal hom for chain complexes Hom (C*(X), A), where A is interpreted as having zero differential, but unfortunately this might give us infinite products in pl* *aces we don't really want them. Instead we'll consider a certain subcomplex. We set Cp,q(X; A) = Hom (Cp(X); Aq) and Cngrd(X; A) = p-q=nCp,q(X; A)_that is, ele- ments of Cp,q(X; A) are regarded as having total degree p - q. For ff 2 Cngrd(X* *; A) we define ffiff by the formula (2.2) (ffiff)(c) = -(-1)nff(@c). MULTIPLICATIVE STRUCTURES ON HOMOTOPY SPECTRAL SEQUENCES II 3 The homology of this complex will be denoted H*grd(X; A); it has a natural dire* *ct sum decomposition into groups Hp,q(X; A). The graded cup-product will be defined on the chain complex C*grd(X; A) as follows: if ff 2 Cp,q(X; A) and fi 2 Cs,t(X; A) then (ff [ fi)(c d) = (-1)(s-t)pff(c) . fi(d) (where c is a p-chain and d is a q-chain). The sign is again just the one dicta* *ted by the usual Koszul convention, and ffi becomes a derivation with respect to th* *is product. If C* D* ! E* is a pairing of graded abelian groups, one also has an extern* *al graded cup-product H*grd(X; C) H*grd(Y ; D) ! H*grd(XxY ; E) defined in a simil* *ar fashion. It is this product which arises naturally in pairings of spectral sequ* *ences. Exercise 2.3. Construct a bi-graded family of isomorphisms jp,q:Hp,q(Z; A) ! Hp(Z; Aq), natural in both Z and A, which makes the diagrams Hp,q(X; C) Hs,t(Y ;_D)__//Hp+s,q+t(X x Y ; E) j j|| j || fflffl| fflffl| Hp(X; Cq) Hs(Y ; Dt)____//_Hp+s(X x Y ; Eq+t) commute up to the sign (-1)sq. Here Hn(Z; Am ) denotes singular cohomology with coefficients in Am as defined via the formulas in (2.1), and the bottom ma* *p is the cup-product pairing associated to Cq Dt ! Eq+t (again with the signs from (2.1)). Exercise 2.4. Repeat the above exercise, but this time show that the isomorphis* *ms jp,qcan be chosen to make the squares commute up to the sign (-1)pt. Convince yourself that it is not possible to choose the jp,q's so that the squares commu* *te on the nose. 3.Spectral sequences for filtered spaces In this section we treat the Atiyah-Hirzebruch spectral sequence, the Serre s* *pec- tral sequence, and spectral sequences coming from geometric realizations. Some other references for the former are [K ], [GM , Appendix B], [V ]. For the Se* *rre spectral sequence see [K ], [Mc , Chap. 5], [Sp, Chap. 9.4], and [Wh , XIII.8]. 3.1. Generalities. Suppose given a sequence of cofibrations ; æ A0 æ A1 æ . . . and let A denote the colimit. If ; æ B0 æ B1 æ . .æ.B is another sequence of cofibrations, we may form the product sequence whose nth term is [ (A x B)n = (Aix Bj). i+j=n This is a sequence of cofibrations whose colimit is A x B. Given a fibrant spec* *trum E, one can look at the induced tower . .!.F(A2+, E) ! F(A1+, E) ! F(A0+, E) and identify the homotopy fibers as F (An=An-1, E). This is a lim-tower rather than a colim-tower, and is not convenient for seeing multiplicative structures;* * one doesn't have reasonable pairings F (Ak, E) ^ F(Bn, E) ! F ((A x B)k+n, E), for instance. Instead we have to use a slightly different tower. 4 DANIEL DUGGER The cofiber sequences An=An-1 ,! A=An-1 ! A=An induce rigid homotopy fiber sequences F (A=An, E) ! F (A=An-1, E) ! F (An=An-1, E). We define an augmented colim-tower by setting W(A, E)n = F (A=An-1, E) and B(A, E)n = F(An=An-1, E). The associated spectral sequence E*(A, E) might be called the E-spectral sequence for the filtered space A. Exercise 3.2. Verify that the tower W(A, E) is weakly equivalent to the tower F(A*, E) via a canonical zig-zag of towers. So the homotopy spectral sequences can be identified. Now assume that E had a multiplication E ^ E ! E. Then for any two pointed spaces X and Y we have the map F(X, E) ^ F(Y, E) ! F(X ^ Y, E ^ E) ! F(X ^ Y, E). Using this, the obvious maps of spaces (A x B)=(A x B)q+t-1! A=Aq-1 ^ B=Bt-1, and (A x B)q+t=(A x B)q+t-1! Aq=Aq-1 ^ Bt=Bt-1 give pairings W(A, E)^W(B, E) ! W(AxB, E) and B(A, E)^B(B, E) ! B(AxB, E) which are compatible with the maps in the towers. So [D1 , Thm 6.1] gives us a pairing of spectral sequences E*(A, E) E*(B, E) ! E*(A x B, E). This is the `formal' part of the construction. 3.3. The Atiyah-Hirzebruch spectral sequence. Now we specialize to where A and B are CW-complexes which are filtered by their skeleta. In this case we c* *an identify the E1- and E2-terms, and we will need to be very explicit about how we do this. For convenience we take A and B to be labelled CW-complexes, meaning that they come with a chosen indexing of their cells. Let Iq be the indexing se* *t for the q-cells in A, and let Cq(A) = oe2IqZ. Recall that Ep,q1= ßpF (Aq=Aq-1, E) ~=Ho(Sp ^ Aq=Aq-1, E). An element oe 2 Iq specifies a map Sq ! Aq=Aq-1 which we will also call oe. Giv* *en an element f 2 Ep,q1= Ho (Sp ^ Aq=Aq-1, E), restricting to each oe specifies an element in Ho(Sp ^ Sq, E). We therefore get a cochain in Hom (Cq(A), ßp+qE) = Cq,p+qgrd(A; E*), and we'll choose this assignment for our isomorphism Ep,q1~=Cq,p+qgrd(A; E*). Note that this isomorphism is completely natural with respect to maps of labell* *ed CW-complexes. We claim that the d1-differential corresponds under this isomorphism to the differential on C*grddefined in Section 2. By naturality (applied a couple of t* *imes) it suffices to check this when A is the CW-complex with Aq-1 = *, Aq = Sq, and Ak = Dq+1 for k q + 1. In this case our d1 is the boundary map in the long ex* *act homotopy sequence of F(Sq+1, E) ! F(Dq+1, E) ! F(Sq, E), which takes the form Ho(Sp^Sq, E) = Ho(Sp, F(Sq, E)) -@!Ho (Sp-1, F(Sq+1, E)) = Ho(Sp-1^Sq+1, E). We know from [D1 , C.6(d)] that the composite is (-1)p-1 times the canonical ma* *p. Via our identification with cochains, we are looking at a map Cq(A; ßp+qE) ! Cq+1(A; ßp+qE), and the sign (-1)p-1 is precisely the one for the coboundary ffi defined in (2.2). MULTIPLICATIVE STRUCTURES ON HOMOTOPY SPECTRAL SEQUENCES II 5 In a moment we will identify the pairing on E2-terms, but before that we make a brief remark on the case A = B. The diagonal map A ! A x A is homotopic to a map 0 which preserves the cellular filtration, and so 0 induces a map of towers W? (A x A, E) ! W? (A, E). Composing this with our above pairing gives W? (A, E) ^ W? (A, E) ! W? (A, E), and so we get a multiplicative structure on * *the spectral sequence E*(A, E). Theorem 3.4 (Multiplicativity of the Atiyah-Hirzebruch spectral sequence). There is a natural pairing of spectral sequences E*(A, E) E*(B, E) ! E*(A x B, E) together with natural isomorphisms p,qEp,q2(?, E) ~= p,qHq(?, E-p-q) (for ? = A, B, A x B) which make the diagrams Ep,q2(A, E) Es,t2(B,_E)____//_Ep+s,q+t2(A x B, E) | | | | fflffl| fflffl| Hq(A; E-p-q) Ht(B; E-s-t)___//Hq+t(A x B; E-p-q-s-t) commute, where the bottom map is the graded cup product from Section 2. In the diagonal case, there is a natural isomorphism of rings p,qEp,q2(A, E)* * ~= p,qHq(A; E-p-q), where the latter is again given the graded cup product. Remark 3.5. Rather than repeat the above statement for every multiplicative spectral sequence we come across, we'll just say that the E2-term is globally i* *so- morphic to the graded cup product (that is, they are naturally isomorphic as pairings of bigraded abelian groups). Proof.We have done everything except identify the product. The pairing on E1- terms is the map ßpF (Aq=Aq-1, E) ^ ßsF (Bt=Bt-1, E) ! ßp+sF (Aq=Aq-1 ^ Bt=Bt-1, E). Recall that this sends ff: Sp ^ Aq=Aq-1 ! E and fi :Ss ^ Bt=Bt-1 ! E to the composite fffi :Sp ^ Ss ^ Aq=Aq-1^ Bt=Bt-1! Sp ^ Aq=Aq-1^ Ss ^ Bt=Bt-1! E ^ E ! E. Choosing a q-cell oe of A yields a map Sq ! Aq=Aq-1, and a t-cell ` of B gives a map St ! Bt=Bt-1. Under our identification with cochains, the `value' of fffi on the cell oe ^ ` is the restriction of fffi to Sp ^ Ss ^ Sq ^ St. If, on the other hand, we compute ff(oe).fi(`) in the ring ß*E, we get the co* *mposite [Sp ^ Sq] ^ [Ss ^ St] ! [Sp ^ Aq=Aq-1] ^ [Ss ^ Bt=Bt-1] ! E ^ E ! E. By inspection, this differs from (fffi)(oe ^ `) by the sign (-1)sq, which is th* *e same sign that was used in defining the graded cup product from section 2.2 (remember that under our isomorphism ff lies in Cq,p+q(A; E*) and fi lies in Ct,s+t(B; E** *)). Remark 3.6. In the square from the statement of the theorem, the bottom map is (-1)t(p+q)times the `un-graded' cup product on cohomology induced by the pairing E-p-q E-s-t ! E-p-q-s-t. This follows from Exercise 2.3. The signs are easy to remember, because they follow the usual conventions: The index `t' is commuted across the index `-(p + q)', and as a result the sign (-1)-t(p+q)is picked up (* *the minus sign can of course be left off the exponent). Note that for most of the f* *amiliar cohomology theories, like K-theory or complex cobordism, the signs end up being irrelevant because the coefficient groups are concentrated in even dimensions. 6 DANIEL DUGGER 3.7. The Serre spectral sequence. Let p: X ! B be a fibration with fiber F , where B is a pointed, connected CW-complex. Let B0 B1 . . .be the skeletal filtration of B, and define Xi = p-1Bi. We'll assume that the inclusio* *ns Xi,! Xi+1are cofibrations between cofibrant objects, and consider the augmented tower Wn = F(X=Xn-1, HZ), Bn = F(Xn=Xn-1, HZ). The associated homotopy spectral sequence E*(X) is the Serre spectral sequence for the fibration. It is easy to see that there is a natural identification ` h i Xn=Xn-1 ~= p-1enff=p-1@(enff) , ff where the wedge ranges over the n-cells enffof B. The interior of a cell en is * *just the interior of Dn, so we can take a closed disk around the origin with radius * *1_2_ call this smaller disk U. Then ~H*(p-1en=p-1@(en)) may be canonically identified with ~H*(p-1U=p-1@U), and we are better off than before because @U is actually a sphere (rather than just the image of one). The diagram p-1(0)__~___//_p-1U fflffl __99______ ~| _~______||___ fflffl|______fflfflfflffl| U x p-1(0)_____////_U has a lifting as shown, and this lifting will be a weak equivalence. It restric* *ts to a weak equivalence @U x p-1(0) ! p-1(@U) (because this is a map of fibrations over @U, and it is a weak equivalence on all fibers). Therefore we have the diagram *oo_____p-1(@U) //_____//_p-1(U) || OO OO || |~ ~| || | | || | | *oo___@U x p-1(0) //__//U x p-1(0) and this necessarily induces a weak equivalence on the pushouts. In this way we get an identification ~Hk(p-1U=p-1(@U)) ~=~Hk([U=@U]^p-1(0)+ ) ~=~Hk(Sn^p-1(0)+ ) ~=Hk-n (p-1(0)). Of course the first isomorphism depended on the lifting ~, and so is not canoni* *cal. We refer to [Mc , Chap. 5] for a detailed discussion of local coefficient sys* *tems and their use in this particular context. But once the right definitions are in pla* *ce the argument we gave for the Atiyah-Hirzebruch spectral sequence in the last section adapts verbatim to naturally identify the (E1, d1)-complex as Ep,q1~=ßpF (Xq=Xq-1, HZ) ~=Cq(B; Hp+q(F )) where H*(F ) denotes the appropriate system of coefficients. The differential o* *n the cochain complex is still the one from section 2, appropriate for cellular cohom* *ology with graded coefficients. Now suppose X0! B0is another fibration satisfying the same basic assumptions as X ! B. We give B x B0 the product cellular filtration, and then pull it back to get a corresponding filtration of X x X0. This coincides with the product of the filtrations on X and X0, and so we get a pairing of spectral sequences by t* *he discussion in section 3.1. The identification of the pairing with the graded c* *up product again follows exactly as for the Atiyah-Hirzebruch spectral sequence. MULTIPLICATIVE STRUCTURES ON HOMOTOPY SPECTRAL SEQUENCES II 7 Theorem 3.8 (Multiplicativity of the Serre spectral sequence). There is a natur* *al pairing of Serre spectral sequences E*(X) E*(X0) ! E*(X x X0) such that the pairing of E2-terms is globally isomorphic to the graded cup product on singular cohomology with local coefficients. 3.9. Spectral sequences for simplicial objects. Filtered spaces also arise in t* *he context of geometric realizations. Let X* be a Reedy cofibrant simplicial space* *, in which case the skeletal filtration of the realization |X| is a sequence of cofi* *brations. There is a resulting tower of rigid homotopy cofiber sequences with Wq(X*, E) = F(|X|= skq-1|X|, E) and Bq(X*, E) = F(skq|X|= skq-1|X|, E). If Y* is another Reedy cofibrant simplicial space, we can equip |X| x |Y | wi* *th the product filtration. We also have the product simplicial space X*xY*, equipp* *ed with its skeletal filtration. There is a natural map j :|X x Y | ! |X| x |Y |, * *and this is actually an isomorphism (using adjointness arguments one reduces to the case where X and Y are the simplicial sets m and n, where it is (T2) from [D1* * , Appendix C]). Unfortunately j does not preserve the filtrations, as can be seen* * by taking X and Y both to be the simplicial set 1 (regarded as a discrete simplic* *ial space). The product filtration on | 1| x | 1| is smaller than the skeletal filt* *ration coming from | 1 x 1|. The formal machinery of section 3.1 gives a pairing from E*(|X|, E) and E*(|Y |, E) to the spectral sequence for the product filtration on |X|x|Y |_let* *'s call this E*(|X|x|Y |, E). Often one would like to have a pairing into E*(|X xY |, E* *), but this doesn't seem to follow from our basic results. Here are two ways around th* *is. One can replace E*(|X|, E) with the homotopy spectral sequence for the cosimpli- cial spectrum [n] 7! F(Xn, E), and similarly for E*(|Y |, E) and E*(|X xY |, E)* *. The paper [BK ] proves that if M* ^ N* ! Q* is a (level-wise) pairing of cosimplici* *al spaces, then there is an associated pairing of spectral sequences_this gives us* * what we wanted. Having not checked the details in [BK ], I can say nothing more about this approach; their results clearly depend on more than the formal theorems of [D1 ], but I couldn't tell from their paper exactly what the important ingredie* *nt is. Here is another approach which sometimes works. While j :|X xY | ! |X|x|Y | does not preserve the filtrations, j-1 is filtration-preserving (by functoriali* *ty and adjointness arguments it suffices to check this when X and Y are the simplicial sets n and m ). So j-1 induces a map of spectral sequences E*(|X x Y |, E) ! E*(|X| x |Y |, E). We have the following: Proposition 3.10. If X* and Y* are simplicial sets, the natural map of spectral sequences E*(|X x Y |, E) ! E*(|X| x |Y |, E) is an isomorphism on E2-terms. Proof.This follows from the work in section 3.3, since it identifies both E1-te* *rms as cellular chain complexes computing H*(|X x Y |, E*), but for different CW- decompositions. It follows that when X* and Y* are simplicial sets we get our desired pairing E*(|X|, E) E*(|Y |, E) ! E*(|X x Y |, E) from the E2-terms onward. This obser- vation will be used in section 7. Exercise 3.11. Is Proposition 3.10 true for simplicial spaces? I haven't worked out the answer to this. 8 DANIEL DUGGER 4.The Postnikov/Whitehead spectral sequence For each spectrum E and each n 2 Z, let PnE denote the nth Postnikov section of E; this is a spectrum obtained from E by attaching cells to kill off all hom* *otopy groups from dimension n + 1 and up. The construction can be set up so that if E is fibrant then all the PnE are also fibrant, and there are natural maps E ! PnE and PnE ! Pn-1E making the obvious triangle commute. So we have a tower of fibrant spectra . .!.P2E ! P1E ! P0E ! . . . and the homotopy cofiber of Pn+1E ! PnE is an Eilenberg-MacLane spectrum of type n+2H(ßn+1E). If A is a cofibrant, pointed space, we can map 1 A into this tower and there* *by get a tower of function spectra . .!.F(A, P2E) ! F(A, P1E) ! F(A, P0E) ! . . . The homotopy cofiber of F (A, Pn+1E) ! F (A, PnE) is weakly equivalent to F_(A, n+2H(ßn+1E)), and the resulting homotopy spectral sequence has Ep,q1~=ßpF_(A, q+2H(ßq+1E)) ~=Hq-p+2(A; ßq+1E). The spectral sequence abuts to ßp-1F_(A, E) = E~1-p(A). This turns out to be another construction of the Atiyah-Hirzebruch spectral sequence_see [GM , Ap- pendix] for some information about how the two spectral sequences are related. Assume that E, F , and G are fibrant spectra, and that there is a pairing E ^ F ! G. There do not exist reasonable pairings PnE ^ Pm F ! Pn+m G, and so Postnikov towers are not convenient for seeing multiplicative structures on spe* *ctral sequences. This is related to the Postnikov tower being a lim-tower rather than* * a colim-tower. Instead we will use the `reverse' of the Postnikov tower, sometim* *es called the Whitehead tower. If WnE denotes the homotopy fiber of E ! Pn-1E, then there are natural maps WnE ! Wn-1E and so we get a new tower. The homotopy cofiber of Wn+1E ! WnE is weakly equivalent to nH(ßnE). We will modify these towers in an attempt to produce a pairing W*E ^ W*F ! W*G. To explain the idea, let's forget about cofibrancy/fibrancy issues for just a* * mo- ment. Consider the following maps: Wm E ^_WnF _____//E ^ F ___ | ~ ______ | fflffl___p fflffl|j Wm+n G ________//_G_____//_Pm+n-1 G. The horizontal row is a homotopy fiber sequence. The spectrum Wm E ^ WnF is (m + n - 1)-connected, and so the composite Wm E ^ WnF ! Pm+n-1 G is null- homotopic. Choosing a null-homotopy lets us construct a lifting ~: Wm E ^WnF ! Wm+n G. If we had two different liftings ~ and ~0, their difference would lift * *to a map Wm E ^ WnF ! Pm+n-1 G and so would be null-homotopic (again, because the domain is (m + n - 1)-connected). So the lift ~ is unique up to homotopy. The situation, then, is that we can produce pairings Wm E ^ WnF ! Wm+n G, but so far they don't necessarily commute with the structure maps in the towers. They certainly commute up to homotopy_this follows from the `uniqueness' con- siderations in the above paragraph_ but we need them to commute on the nose. MULTIPLICATIVE STRUCTURES ON HOMOTOPY SPECTRAL SEQUENCES II 9 By using obstruction theory we will be able to alter these maps so that the rel* *evant diagrams do indeed commute. The argument proceeds in a few steps. Lemma 4.1. For each fibrant spectrum E there is a natural tower of rigid homoto* *py cofiber sequences (fW*E, eC*E) such that every fWnE and eCnE is cofibrant-fibra* *nt, together with a natural zig-zag of weak equivalences from fW*E to W*E. Proof.First take W*E and apply a cofibrant-replacement functor Q to all the lev* *els: this produces QW*E, a tower of cofibrant spectra. Then perform the telescope construction from [D1 , B.4] to get a tower of cofibrations between cofibrant o* *bjects T W*E and a weak equivalence T W*E ! QW*E. Let CnE denote the cofiber of T Wn+1E ! T WnE. Finally, let F be the fibrant-replacement functor for Spectra such that F (*) = * given in [D1 , C.3(c)]. Applying F to the rigid tower (T W * *E, CE) gives a new rigid tower which has the desired properties. At this point we have towers where everything is cofibrant-fibrant, so the ar- gument we have already explained will construct maps fWmE ^ fWnF ! fWm+nG which commute up to homotopy with the maps in the towers. By considering the diagram fWmE ^ fWnF ____//_fWm+nG | | | | fflffl| fflffl| eCmE ^ eCnF eCm+nG one can see that there is a unique homotopy class eCmE ^ eCnF ! eCm+nG which makes the square commute. This is because Cem+nG is an Eilenberg-MacLane spectrum of type n+m H(ßn+m G), and fWmE ^ fWnF ! CemE ^ eCnF induces an isomorphism on the corresponding cohomology group (since both the domain and codomain are (m + n - 1)-connected). So at this point we have produced a homotopy-pairing (fWE, eCE) ^_(fWF, eCF ) ! (fWG, eCG) (see [D1 , 6.3]). We wi* *ll prove: Proposition 4.2. The homotopy-pairing (fWE, eCE) ^_(fWF, eCF ) ! (fWG, eCG) is locally realizable. The following lemma encapsulates the basic facts we will need. The proof will be left to the reader. Lemma 4.3 (Obstruction theory). Suppose that X ! Y is a fibration of spectra which induces isomorphisms on ßk for k n. Let A æ B be a cofibration which induces isomorphism on ßk for k < n. Then any diagram Af_____//flfflX>>__ ___|____ | _____|_ fflffl|fflfflfflffl|___ B _____//Y has a lifting as shown. Proof of Proposition 4.2.First we truncate the towers, and we might as well ass* *ume we are dealing with truncations ø0 k(fWE, eCE) and ø0 l(fWF, eCF ) because the argument will be the same no matter what the lower bounds are. For the rest of 10 DANIEL DUGGER the argument we will only be dealing with these finite towers, and will omit th* *e ø's from the notation. We replace (fWE, eCE) and (fWF, eCF ) by the equivalent towers (T W E, CE) and (T W F, CF ) constructed in the proof of Lemma 4.1, because these consist of cofibrations between cofibrant spectra. It is easy to see that one can also fin* *d a tower cW*G consisting of fibrant spectra and fibrations, together with a weak equival* *ence fW*G ! cW*G (remember that all our towers are finite). We will construct a pairing of towers T W*E ^ T W*F ! cW*G which realizes the homotopy-pairing. For T W0E ^ T W0F ! cW0G we choose any map in the correct homotopy class. Next consider the diagram cW1G | | fflffl| T W0E ^ T W1F _____//T W0E ^ T W0F___//_cW0G. The vertical map is a fibration inducing isomorphisms on ß1 and higher, and the spectrum T W0E ^ T W1F is 0-connected; so there is a lifting ~(0,1). Next, look* * at the diagram T W1E ^ T W1F _____//T W0E ^ T W1F____//cW1G | | | | fflffl| fflffl| T W1E ^ T W0F _____//T W0E ^ T W0F___//_cW0G. This diagram commutes (because the missing vertical arrow in the middle may be filled in). The right vertical map is a fibration which induces isomorphisms on* * ß1 and higher. The left vertical map induces isomorphisms on ß0 and lower (because both domain and range are 0-connected), and is a cofibration. So there is a lif* *ting ~(1,0):T W1E ^ T W0F ! cW1G. This process may be continued to inductively define ~(0,2), ~(1,1), and ~(2,0* *), and then onward from level three. In this way, we construct the required pairing of towers. This pairing agrees with the original homotopy-pairing because of the `uniqueness' of the liftings ~ in our original discussion; the details are left* * for the reader. At this point we have a pairing of towers, but we need a pairing of augmented rigid towers. We can't just take~cofibers in cW*G because the maps are fibratio* *ns, not cofibrations. So let Q(cW*G) -i cW*G be the cofibrant approximation guaranteed in [D1 , Lemma B.2]. We have a diagram Q(cW*G) |~| fflfflfflffl| T W E ^ T W F______//cW*G and by [D1 , Lemma B.3] the lower left corner is a cofibrant tower; so there is a lifting. The tower Q(cWG) consists of cofibrations, and so augmenting by the cofibers gives a rigid tower. The new pairing automatically passes to cofibers* * to give (T W E, CE) ^ (T W F, CF ) ! (Q(cWG), CQ(cWG)). MULTIPLICATIVE STRUCTURES ON HOMOTOPY SPECTRAL SEQUENCES II 11 Finally, consider Q(cW*G) |~| fflfflfflffl| fW*G __~__//_cW*G. The tower fW*G was cofibrant, so there is a lifting. This will be a levelwise e* *quiv- alence, and therefore induces an equivalence on the cofibers. So we get an equi* *v- alence of augmented towers (fW*G, eC*G) ! (Q(cWG), CQ(cWG)). Thus, we have constructed the required realization of our homotopy-pairing. For each cofibrant space X consider the tower of function spectra F(X+ , fWE?* * ) (recall that the fW*E are all fibrant). This is a tower of rigid homotopy cofib* *er se- quences, and we will call the associated spectral sequence E*(X, E) the Whitehe* *ad spectral sequence for X based on E. The homotopy-pairing fWE? ^_fWF? ! fWG? induces a homotopy-pairing on towers of function spectra, and by [D1 , Pro* *p. 6.10] this is locally-realizable and so induces a pairing of spectral sequences* *: for any cofibrant spaces X and Y we have E*(X, E) E*(Y, F ) ! E*(X x Y, G). What is left is to identify the pairing on E1-terms with the pairing on singular cohomo* *logy (up to the correct sign). If X is a spectrum with a single non-vanishing homotopy group in dimension m, there is a unique isomorphism in the homotopy category Sm ^_H(ßm X) ! X with the property that the composite ßm X ! ß0H(ßm X) -oel!ßm (Sm ^ H(ßm X)) ! ßm X is the identity map (the first map in the composite is the one provided by [D1 , Section C.7]). If X ^ Y ! Z is a pairing of spectra where X, Y , and Z each have a single non-vanishing homotopy group in dimensions m, n and m + n, then the diagram in Ho(Spectra) Sm ^_H(ßm X) ^_Sn ^ H(ßnY )___//_Sm+n ^_H(ßm+n Z) | | | | fflffl| fflffl| X ^ Y ______________________//Z is commutative. Here the top map interchanges the Sn and the H(ßm X) and then uses the map H(ßm X) H(ßnY ) ! H(ßm+n Z) induced by the pairing ßm X ßnY ! ßm+n Z (cf. [D1 , Section C.7]). The above observations are simple calculations in the homotopy category of spectra. In our situation we have specific isomorphisms ßm eCmE ~=ßm E, and the same for F and G. This is because fWmE ! CemE induces an isomorphism on ßm , fWmE is connected by a chosen zig-zag of weak equivalences to Wm E, and the map Wm E ! E induces an isomorphism on ßm as well. The pairing eCmE ^ eCnF ! eCm+nG induces a pairing on homotopy groups which corresponds to the expected pairing ßm E ßnF ! ßm+n G under these isomorphisms. Putting all the above statements together, we have proven: 12 DANIEL DUGGER Lemma 4.4. In Ho (Spectra) there exist isomorphisms Sn ^ H(ßnE) ! CenE, Sn ^ H(ßnF ) ! CenF , and Sn ^ H(ßnG) ! CenG for all n 2 Z, such that the diagrams Sm ^ H(ßm E) ^ Sn ^ H(ßnF )___//_Sm+n ^ H(ßm+n G) | | | | fflffl| fflffl| CemE ^ eCnF_________________//eCm+nG all commute (in the homotopy category). The lemma tells us that if A and B are spectra the pairing ~ ~ ~ ~ p,qHo(Sp^_A, eCqE) s,tHo(Ss^_B, eCtF ) ! u,vHo(Su ^_A ^_B, eCvG) is globally isomorphic to the pairing obtained from the maps Ho(Sp^_A, Sq^_H(ßqE)) Ho(Ss^_B, St^_H(ßtF )) | | fflffl| Ho (Sp+s^_A ^_B, Sq+t^_H(ßq+tG)). Now, the left-suspension map gives an isomorphism Ho(Sp^_B, Sq^_H(ßqE)) ~=Ho(Sp-q^_B, H(ßqE)) = Hq-p(B; ßqE), and similarly for the F and G terms. This allows us to rewrite the above pairin* *g as a pairing of singular cohomology groups, but the suspension maps introduce sign* *s. For any spectra M and N, the diagram oeql^oetl Ho (Sa^_A, M) Ho(Sb^_B, N)___//Ho(Sq+a^_A, Sq^_M) Ho(St+b^_B, St^_N) | | | | fflffl| oeq+tl fflffl| Ho (Sa+b^_A ^ B, M ^_N)________//Ho(Sq+t+a+b^ A ^ B, Sq+t^ M ^ N) commutes up to the sign (-1)ta(one compares the string `qatb' to the string `qt* *ab' and sees that the t and a must be commuted). Taking A = 1 X+ and B = 1 Y+ , we now conclude: Theorem 4.5 (Multiplicativity of the Postnikov/Whitehead spectral sequence). For cofibrant spaces X and Y there is a pairing of Whitehead spectral sequences* * in which the E1-term Ep,q1 Es,t1! Ep+s,q+t1is globally isomorphic to the pairing Hq-p(X; ßqE) Ht-s(Y ; ßtF ) ! Hq+t-p-s(X x Y ; ßq+tG) up to a sign of (-1)t(p-q). Remark 4.6. At first glance the sign given here doesn't agree with the sign we obtained in Theorem 3.4: if we were to re-index the Atiyah-Hirzebruch spectral sequence in the above form, the sign would be (-1)q(t-s). While this is not the same as the above sign, the two are consistent. Using the family of isomorphisms (-1)pq:Hq-p(X; ßqE) ~=Hq-p(X; ßqE) and similarly for the F and G terms, the sign (-1)t(p-q)transforms into (-1)q(t-s). See Exercise 2.4. MULTIPLICATIVE STRUCTURES ON HOMOTOPY SPECTRAL SEQUENCES II 13 5. Bockstein spectral sequences In this section we consider two spectral sequences: one is the classical Bock* *stein spectral sequence for the homotopy cofiber sequence HZ xn-!HZ ! HZ=n. The other is the Bockstein spectral sequence for inverting the Bott element in conn* *ective K-theory. 5.1. The Bockstein spectral sequence for HZ. Consider the following tower (W*, B*)* 0: HZ=nOO HZ=nOO HZ=nOO | | | | | | | | n | . ._._n__//HZ___n__//_HZ______//_HZ. We extend this to negative degrees by taking Wq = HZ, Bq = *, and Wq+1 ! Wq to be the identity map. This is a tower of rigid homotopy cofiber sequences, and there is an obvious pairing (W, B) ^ (W, B) ! (W, B) which comes from the multiplications on HZ and HZ=n (cf. [D1 , Appendix C.7]). For any cofibrant space X, let W? X denote the augmented tower whose levels are F (X+ , Wn+1) ! F (X+ , Wn) ! F (X+ , Bn); these are rigid homotopy cofiber sequences, since HZ and HZ=n are fibrant. The homotopy spectral sequence for W? X is called the mod n Bockstein spectral sequence, and has the form Ep,q1= H-p (X; Z=n) ) H-p (X; Z). The d1-differential is the usual Bockstein homomorphism. The multiplication on (W, B) gives rise to pairings of towers W? X ^W? Y ! W? (X xY ), and therefore * *to pairings of spectral sequences by [D1 , Thm 6.1]. The following result is immed* *iate, and unlike the examples in sections 3 and 4 there are no extra signs floating a* *round. Theorem 5.2 (Multiplicativity of the Bockstein spectral sequence). For cofibrant spaces X and Y there is a pairing of Bockstein spectral sequences whose E1-term* * is isomorphic to the usual pairing H-p (X; Z=n) H-s(Y ; Z=n) ! H-p-s(XxY ; Z=n) of singular cohomology groups. The spectral sequence converges to the usual pai* *ring on H*(-; Z). 5.3. The Bockstein spectral sequence for bu. Let bu denote a commutative ring spectrum representing connective K-theory, and assume we have a map of ring spectra bu ! HZ. Assume there is a map S2 ! bu which represents a generator in Ho(S2, bu) ~=Z (this is automatic if bu is a fibrant spectrum). Consider the in* *duced map fi :S2 ^ bu fi^id-!bu ^ bu -~!bu. It can be shown that S2^ bu ! bu ! HZ is a homotopy cofiber sequence. If we let (W, B) be the tower S4 ^OHZO S2 ^OHZO HZOO S-2 ^OHZO | | | | | | | | 4^fi | S2^fi | fi | S-2^fi | S-4^fi . ._.S___//_S4 ^_bu____//S2 ^ bu_____//_bu_____//_S-2 ^ bu____//_//_. . . 14 DANIEL DUGGER then one sees that there is a pairing (W, B) ^ (W, B) ! (W, B) (this uses that * *bu is commutative). Unfortunately we are not yet in a position to apply [D1 , Thm 6.1* *]: (W, B) is not a rigid tower, because we don't know that S2^ bu ! bu ! HZ is null rather than just null-homotopic. We don't get a long exact sequence on homotopy groups until we choose null-homotopies for each layer, and these must be accoun* *ted for. In this particular case any two null-homotopies are themselves homotopic, * *and so there should be no problems with compatibility, but it's awkward to formulate results along these lines. The best way I know to proceed is actually to discar* *d the HZ's and consider just the tower W* consisting of suspensions of bu. We are then in a position to apply [D1 , Thm 6.2], but we must work a little harder to iden* *tify what's happening on the E1-terms. First, one can replace bu by a cofibrant commutative ring spectrum, and this will be cofibrant as a symmetric spectrum [SS, Thm. 4.1(3)]. Because of this, we will just assume that our bu was cofibrant in the first place. By [D1 , B.4] t* *here is an augmented tower T W* consisting of cofibrations between cofibrant spectra, a weak equivalence T W* ! W*, and a pairing T W ^ T W ! T W which lifts the pairing W ^ W ! W . We let Cn denote the cofiber of T Wn+1 ! T Wn and note that our pairing T W ^ T W ! T W extends to (T W, C) ^ (T W, C) ! (T W, C). For any cofibrant space X, we consider the derived tower of function spec- tra F der(X+ , T W? ) from [D1 , B.7]. By [D1 , Theorems 6.1,6.10] there is an* * in- duced pairing of spectral sequences. Note that we have Ep,q1= ßpF_(X+ , Cq) ~= ßpF (X, 2qHZ) ~=H2q-p(X; Z). As usual, what we want is to identify the pairing on E1-terms with a pairing on singular cohomology. The following does this: Lemma 5.4. In Ho(Spectra) it is possible to choose a collection of isomorphisms Cn ! S2n^_HZ such that the following diagrams commute: Cm ^_Cn _____________________________________//_Cm+n | | | | fflffl| 1 ^_t ^_1 1 ^_~ fflffl| S2m ^_HZ ^_S2n^_HZ_____//S2m ^_S2n^_HZ ^_HZ____//S2(m+n)^_HZ. Proof.This is very similar to the proof of Lemma 4.4, making use of the fact th* *at the graded ring nHo (S2n, Cn) is isomorphic to the ring of Laurent polynomials Z[t, t-1]. Details are left to the reader. The spectral sequence converges to fi. fi. colimß*F_(X+ , bu) -! ß*F_(X+ , bu) -! . . . which is fi-1[ß*F_(X+ , bu)] = fi-1bu*(X). If we write fi-1bup(X) for the pth g* *raded piece of fi-1(bu*X) then we can state the final result as follows: Theorem 5.5. For any space X there is a conditionally convergent spectral se- quence Ep,q1= H2q-p(X; Z) ) fi-1bu-p(X). If X and Y are two cofibrant spaces there is a pairing of spectral sequences wh* *ose E1-term is globally isomorphic to the usual pairing on singular cohomology, and which converges to the pairing fi-1bu*(X) fi-1bu*(Y ) ! fi-1bu*(X x Y ). The above spectral sequence of course has the same form (up to re-indexing) as the Atiyah-Hirzebruch spectral sequence for complex K-theory. One finds in the end that fi-1bu*(X) ~=K*(X). MULTIPLICATIVE STRUCTURES ON HOMOTOPY SPECTRAL SEQUENCES II 15 6.The homotopy-fixed-point spectral sequence Suppose E is a fibrant spectrum with an action of a discrete group G. The homotopy fixed set EhG is defined to be F(EG+ , E)G . Theorem 6.1. If E is a fibrant ring spectrum and G is a discrete group acting via ring automorphisms, then there is a multiplicative spectral sequence of the* * form Ep,q2= Hq(G; ßp+qE) ) ßp(EhG). Here the pairing on E2-terms is the pairing on group cohomology with graded coefficients, defined analagously to what was done* * in section 2. Remark 6.2. The above means that the pairing on E2-terms Hq(G; ßp+qE) Ht(G; ßs+tE) ! Hq+t(G; ßp+q+s+tE) is (-1)t(p+q)times the `standard' pairing on group cohomology induced by ß*E ß*E ! ß*E. Proof.Take any model for EG which is an equivariant G-CW-complex (with G acting freely on the set of cells in each dimension). Write EGk for skkEG, and let Wn = F(EG=EGn-1, E), Bn = F(EGn=EGn-1, E). The spectral sequence for this tower is just the Atiyah-Hirzebruch spectral sequence based on EG and E, from section 3.3. The sequences Wn+1 ! Wn ! Bn are actually G-equivariant homotopy fiber sequences, which implies that they give homotopy fiber sequences on H-fixed sets for any H. We'll consider the associated tower (WG*, BG*). The E1-term of the spectral sequence is Ep,q1= ßp[F (EGq=EGq-1, E)G ]. The map ßp[F (EGq=EGq-1, E)G ] ! ßpF (EGq=EGq-1, E) = Ho (Sp ^ [EGq=EGq-1], E) has its image in the G-fixed set of the right-hand-side, and one can check that* * this gives an isomorphism Ep,q1~=Ho (Sp ^ [EGq=EGq-1], E)G . Since EGq=EGq-1 is a wedge of q-spheres indexed by the set of q-cells, with G action induced by that on the indexing set, one obtains a natural isomorphism with the group Hom G(C*(EG), ßp+qE). The identification works the same as in section 3.3, and the description of the differential carries over as well. In fact the map of t* *owers (WG*, BG*) ! (W*, B*) lets us compare the differential with the one on the Atiy* *ah- Hirzebruch spectral sequence, which we have already analyzed. So the identifica* *tion of the E2-term follows. The pairing of augmented towers W(EG, E) ^ W(EG, E) ! W(EG x EG, E) from section 3 obviously restricts to fixed sets, giving W(EG, E)G ^ W(EG, E)G ! W(EG x EG, E)G . This uses that E ^ E ! E is G-equivariant. If one accepts that the diagonal map EG ! EGxEG is homotopic to a map 0which is both cellular and G-equivariant, then we get an equivariant map of towers W(EG x EG, E) ! W(EG, E) and can restrict to fixed sets. The existence of 0 follows from the G- equivariant cellular approximation theorem. Therefore we have an induced pairing of augmented towers WG?^ WG? ! WG?. The identification of the product on E2-terms again can be done by comparing with the Atiyah-Hirzebruch spectral sequence for the tower (Wn, Bn). Everything follows exactly as in section 3.3. Remark 6.3. Instead of filtering EG by skeleta, the spectral sequence can also be constructed via a Postnikov tower on E, just as in section 4 (by functoriali* *ty, the Postnikov sections of E inherit G-actions). Instead of needing a G-equivari* *ant cellular approximation for the diagonal map, one instead needs to carry out G- equivariant obstruction theory. 16 DANIEL DUGGER 7.Spectral sequences from open coverings In this section we give a second treatment of the Atiyah-Hirzebruch spectral * *se- quence, together with a similar approach to the Leray-Serre spectral sequence. * *Our towers are obtained by using open coverings and their generalization to hyperco* *vers. We will assume a basic knowledge of hypercovers, for which the reader can consu* *lt [DI]. Basically, one starts with an open cover {Ua} of a space X and then choos* *es another open cover for every double intersection Ua \ Ub; for every resulting `* *triple intersection' another covering is chosen, and so on. All of this data is compil* *ed into a simplicial space U*, called a hypercover. The discussion in this section is much sloppier than the previous ones, and s* *hould probably be improved at some point... 7.1. The descent spectral sequence. Given a hypercover U* of a space X and a sheaf of abelian groups F on X, we let F (U*) denote the cochain complex one gets by applying F to the open sets in U*. If all the pieces of the hypercover * *Un are such that Hkshf(Un, F ) = 0 for k > 0, then Verdier's Hypercovering Theorem giv* *es an isomorphism Hp(F (U*)) ! Hpshf(X, F ) which is functorial for X, U, and F . * *It is easy to explain how to get this. First, choose a functorial, injective resol* *ution I* for F , and look at the double complex of sections Dmn = In(Um ). This double complex has two edge homomorphisms I*(X) ! TotD** and F (U*) ! TotD**; the two spectral sequences for the homology of a double complex immediately show that these maps give isomorphisms on homology. Composing them appropriately gives our natural isomorphism Hp(F (U*)) ! Hp(I*(X)) = Hpshf(X, F ). Given a fibrant spectrum E and a hypercover U* of a X, we use the simplicial space U* to set up a tower as in section 3.9. We let Wn = F(|U|= skn-1|U|, E), * *Bn = F(skn|U|=skn-1|U|, E), and denote the resulting spectral sequence by E*(U, E). * *It is a theorem of [DI] that the natural map |U*| ! X is a weak equivalence, and so the spectral sequence converges to E*(X). We'll call this the descent spectral sequence based on U and E. Note that the spectral sequence is functorial in several ways. It is clearly * *func- torial in E, and if V* ! U* is a map of hypercovers of X then there is a natural map E*(U, E) ! E*(V, E). Also, if f :Y ! X is a map then there is a pullback hypercover f-1 U* ! Y , and a map of spectral sequences E*(U, E) ! E*(f-1 U, E). So far we have said nothing about the E2-term. A space X is locally con- tractible if given any point x and any open neighborhood x 2 V , there is an open neighborhood x 2 W V such that W is contractible. Given such a space X, one can build a hypercover U* of X in which every level is a disjoint union * *of contractible opens. We'll call U* a contractible hypercover. If U* ! X is a hypercover there is a natural isomorphism Ep,q2(U, E) ~= Hq(E-p-q(U*)). If we assume X is locally contractible and U* is a contractible hypercover, we can simplify this further. Let ~Eqdenote the sheafification of * *the presheaf V 7! Eq(V ), and note that this is a locally constant sheaf on X. It follows that Eq(V ) ! ~Eq(V ) is an isomorphism for every contractible V . In p* *ar- ticular, Hq(Ep-q(U*)) ! Hq(~Ep-q(U*)) is an isomorphism. But since the sheaves ~E*are locally constant, it follows that if V is a contractible open set in X t* *hen Hpshf(V, ~Eq) = 0 for p > 0 (see [Br, II.11.12]). The fact that U* is a contra* *ctible MULTIPLICATIVE STRUCTURES ON HOMOTOPY SPECTRAL SEQUENCES II 17 hypercover now shows that Ep,q2(U, E) ~=Hq(Ep-q(U*)) ~=Hq(~Ep-q(U*)) ~=Hqshf(X, ~E-p-q) (and all these isomorphisms are natural). Remark 7.2. Let us for a moment forget about contractible hypercovers, and look at all of them. For each map of hypercovers V* ! U* there is an induced map of spectral sequences E*(U, E) ! E*(V, E). Two homotopic maps V* ' U* induce the same map of spectral sequences from E2 on. So if we forget about E1-terms, we have a diagram of spectral sequences indexed by the homotopy category of hypercovers HCX . We can take the colimit of these spectral sequences, and the Verdier Hypercovering Theorem identifies the E2-term with sheaf cohomology. So the hypothesis that the base X be locally contractible is not really necessary * *for the development of our spectral sequence. It does make things easier to think about, though, so we will continue to specialize to that case. Now suppose that E is a fibrant ring spectrum, Y is another locally contracti* *ble space, and V* ! Y is a contractible hypercovering. The product simplicial space U* x V* is a contractible hypercover of X x Y . By the discussion in section 3* *.9 we get a pairing of spectral sequences E*(U, E) E*(V, E) ! E*(|U| x |V |, E) and a map of spectral sequences E*(U x V, E) ! E*(|U| x |V |, E). It follows fr* *om Proposition 3.10 that the latter map is an isomorphism (from the hypercover U o* *ne defines a simplicial set ß0U by applying ß0(-) in every dimension, and in our c* *ase U ! ß0(U) is a levelwise weak equivalence; the same holds for V and U x V , and so we are really just dealing with simplicial sets). Therefore we get the follo* *wing: Theorem 7.3 (Multiplicativity of the descent spectral sequence). Let X and Y be locally contractible spaces, with contractible hypercovers U* ! X and V* ! Y . * *Then there is a pairing of descent spectral sequences E*(U, E) E*(V, E) ! E*(U x V* *, E) in which the E2-term is globally isomorphic to Hqshf(X, E-p-q) Htshf(Y, E-s-t) ! Hq+tshf(X x Y, E-p-q-s-t) up to a sign difference of (-1)t(p+q). Proof.To identify the product on E2-terms we use the fact that if F and G are two sheaves on X and Y , and U and V contractible hypercovers of X and Y , then the pairing of cosimplicial abelian groups F (U*) G(V*) ! (F G)(U* x V*) induces the cup product on sheaf cohomology via an Eilenberg-Zilber map and the isomorphisms Hp(F (U*)) ~=Hp(X, F ), etc. Someone should write down a careful proof of this someday, but I have proven enough `trivial' things for one paper. The sign comes from an implicit use of the suspension isomorphism. The E2-term of E*(U, E) is most naturally identified with Hqshf(X, Gp,q) where Gp,qdenotes * *the sheafification of V 7! E-p(Sq ^ V+ ). Clearly Gp,qis isomorphic to the sheaf ~E* *-p-q via the left-suspension isomorphism, but this introduces signs into the pairings much like in Theorem 4.5. We leave the details to the reader. The assumption in the above theorem that X and Y be locally contractible is probably not really necessary, as in Remark 7.2; but I have not worked out the details. Also, when X and Y are both locally contractible and paracompact, sheaf cohomology with locally constant coefficients is isomorphic to singular cohomol* *ogy; 18 DANIEL DUGGER the pairing on sheaf cohomology corresponds to the cup product under this isomo* *r- phism [Br, Theorem III.1.1]. So the descent spectral sequence takes the same fo* *rm as the Atiyah-Hirzebruch spectral sequence in this case. Exercise 7.4. If U* ! X is a contractible hypercover, let ß0(U) be the simplici* *al set defined above_obtained by replacing each Un by its set of path components. There is a natural map U ! ß0(U), and this is a levelwise weak equivalence. Con- vince yourself that the descent spectral sequence E*(U, E) is canonically isomo* *rphic to the Atiyah-Hirzebruch spectral sequence for the space |ß0(U)| based on its s* *kele- tal filtration. Conclude that everything in this section is just a restatement * *of the results in section 3.3. 7.5. The Leray spectral sequence. Now assume that ß :E ! B is a fibration with fiber F , and that B is locally contractible. Choose a hypercover U* of B and consider the pullback hypercover ß-1U* of E. Once again, we know by [DI] that |ß-1U*| ! E is a weak equivalence. Taking E = HZ, the skeletal filtration of |ß-1U*| gives a spectral sequence for computing the homotopy groups of the function space F (E, HZ)_i.e., it computes the singular cohomology groups of E. This is the Leray (or Leray-Serre) spectral sequence. Exercise 7.6. Let Hnß-1 denote the sheaf on B obtained by sheafifying V 7! Hnsing(ß-1V ). Check that Hnß-1 is a locally constant sheaf on B whose stalks a* *re isomorphic to Hnsing(F ). So if B is simply-connected, it is a constant sheaf. * *We will abbreviate Hnß-1 as Hn(F ). It follows that if U is a contractible hypercover then the E2-term of the spe* *ctral sequence is isomorphic to Hqshf(B, H-p-q(F )). When B is simply connected this * *is Hqshf(B, H-p-qF ), and when B is paracompact this can be identified with singul* *ar cohomology. Once again, if E0! B0 is a second fiber bundle with fiber F 0, there is a pai* *ring of spectral sequences which on E2-terms has the form Hqshf(B, H-p-qF ) Htshf(B0, H-s-tF 0) ! Hq+tshf(B x B0, H-p-q-s-t(F x F 0)). One has the usual sign difference from the canonical pairing, for the same reas* *ons as in Theorem 7.3. If E0! B0is the same as E ! B, then we can compose with the diagonal map to get a multiplicative structure on the Leray-Serre spectral sequ* *ence for E ! B. Theorem 7.7 (Multiplicativity of Leray-Serre). Let E ! B and E0 ! B0 be fibrations where B and B0are locally contractible. Then there is a pairing of L* *eray- Serre spectral sequences for which the E2-term is globally isomorphic to the pa* *iring Hqshf(B, H-p-qF ) Htshf(B0, H-s-tF 0) ! Hq+tshf(B x B0, H-p-q-s-t(F x F 0)). except for a sign difference of (-1)t(p+q). Exercise 7.8. Fill in the many missing details from this section. Exercise 7.9. Of course we didn't need to take E = HZ in the above discussion. We could have used any ring spectrum, in which case we would obtain the combina- tion Atiyah-Hirzebruch-Leray-Serre spectral sequence. Think through the details of this. MULTIPLICATIVE STRUCTURES ON HOMOTOPY SPECTRAL SEQUENCES II 19 References [AH] M. Atiyah and F. Hirzebruch, Vector bundles and homogeneous spaces, Proc.* * Symp. Pure Math. 3, pp. 7-38. [BK] A. Bousfield and D. Kan, A second quadrant homology spectral sequence, Tr* *ans. Amer. Math. Soc. 177 (1973), 305-318. [Br] G. Bredon, Sheaf Theory, Springer-Verlag New York, 1997. [Do] A. 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Whitehead, Elements of Homotopy Theory, Springer-Verlag, New York, 1* *978. Department of Mathematics, University of Oregon, Eugene, OR 97403 E-mail address: ddugger@math.uoregon.edu