A spectral sequence for string cohomology Marcel Bökstedt & Iver Ottosen* 5 December 2002 Abstract Let X be a 1-connected spaces with free loop space X. We in- troduce two spectral sequences converging towards H*( X; Z=p) and H*(( X)hT; Z=p). The E2-terms are certain non Abelian derived func- tors applied to H*(X; Z=p). When H*(X; Z=p) is a polynomial alge- bra, the spectral sequences collapse for more or less trivial reasons. If X is a sphere it is a surprising fact that the spectral sequences collapse for p = 2. AMS subject classification (2000): 55N91, 55P35, 18G50 1 Introduction Let X be a space and let X denote its free loop space. The circle group T acts on X by rotation of loops. The associated homotopy orbit space XhT is sometimes called the string space. Consider the cohomology H*X as given. The purpose of this paper is to study the cohomology of the free loop space and of its homotopy orbit space. In some cases, it is relatively easy to compute this cohomology. For instance, suppose that X is an Eilenberg-Maclane space. Then there is a homotopy splitting X ' X x X. The space X is also a Eilenberg- MacLane space, so that the cohomology of X is known. The cohomology of the homotopy orbits XhT is more difficult to com- pute. However, this is achieved in [BO ] and [O2 ]. The main idea of the present paper is to use these computations to study the case of a general X. In essence, this application is done using a Postnikov ______________________________ *The second author was supported by the University of Copenhagen and by the * *Euro- pean Union TMR network ERB FMRX CT-97-0107: Algebraic K-theory, Linear Algebraic Groups and Related Structures. 1 decomposition of X. From our point of view, the simplest case is when X is a product of Eilenberg-MacLane spaces, and correspondingly, the more k-invariants a space X has, the more complicated it appears. In particular, the spheres are very complicated spaces for this approach. Formally, we will study two spectral sequences converging towards the cohomology groups H*( X; Fp) and H*( XhT; Fp). Both spectral sequences have origin in the Bousfield homology spectral sequence [B1 ]. This is a remarkable spectral sequence that under fortunate circumstances converges to the homology of the total space of a cosimplicial space. Let X be a simply connected space. We re-write its Postnikov tower as a cosimplicial space, whose total space is the p-completion of X. This cosimplicial space is the cosimplicial resolution R X of X with R = Fp. Given this, we can form two cosimplicial spaces RX and ( RX)hT by applying the functors (-) and (-)hT in each codegree. The total space of these new cosimplicial spaces are the completions of X respectively ( X)hT. These cosimplicial spaces have associated Bousfield homology spectral sequences {E^r} and {Er} respectively. For 1-connected X it is well known that {E^r} converges strongly towards H*( X; Fp). We show that {Er} converges strongly towards H*( XhT; Fp) under the additional assumption that H*(X; Fp) is of finite type. For the dual cohomology spectral sequences, {E^r} and {Er}, we give an interpretation of the E2-terms. The idea is that the E1-terms are given by the cohomology of the respective functors (from spaces to spaces) ap- plied to the Eilenberg-MacLane spaces. This cohomology_can,_according to [BO ],[O1 ] and [O2 ] be written as certain functors respectively ` (from al- gebras with a certain extra structure to algebras), applied to the cohomology of the Eilenberg-MacLane spaces. This means that the E2-term is the homology of a chain complex, where the chains are given by these functors applied to the cohomology of Eilenberg- MacLane spaces. Since the cohomology of an Eilenberg-MacLane space turns out to be a free object, we can compute the E2-terms as derived_functors. To be precise, they are the non Abelian derived functor of applied to H*(X; Fp) respectively the non Abelian derived functor of ` applied to H*(X; Fp). When H*(X; Fp) is a polynomial algebra the higher derived functors vanish so the spectral sequences collapse at the E2-terms. So far, the results are of a theoretical nature. As a concrete example, we finally study the case X = Sn and p = 2. We develop homological algebra sufficient for computing the relevant E2-terms. For these spaces, there are other methods for computing H*( X; Z=p) and H*(( X)hS1; Z=p). 2 Comparing our E2 terms with these results, we show that for X = Sn with n 2 and p = 2 the spectral sequences collapse at the E2-terms. We emphasize that this collapsing is not something to be expected a priory. Since spheres have complicated Postnikov systems, from the point of view of our spectral sequence, one would naively expect that these spectral sequence could have many nontrivial differentials. A natural question is: For how large a class C of 1-connected spaces do the two spectral sequences collapse at the E2-terms ? As a first approach we conjecture that C contains any suspension X = Y where Y is path connected and H*(Y ; F2) is of finite type. Finally, we want to thank the referee of [BO ] for suggesting that we look at the Bousfield spectral sequence in this connection. 2 Bousfield homology spectral sequences Let X be a fibrant cosimplicial space and let A be an Abelian group. In [B1 ] Bousfield constructs a spectral sequence {Er(X; A)} with the homology of the total space H*(Tot X; A) as expected target. The precise convergence statement is as follows. Recall that there is a tower of fibrations . . .! Totm X ! Totm-1 X ! . .!.Tot0X with inverse limit TotX. Hence for each n 0 there is a tower map Pn(X) : {Hn(Tot X; A)}m 0 ! {Hn(Tot mX; A)}m 0 where the domain tower is constant. Let A X denote the cosimplicial simplicial Abelian group with (A X)mt= A Xmtwhere A S = x2SA for a set S. Bousfield forms the double normalized complex and let T (A X) denote its total complex. It is filtered by subcomplexes F mT (A X) and the quotient complex T (A X)=F m+1T (A X) is denoted Tm (A X). A comparison map is defined n(X) : {Hn(Tot mX; A)}m 0 ! {HnTm (A X)}m 0 and the following result is proved: Lemma 2.1. {Er(X; A)} converges strongly to H*(Tot X; A) if and only if the tower map n(X) O Pn(X) is a pro-isomorphism for each n. 3 If n(X) is a pro-isomorphism for each n then X is called an A-pro- convergent cosimplicial space and {Er(X; A)} is called pro-convergent. We are interested in two special cases of this spectral sequence. Let R = Fp be the field on p elements where p is a fixed prime. For a space X we let RX denote the cosimplicial resolution of X in the sense of [BK ]. Note that (RX)n = Rn+1X. The free loop space on X is by definition the simplicial mapping space X = map (T, X) where we take T = BZ. By applying codegree wise we get a cosimplicial space RX. We can also form the T homotopy orbit space codegree wise and get the cosimplicial space ( RX)hT. We are interested in the Bousfield homology spectral sequences for these two spaces. As a corollary of [B2 ] Proposition 9.7 we have Proposition 2.2. If X is a 1-connected and fibrant space then Pn( RX) and n( RX) are pro-isomorphisms for each n and the spectral sequence {Er( RX; Fp)} converges strongly to H*( (X^p); Fp) ~=H*( X; Fp). In order to handle the other spectral sequence we need some results on cosimplicial spaces which are presented in the next section. 3 Cosimplicial spaces with group actions In this section the category of simplicial sets is denoted S and the category of cosimplicial spaces cS . For A, B 2 S we let map (A, B) = BA denote the simplicial mapping space. We write cA for the constant cosimplicial space with (cA)n = A for each n. The category cS is a model category with weak equivalences, cofibrations and fibrations as described in [BK ] X x4. The fibrations are here defined in terms of matching spaces. By this definition it is clear that if f : A ! B is a fibration in S then c(f) : cA ! cB is a fibration in cS . The category cS is in fact a simplicial model category in the sense of [Q ] with X K 2 cS, XK 2 cS and Map (X, Y) 2 S defined as follows for K 2 S and X, Y 2 cS: (X K)(ff) = X(ff) x K (XK )(ff) = X(ff)K Map (X, Y)n = Hom cS(X n, Y) where ff is a morphism in the simplicial category and n = [n] 2 S de- notes the standard n-simplex. In case K is a simplicial group, this notation potentially clashes with the usual notation for fixed points. In this paper, we are not going to consider fixed points. 4 Let be the cosimplicial space which in codegree n equals n. We write [m]for the simplicial m-skeleton and put [1]= . By [BK ] X.4.3 we have that [m]is a cofibrant cosimplicial space for each 0 m 1. The total space of a cosimplicial space X is defined as TotX = Map ( , X) If X is not fibrant, the total space might not give you the "right" homotopy type. In this case, we have to choose a fibrant replacement_Z_of X, that is a weekly equivalent, fibrant cosimplicial space, and define TotX = TotZ. When the cosimplicial space has a group action one can choose an equiv- ariant fibrant replacement in the following sense: Lemma 3.1. Let G be a simplicial group and X a cosimplicial G-space. Assume that Xn is a fibrant simplicial set for each n 0. Then there is a cosimplicial G-space E(X) such that both E(X) and E(X) =G are fibrant cosimplicial spaces and such that the following diagram commutes: EG x X --~-! E(X) ? ? ? ? y y (1) EG xG X --~-! E(X) =G Here the vertical maps are the obvious quotient maps, and the horizontal maps are weak equivalences. The map E(X) ! E(X) =G is the pullback of the principal G-fibration cEG ! cBG over a fibration E(X) =G ! cBG. Proof. By the model category properties we can factor the projection map EGxG X ! cBG as a composite pOi where i : EGxG X ! Y is a cofibration which is simultaneously a weak equivalence, and p : Y ! cBG is a fibration. BG is a fibrant space by [GJ ] Lemma I.3.5 so cBG is a fibrant cosimplicial space. Thus Y is fibrant. We form the codegree wise pullback of ß : cEG ! cBG over p. ~p E(X) _______E(X) --- ! cEG ? ? ? ? p? ? y i y iy ~= p E(X) =G -- - ! Y --- ! cBG The principal G-action (in the sense of [M3 ]) of G on EG gives a principal G-action on E(X) n for each n and an isomorphism of cosimplicial spaces E(X) =G ~=Y as written in the diagram. By [B1 ] Lemma 7.1 it follows that ßp is a fibration so E(X) is fibrant. By the pullback property we can lift the map i to a map EGxX ! E(X) . This constructs the missing map in the statement of the lemma. In each codegree (1) is a map of fibrations over BG and we conclude that the lifting __ is also a weak equivalence. |__| 5 Theorem 3.2. Let X be a fibrant cosimplicial space and G a simplicial group. Then XG is a cosimplicial G-space and we can form its equivariant fibrant replacement E(XG ). There is a natural map of fibrations of simplicial sets for each m with 0 m 1: (Tot mX)G --- ! EG xG (Tot mX)G -- - ! BG ? ? ? ~?y ~?y ~=?y Totm (E(XG )) --- ! Totm (E(XG )=G) -- - ! Totm (cBG) The first and middle vertical maps are weak equivalences and the right vertical map is an isomorphisms of simplicial sets. Proof. Since X is fibrant each Xn is fibrant such that (XG )n = (Xn)G is fibrant by [M3 ] Theorem 6.9. Hence we can form E(XG ). By [M3 ] Definition 20.3 and Theorem 20.5 we have that the top vertical line in the diagram is a fiber bundle. By [BK ] X.5. SM7 and the fact that [m] 2 cS is cofibrant we see that if p : A ! B is a fibration in cS then Tot m(p) : Tot mA ! Tot mB is a fibration in S. In particular Tot mX is fibrant since X is fibrant and by [M3 ] Theorem 6.9 we have that (Tot mX)G is fibrant. Thus the top vertical line is a Kan fiber bundle and hence a fibration by [M3 ] Lemma 11.9. The lower vertical line is Totm of a fibration and hence a fibration. There is a commutative diagram as follows: (Tot mX)G --- ! EG xG (Tot mX)G -- - ! BG ? ? ? ~=?y fm?y ~=?y Totm (XG ) --- ! Tot m(EG xG XG ) -- - ! Totm (cBG) ? ? fl ~?y ?y flfl Totm (E(XG )) --- ! Totm (E(XG )=G) -- - ! Totm (cBG) The isomorphism (Tot mX)G ~=Tot m(XG ) is one of the axiomatic isomor- phisms in a simplicial model category. We examine it closer in order to define fm . A cosimplicial space is a diagram in S and the axiomatic isomor- phism comes from the corresponding isomorphism in the simplicial model category S. For A, B, C 2 S this isomorphism is the composite F : (AB )C ~= ABxC ~=ACxB ~=(AC )B The following commutative diagram shows that F is equivariant with 6 respect to actions of the monoid CC . CC x (AB )C _______________O______________//_(AB )C | | | | fflffl|i2x1 O fflffl| CC x ABxC _____//(B x C)BxC x ABxC _____//ABxC | | | | fflffl|i1x1 O fflffl| CC x ACxB _____//(C x B)CxB x ACxB _____//ACxB | | | | fflffl| ix1 (O)B fflffl| CC x (AC )B _______//_(CC )B x (AC )B_____//_(AC )B For Z 2 S the action of G on the mapping space ZG is defined by ad(~)x1 G G O G G x ZG -- - - ! G x Z --- ! Z where ad(~) denotes the adjoint of the product ~ : G x G ! G. So taking C = G in the above we see that F is G-equivariant such that we have a map 1 xG F : EG xG (AB )G ! EG xG (AG )B The composite EG x (AG )B --ix1-!(EG x AG )B --- ! (EG xG AG )B factors through EG xG (AG )B and we compose with 1 xG F to get a map EG xG (AB )G ! (EG xG AG )B The morphism fm in the theorem is codegree wise given by this map. The lover part of the diagram is induced by (1). The functor (-)K : cS ! cS where K 2 S preserves fibrations as one sees from the right lifting property by taking adjoints. Hence XG is fibrant since X is fibrant. By [BK ] X.5.2 we get a weak equivalence when applying Totm to a weak equivalence between fibrant cosimplicial spaces. Thus the left vertical map is a weak __ equivalence. The result follows. |__| 4 Strong convergence In this section we discuss convergence of the Bousfield homology spectral sequence associated with ( RX)hT where R = Fp, the field on p elements. We use Fp coefficients everywhere unless stated otherwise. 7 Proposition 4.1. If X is a 1-connected space then X and XhT are nilpo- tent spaces. In fact we have ß1( X)- respectively ß1( XhT)-central series as follows for each i 1: ßi( X) ßi( X) 0, (2) ßi( XhT) ßi( X) ßi( X) 0. (3) Proof. (2) The fibration X ! X ! X splits by the constant loop inclu- sion X ! X. So we have ßi( X) ~= ßi( X) ßi(X) for i 1. Since the action of the fundamental group is natural there is a commutative diagram ß1( X) x ßi( X) --- ! ßi( X) ? ? ? ? y y ß1( X) x ßi( X) --- ! ßi( X) ? ? ? ? y y ß1(X) x ßi(X) --- ! ßi(X) We have ß1( X) ~=ß1( X) since X is simply connected. Further ß1( X) acts trivially on ßi( X) since X is an H-space. From the upper square we see that the filtration (2) is ß1( X)-stable and that the action on ßi( X) is trivial. Since ß1(X) = 0 the lower square shows that the action on the quotient ßi( X)=ßi( X) is trivial. (3) The fibration X ! XhT ! BT splits by a map constructed from a constant loop. So for i 1 we have ßi( XhT) ~=ßi( X) ßi(BT). Especially ß1( XhT) ~=ß1( X). By naturality there is commutative diagram ß1( X) x ßi( X) -- - ! ßi( X) ? ? ? ? y y ß1( XhT) x ßi( XhT) -- - ! ßi( XhT) ? ? ? ? y y ß1(BT) x ßi(BT) -- - ! ßi(BT) From the upper square we see that the inclusion ßi( XhT) ßi( X) is ß1( XhT)-stable. The lower square shows that the action on the quotient ßi( XhT)=ßi( X) is trivial. The rest of the sequence (3) has the desired __ properties since (2) is a ß1( X)-central series. |__| Proposition 4.2. If X is a 1-connected space then the cosimplicial space E( RX)=T is R-pro-convergent. 8 Proof. This is a consequence of [B1 ] 3.3. Via the weak equivalences from Lemma 3.1 we can use the filtrations from Proposition 4.1 in each codegree. Then the quotients are ßi(cBT), ßi(RX) and ßi+1(RX). Hence it suffices to show that when n 0 the following holds for all m 0: ßm ßm+n (cBT) = 0, ßm ßm+n (RX) = 0, ßm ßm+n+1 (RX) = 0. (4) Clearly ßm ßm+n (cBT) = 0 unless m + n = 2 and ß2-nß2(cBT) = 0 since the differentials in the complex ß2(cBT) are alternating zeros and ones. By the proof of 6.1 in [BK ], Ch. I and Proposition 6.3 in Ch. X The following holds for any space Y : If eHi(Y ; R) = 0 for i k then ßjßi(RY ) = 0 __ for i k + j. So the last two groups in (4) are also zero. |__| Lemma 4.3. Let X be a 1-connected space with H*X of finite type. Then RsX is 1-connected and H*RsX is of finite type for each 0 s < 1. Proof. By [BK ] I.6.1 we have thatQRsX is 1-connected for each s. Recall that R(Y ) is weakly equivalent to 1n=0K(H~n(Y ), n) for any space Y . So if H*Y is of finite type then H*R(Y ) is also of finite type and ßiR(Y ) = H~iY is finite for each i. Hence ßi((RX)m ) is finite for each i, m. From [S] Lemma 2.6 we see that ßi(RsX) is finite for each i, s. By the Postnikov tower for __ RsX we conclude that H*RsX is of finite type for each s. |__| Lemma 4.4. Let . . .! C*(2) ! C*(1) ! C*(0) be a sequence of maps of chain complexes. If for all n and m the group Cn(m) is finite, then there is an isomorphism Hn(lim C*(m)) ~=lim Hn(C*(m)) for all n. Proof. This is a consequence of the lim1-sequence which can be found in e.g. __ [M2 ] Appendix A5. |__| Proposition 4.5. Let G be a simplicial group such that Hn(BG) is finite for all n. Let {Zm } be a tower of G-spaces and put Z1 = limZm . Assume that {H*(Z1 )}m 0 ! {H*(Zm )}m 0 is a pro-isomorphism and that Hn(Zm ) is finite for all integers n, m. Then {H*((Z1 )hG )}m 0 ! {H*((Zm )hG )}m 0 is also a pro-isomorphism. Proof. We have Leray-Serre spectral sequences for 0 m 1 as follows: E2(m) = H*(BG; H*(Zm )) ) H*((Zm )hG ). The tower map {E2i,j(1)}m 0 ! {E2i,j(m)}m 0 is a pro-isomorphism for all i and j by the pro-isomorphism in the assumption, so E2i,j(1) ~=lim E2i,j(m). By the assumptions on the homology of BG and Zm , the groups E2i,j(m) with m < 1 are all finite so by Lemma 4.4 we have E3i,j(1) ~=lim E3i,j(m). 9 By induction Eri,j(1) = limEri,j(m) for each r and since we have only finite filtrations E1i,j(1) ~= lim E1i,j(m). Since E1i,j(m) is finite for all i, j, m* * it follows that {E1 (1)}m 0 ! {E1 (m)}m 0 is a pro-isomorphism. The result __ follows by the five lemma [BK ] III 2.7. |__| Theorem 4.6. If X is a 1-connected fibrant space with H*(X; Fp) of fi- nite type, then the Bousfield spectral sequence {Er( RXhT; Fp)} converges strongly to H*( (X^p)hT; Fp) ~=H*( XhT; Fp). Proof. Let Y = RX. The spectral sequence abuts to the homology of the total space of a fibrant replacement of YhT. We choose the fibrant replace- ment E(Y)=T from Lemma 3.1. The total space of this fibrant replacement is weakly equivalent to (X^p)hT by Theorem 3.2. Thus the spectral se- quence converges to the stated result if it converges. (A Leray-Serre spectral sequence argument shows that we can remove the p-completion inside the homology group.) We have shown in Proposition 4.2 that the spectral sequence is pro- convergent. Hence it suffices to show that Pn(E(Y)=T) or equivalently Pn(YhT) is a pro-isomorphism. By the Eilenberg-Moore spectral sequence and Lemma 4.3 we see that H*(Tot sY) ~= H*( RsX) is of finite type for each 0 s < 1. By Proposition 4.5 and Proposition 2.2 the result fol- __ lows. |__| We now change to cohomology. The dual of Proposition 2.2 and of The- orem 4.6 is as follows: Theorem 4.7. If X is a 1-connected and fibrant space with H*X of finite type then we have strongly convergent Bousfield cohomology spectral sequences E^r) H*( X), ^E-m,t2= (ßm H*( RX))t Er ) H*(( X)hT), E-m,t2= (ßm H*(( RX)hT))t. We are going to give a description of the E2-terms as certain non Abelian derived functors evaluated at H*X. In the next section we set up categories relevant for this purpose. 5 The category F and the simplicial model category sF For a fixed prime p we let A denote the mod p Steenrod algebra and K the category of unstable A-algebras. The category of non-negatively graded 10 unital Fp-algebras with the property that A0 is a p-Boolean algebra (ie. x = xp for all x 2 A0) is denoted Alg . In [O1 ], [O2 ] we defined a category F with forgetful functors K ! F ! Alg as follows: Definition 5.1. An object in F consists of an object A in Alg which is equipped with an Fp-linear map ~ : A ! A with the following properties: o |~x| = p(|x| - 1) + 1 for all x 2 A. o ~x = x when |x| = 1 and if p is odd and |x| is even then ~x = 0. o ~(xy) = ~(x)yp + xp~(y) for all x, y 2 A. Furthermore A is equipped with an Fp-linear map fi : A ! A with the following properties: o |fix| = |x| + 1 for all x 2 A. o fi O fi = 0 and if |x| = 0 then fix = 0. o fi(xy) = fi(x)y + (-1)|x|xfi(y) for all x, y 2 A. If p = 2 we require that fi = 0. A morphism f : A ! A0 in F is an algebra homomorphism such that f(~x) = ~0f(x) and f(fix) = fi0f(x). Remark 5.2. For an object K 2 K the map ~ : K ! K is defined by ~x = Sq|x|-1x when p = 2 and ~x = P (|x|-1)=2x when p is odd and |x| is odd. The map fi is the Bockstein operation when p is odd. There is an obvious product on F. There is also a coproduct. For two objects A and A0 in F the coproduct A A0 is the tensor product of the underlying objects in Alg equipped with maps ~ * ~0 and fi * fi0 as follows; ~ * ~0(x y) = ~(x) yp + xp ~0(y) fi * fi0(x y) = fi(x) y + (-1)|x|x fi0(y) In appendices in [O1 ] and [O2 ] we showed that F is complete and cocomplete. It is well known that K and Alg also possess these properties. In the following R denotes any one of the categories K, F or Alg . Let nFp denote the category of non-negatively graded Fp-vector spaces. The free functor SR : nFp ! R is by definition the left adjoint of the forgetful functor R ! nFp. If X is a non-negatively graded set we put SR (X) = SR (Fp X) where Fp X is the free graded Fp-vector space with basis X. In particular we have free objects SR (xn) on one generator xn of degree n. 11 Remark 5.3. Note that SF (V ) = SAlg(V~) where M ~V= V ~iV * 2 , p = 2 i 1 M ~V= V fiV * 1 fi ~i(fiV even,* 2 V odd,* 2) , p > 2. i 1, 2{0,1} In the following we use [Q ] II.4 Theorem 4 to see that the category sR of simplicial objects in R is a simplicial model category. The arguments are standard but we have included them anyhow. We start by verifying that R has enough projectives. Recall that a mor- phism f : X ! Y in a category D is called an effective epimorphism if for any object T and morphism ff : X ! T there is a unique fi : Y ! T with fi O f = ff provided ff satisfies the necessary condition that ff O u = ff O v whenever u, v : S ' X are maps such that f O u = f O v ([Q ] II.4 proof of Proposition 2). Proposition 5.4. Let f be an effective epimorphism in a category D. Then f is an epimorphism. Furthermore if f can be factored as f = i O p where i is a monomorphism then i is an isomorphism. Proof. Assume that f is an effective epimorphism. Let r, s be two parallel arrows such that r O f = s O f. Then for ff = r O f we have fi O f = ff both for fi = r and fi = s. So by uniqueness r = s. Thus f is an epimorphism. Assume that f = i O p where i is a monomorphism. If f O u = f O v for two parallel arrows u, v then i O p O u = i O p O v and p O u = p O v since i is a monomorphism. Hence there exists an arrow j such that p = j O f. Now i O j O f = i O p = f which implies that i O j = id since f is an epimorphism. Furthermore i O j O i = id O i = i which implies that j O i = id since i is a __ monomorphism. |__| Proposition 5.5. A morphism in R is an effective epimorphism if and only if it is a surjection on underlying graded sets. Proof. Any morphism f : X ! Y may be factored as X ! f(X) ! Y where the last map is clearly a monomorphism. So by the previous proposition we see that an effective epimorphism is surjective. Assume that f : X ! Y is a surjection and let fi : X ! T be a map which satisfies fi O u = fi O v whenever f O u = f O v. For a given x 2 kerf let n = |x| and define u, v : SR (xn) ' X by u(xn) = x and v(xn) = 0. Then x 2 kerfi so we have kerf kerfi. Now ff : Y ! T with ff(f(a)) = fi(a) is __ well defined and has ff O f = fi. |__| 12 Recall that in [Q ] an object P in a category D is called projective if Hom D(P, -) sends any effective epimorphism to an surjection of hom-sets. Proposition 5.6. The following statements hold in the category R: 1. SR (V ) is projective for any object V in nFp. 2. R has enough projectives. 3. {SR (xn)|n 0} is a set of small projective generators. Proof. (1) By taking adjoints and applying the previous proposition we see that SR (V ) is projective. (2) Let U : R ! nFp denote the forgetful functor and let X be an object in R. The adjoint j : SR (U(X)) ! X of idU(X) is surjective and hence an epimorphism. Thus R has enough projectives. (3) The object SR (xn) is projective by (1). Since Hom R(SR (xn), X) = Xn we have that Hom R(SR (xn), -) commutes with filtered colimits so SR (xn) is small. Finally, for two different morphisms f, g : X ' Y there exist an x 2 X such that f(x) 6= g(x). Hence the map SR (xn) ! X with xn 7! x where n = |x| separates f and g such that we have a set of generators as __ stated. |__| We now turn to the category sR of simplicial objects in R. The homotopy groups of an object R in R is defined as the homology ß*R = H*(R, @) where @ is the differential given by the alternating sums Xn @ = (-1)idi : Rn ! Rn-1. i=0 Especially ß0(R) = R=(d0 - d1)R and we have a morphism ffl : R ! ß0(R) in sR given by projection where we view ß0(R) as a constant simplicial object. If f : X ! Y is a morphism in R we can form the diagram X -- ffl-!ß0X ? ? f?y i0f?y Y -- ffl-!ß0Y One says that f is surjective on components if the map from X into the pullback (f, ffl) : X ! Y xi0Y ß0X is a surjection. Note that if ß0(f) is an isomorphism then f is surjective on components if and only if f is surjective. Proposition 5.7. There is a simplicial model category structure on sR as follows: 13 o f : X ! Y is a weak equivalence if ß*f : ß*X ! ß*Y is an isomor- phism. o f : X ! Y is a fibration if it is surjective on components and an acyclic fibration if it is both a fibration and a weak equivalence. o f : X ! Y is a cofibration if for any commutative diagram X -- - ! A ? ? f?y p?y (5) Y -- - ! B where p is an acyclic fibration, there exist a map Y ! A making both triangles commute. The solution to the arrow diagram (5) is unique up to simplicial homotopy under X and over B. Proof. This is a special case of [Q ] II x4 Theorem 4. The uniqueness part __ follows from [Q ] II x2 Proposition 4. |__| Note that the cofibrations are described in an indirect way. The concept of an almost free map make up for this weakness. See [Q ] II page 4.11 Remark 4 and the main source [M4 ] x3, [M5 ] x2 or [G ]. Definition 5.8. Let e denote the subcategory of the simplicial category with objects [n] = {0, 1, . .,.n} for n 0 and morphisms the order preserving maps which sends 0 to 0. An almost simplicial object in a category C is a functor from e opto C. Definition 5.9. A morphism f : X ! Y in sR is called almost free if there is an almost simplicial sub vector space V of Y such that for each n 0, the natural map Xn SR (Vn) ! Yn is an isomorphism. Proposition 5.10. (1) Almost free morphisms are cofibrations in sR (2) Any morphism A ! B may be factored canonically and functorially as A ! X ! B where the first map is almost free and the second is an acyclic fibration. (3) Any cofibration is a retract for an almost free map. __ Proof. Similar to the one given in [M4 ]. See also [G ]. |_* *_| Definition 5.11. A simplicial resolution of an object A 2 sR is an acyclic fibration P ! A in sR with P cofibrant. An almost free resolution of A is an acyclic fibration Q ! A such that Fp ! Q is almost free. 14 Note that an almost free resolution is a resolution and that almost free resolutions always exist by the above proposition. Example 5.12. Let RX be the cosimplicial resolution of a space X. Then H*(RX) is an almost free resolution of H*X in each of the categories K, F and Alg . 6 Derived functors In this section R denotes any of the categories K, F or Alg . We use the following notation for non Abelian derived functors: Definition 6.1. The homology of an object R in sR with coefficients in a functor E : R ! Alg is defined by H*(R; E) = ß*E(P ) where P ! R is a simplicial resolution of R. By the uniqueness statement in Proposition 5.7 this homology theory is well defined and functorial in R. For an object R 2 R we also write R for the corresponding constant simplicial object in sR. We are mainly interested in H*(R; E) when R 2 R. These homology groups have certain properties which we now describe. Let E, F and G be functors from R to Alg with natural transformations E ! F ! G. Let V : Alg ! nFp denote the forgetful functor to graded Fp-vector spaces. If the sequence 0 ! V E ! V F ! V G ! 0 is short exact when evaluated on any free object in R then we get a long exact sequence . . .Hi(R; E) Hi(R; F ) Hi(R; G) Hi+1(R; E) . ... The 0th homology group is sometimes given by the following result: Lemma 6.2. Define the category R0 as we defined the category R except that we do no longer require that objects are unital. Let F : R0 ! Alg 0be a functor. Assume that for every surjective morphism f : A ! B in R0 the following two conditions hold: 1. F (f) : F (A) ! F (B) is surjective 2. F (ker f) ! F (A) ! F (B) is exact then H0(C; F ) ~=F (C) for all objects C in R. 15 Proof. Let P ! C be a simplicial resolution of C. From the normalized chain complex N*F (P ) we see that H0(C; F ) = F (P0)=F (d1)(ker F (d0)). The maps d0, d1 : P1 ! P0 are surjective by the simplicial identities. Let i : ker d0 ! P1 denote the inclusion. By condition 2. we have that kerF (d0) = F (i)(F (ker d0)). Thus F (d1)(ker F (d0)) = F (d1) O F (i)(F (ker d0)). There is a commutative diagram d01 kerd0 -- - ! d1(ker d0) ? ? i?y j?y P1 --d1-! P0 where d01denotes the restriction of d1 and j is the inclusion. By this diagram F (d1) O F (i) = F (j) O F (d01). Furthermore F (d01)(F (ker d0)) = F (d1(ker d* *0)) by condition 1. So we have F (d1)(ker F (d0)) = F (j) O F (d01)(F (ker d0)) = F (j)(F (d1(ker d0))) and H0(C; F ) = F (P0)=F (j)(F (d1(ker d0))). Using condition 1. and 2. on the projection map P0 ! P0=d1(ker d0) we __ see that H0(C; F ) ~=F (P0=d1(ker d0)) ~=F (C). |__| The following result can sometimes be used to compute derived functors of pushouts. We denote the pushout of a diagram A0 A ! A00in sR or R by A0 A A00. Proposition 6.3. Let E : R ! Alg be a functor. (1) If there is a natural isomorphism E(A0 A00) ~= E(A0) E(A00) for objects A0, A00in R then there is an isomorphism H*(B0 B00; E) ~=H*(B0; E) H*(B00; E) for B0, B002 R. (2) Assume that there is a natural isomorphism E(A0 A A00) ~=E(A0) E(A) E(A00) for diagrams A0 A ! A00in R. Assume further that B0 B ! B00is a diagram in R such that Tor Bi(B0, B00) = 0 for i > 0. Then there is a first quadrant spectral sequence as follows: E2i,j= TorH*(B;E)i(H*(B0; E), H*(B00; E))j ) Hi+j(B0 B B00; E). 16 Proof. Let P ! B be a simplicial resolution of B. By the factorization axiom we get a diagram P 0oo__oPo__//_//_P 00 | | | |~ |~ |~ fflffl|fflffl|fflffl|fflffl|fflffl|fflffl| B0 oo___B _____//B00 where the vertical maps are acyclic fibrations and the upper horizontal maps are cofibrations as indicated. Since Fp ! P is a cofibration and cofibrations are stable under composition we see that P 0! B0and P 00! B00are simplicial resolutions. Now form the map of pushouts f : P 0 P P 00! B0 B B00and consider the corresponding map of derived tensor products in the sense of [Q ] II x6: Lf : P 0 P P 00= P 0 LPP 00! B0 LBB00. By [Q ] II x6 Theorem 6 there are second quadrant spectral sequences Tor i*Pi(ß*P 0, ß*P 00)j ) ßi+j(P 0 P P 00) Tor i*Bi(ß*B0, ß*B00)j ) ßi+j(B0 LBB00) The above diagram gives a map of spectral sequences which is an isomorphism at the E2-terms. Hence Lf is a weak equivalence. By the Corollary following Quillen's Theorem 6 we have that B0 LBB00! B0 B B00is a weak equivalence. Thus f is itself a weak equivalence. Since f is surjective it is a fibration. Since the pushout of a cofibration is a cofibration P 0! P 0 P P 00is a cofibration and thus the domain of f is cofibrant. So f is a simplicial resolution. For the proof of (1) take B = Fp and apply E codegree wise. The re- sult follows by the Eilenberg-Zilber theorem. For the proof of (2) apply E __ codegree wise. The result follows by Quillen's Theorem 6. |__| If one knows that the higher derived functors vanish on a certain class of objects, they can be used to compute derived functors by the following result. Theorem 6.4. Let E : R ! Alg be a functor and let A 2 R. Assume that Qi~ A is an acyclic fibration in sR and that ( F (Qj) , i = 0 Hi(Qj; E) = 0 , i > 0. Then H*(A; E) ~=ß*E(Q). 17 Proof. We have shown that sR is a simplicial model category. So ssR is a simplicial model category by the Reedy structure [GJ ] VII 2.13. A fibration in ssR is especially a level fibration and a cofibration is especially a level cofibration by [GJ ] VII 2.6. A weak equivalence is a level weak equivalence by definition. We use a dot to denote a simplicial direction in the following. Let cQoo denote the object in ssR defined by (cQ)ij = Qj for all i. Let Poo be a resolution of cQoo ie. (Fp)oo æ Pooi~cQoo. We have that (Fp)o æ Pio~iQo for each i by the above. By composition with the acyclic fibration Qoi~A we see that Piois a resolution of A. So the horizontal homotopy of F (Poo) is given by ßhjF (Pio) = Hj(A; F ). We apply vertical homotopy on this and obtain ( Hj(A; F ) , i = 0 ßvißhjF (Poo) ~= 0 , i > 0. We also have that Poj is a resolution of Qj for each j. So ßviF (Poj) ~= Hi(Qj; F ) which equals F (Qj) for i = 0 and equals 0 for i > 0. We apply horizontal homotopy on this and obtain ( ßjF (Qo) , i = 0 ßhjßviF (Poo) ~= 0 , i > 0. Thus both spectral sequences associated with F (Poo) collapse and the result __ follows. |__| 7 The E2 terms seen as derived functors __ In [BO ], [O1 ] and [O2 ] we introduced a functor : F ! Alg as follows: __ Definition 7.1. (R) is the quotient of the free graded commutative and unital R-algebra on generators {dx|x 2 R} of degree |dx| = |x| - 1, modulo the ideal generated by the elements d(x + y) - dx - dy, d(xy) - d(x)y - (-1)|x|xd(y), d(~x) - (dx)p, d(fi~x). __ __ There is a differential d : (R) ! (R) given by d(x) = dx for x 2 R. __ Note that for p = 2 the Bockstein is trivial so here the functor is the same as the functor which we originally denoted ~. 18 __ It was shown that there is a lift to a functor : K ! K and that this lift is nothing but_Lannes' division functor (- : H*(T))K . In particular there is a morphism (H*X) ! H*( X) for any space X which is an isomorphism when H*X is a free object in K. An other functor ` : F ! Alg was also defined by generators and rela- tions. The explicit definition was rather complicated so we do not repeat it here. The functor ` also lifts to an endofunctor on K and it comes with a natural morphism `(H*X) ! H*( XhT) which is an isomorphism if H*X is a free object in K. Via Example 5.12 we can now restate Theorem 4.7 in an appropriate form. Theorem 7.2. If X is a 1-connected and fibrant space with H*X of finite type then we have strongly convergent Bousfield cohomology spectral sequences E^r) H*( X) and Er ) H*( XhT) with the following E2 terms: ^E-m,t2~=Hm (H*(X); __)t and E-m,t2~=Hm (H*(X); `)t. We now introduce other functors in order to study the derived functors of `. Recall that the functors L and e from F to Alg are defined by L(R) = `(R)=(u) and e (R) = L(R)=(ffi(x)|x 2 R). Proposition_7.3._For each object R 2 F there are isomorphisms as follows: H0(R; ) ~= (R), H0(R; e) ~=e (R) and H0(R; L) ~=L(R). Proof._We use Lemma 6.2 to prove this. By their definitions we may consider , e and L as functors from F0 to Alg 0. Let A be an object in F0 and let I A be an ideal. We must verify condition 1. and 2. in Lemma 6.2 for these functors where f : A ! A=I is the natural projection. We do this for the functor L. The verification for the other functors is similar but easier. The map L(f) is surjective with kernel J = (OE(x) - OE(y), q(x) - q(y), ffi(x) - ffi(y)|x - y 2 I) L(A) so L(A)=J ~= L(A=I). We must check that L(I) = J. The inclusion L(I) J holds since OE(0) = q(0) = ffi(0) = 0. For the inclusion L(I) J assume first that p = 2. Since OE and q are additive we have that ffi(x) - ffi(y) and OE(x) - OE(y) lie in L(I). Furth* *er q(x) - q(y) = q(x - y) + ffi(xy) but ffi(xy) = ffi(x(y - x)) so also q(x) - q(y* *) 2 L(I). Thus the inclusion holds. 19 For p odd ffi is additive, OE is additive on elements of even degree and q is additive on elements of odd degree. For |x| = |y| odd we have p-2X OE(x) - OE(y) = OE(x - y) + ffi(x)iffi(y)p-2-iffi(xy) i=0 and again ffi(x(y - x)) = ffi(xy) such that this lies in L(I). For |x| = |y| ev* *en we have p-1X 1 i p-i q(x) - q(y) = q(x - y) - ffi( __x y ) i=1i so it suffices to see that y - x divides the sum inside the ffi(-). The followi* *ng equation in Fp[x, y] shows that this is the case: p-3X p-1X k 1 i p-i X 1 xy(y - x) akxkyp-3-k = __x y where ak = _____. k=0 i=1 i j=0 j + 1 P p-1 __ The equation holds since by Euler's sum formula n=1 n = 0 modulo p. |__| Definition 7.4. Let Z, B, H : F ! Alg denote the functors given by Z(R) = ker(d), B(R) = im(d), H(R) = Z(R)=B(R) __ where d is the differential on (R). Recall from [BO ], [O1 ] and [O2 ] that there are natural transformations of functors : e ! H and Q : L ! Z. It was shown that if A 2 F is a free object, or its underlying algebra is polynomial, then A and QA are isomorphisms. We can now give a nice interpretation of the functor L, which was originally defined by generators and complicated relations. Theorem 7.5. For any R 2 F one has L(R) ~=H0(R; Z). Proof. The induced map Q* : H*(R; L) ! H*(R; Z) is an isomorphism since Q is an isomorphism on free objects. The 0th derived functor of L was __ computed in Proposition 7.3. |__| For any functor E : F ! Alg we have that Hi(A; F ) = 0 for i > 0 when A is a free object since we can use the trivial almost free resolution to compute the derived functors. For polynomial algebras we also have nice results. Theorem 7.6. Assume that the underlying_algebra of A 2 F is a polynomial algebra. Then one has Hi(A; ) = 0, Hi(A; e) = 0, Hi(A; L) = 0 and Hi(A; `) = 0 for each i > 0. 20 __ Proof. We first prove the statements for and e. Let : Alg ! Alg denote the usual de Rham complex functor. Pick an almost free resolution P 2 sF of A. The forgetful functor U : F ! Alg takes free objects to free objects. So we can apply U to P and get an almost free resolution of U(A) in sAlg . Thus there is an isomorphism HFi(A; U) ~=HAlgi(U(A); ) and the last group is trivial for_i_> 0 since U(A) is a free object in Alg . There is a linear map U(A) ! (A); x0dx1 . .d.xn 7! x0dx1 . .d.xn. The map is not multiplicative and it does not commute with the de Rham __ differential, but it is an isomorphism of graded vector spaces. Thus Hi(A; ) is additively isomorphic to Hi(A; U) which is trivial for i > 0. A similar isomorphism gives the result for the functor e . __ Next we consider the functor L. The short exact sequence 0 ! Z ! ! B ! 0 gives a long exact sequence of derived functors. By the above this sequence breaks up into the exact sequence __ 0 ! H1(A; B) ! H0(A; Z) ! H0(A; ) ! H0(A; B) ! 0 together with the isomorphisms Hi(A; Z) ~=Hi+1(A; B) for i 1. There is also a short exact sequence 0 ! B ! Z ! H ! 0 with corre- sponding long exact sequence of derived functors. Since is an isomorphism on free objects we have a natural isomorphism * : H*(-; e) ~= H*(-; H). By the above vanishing result for H*(A; e) the long exact sequence breaks up into the short exact sequence 0 ! H0(A; B) ! H0(A; Z) ! H0(A; H) ! 0 and the isomorphisms Hi(A; B) ~=Hi(A; Z) for i 1. Using Proposition 7.3 and Theorem 7.5 we can rewrite the exact sequences involving 0th derived functors as __ 0 ! H1(A; B) ! L(A) ! (A) ! H0(A; B) ! 0 0 ! H0(A; B) ! L(A) ! H(A) ! 0. Since Q : L(A) ! Z(A) is an isomorphism we see that H1(A; B) = 0. By the isomorphisms H1(A; B) ~=H1(A; Z) ~=H2(A; B) ~=. . .we conclude that Hi(A; Z) is trivial for i > 0. But H*(-; L) is isomorphic to H*(-; Z) so we are done. Finally we consider the functor `. By definition of L there is a short exact sequence 0 ! u` ! ` ! L ! 0. From the corresponding long exact 21 sequence of derived functors we find that Hi(A; u`) ~=Hi(A; `) for i > 0. By Theorem 10.3 from the appendix and Proposition 4.4 from [O2 ] there is a short exact sequence 0 ! uj+1`(B) ! uj`(B) ! uj e(B) ! 0 , j > 0 (6) when B is a free object in F or when the underlying algebra of B is a poly- nomial algebra. The corresponding long exact sequence of derived functors shows that Hi(A; uj`) ~= Hi(A; uj+1`) so we have Hi(A; `) ~= Hi(A; uj`) for all j 0. But (uj`)k = 0 for k < 2j so Hi(A; uj`)k = 0 for k < 2j and the __ result follows. |__| Proposition 7.7. If the underlying algebra of an object A 2 F is a polyno- mial algebra, then H0(A; `) ~=`(A). Proof. The short exact sequence 0 ! u` ! ` ! L ! 0 gives a short exact sequence of 0th derived functors since H1(A; L) = 0. Furthermore, there is a natural map H0(-; F ) ! F for any functor F : F ! Alg . So we have a commutative diagram with exact rows as follows: 0 --- ! H0(A; u`) -- - ! H0(A; `) -- - ! H0(A; L) --- ! 0 ? ? ? ? ? ? y y y 0 --- ! u`(A) -- - ! `(A) -- - ! L(A) --- ! 0. The right vertical map is an isomorphism so it suffices to show that the left vertical map is also an isomorphism. Since H1(A; e) = 0 the short exact sequence (6) gives a commutative diagram as follows for j > 0: 0 --- ! H0(A; uj+1`) -- - ! H0(A; uj`) -- - ! H0(A; uj e) --- ! 0 ? ? ? ? ? ? y y y 0 --- ! uj+1`(A) -- - ! uj`(A) -- - ! uj e(A) --- ! 0. where the right vertical map is an isomorphism. Fix a degree n. For j + 1 > n=2 the map H0(A; uj+1`)n ! (uj+1`(A))n is an isomorphism since both __ domain and target space are zero. The result follows by induction. |__| Corollary 7.8. Let X be a 1-connected space such that H*X is of finite type and H*X is a polynomial algebra. Then then the spectral sequences of Theorem 7.2 collapses at the E2 terms. So there are isomorphisms __* * * * * H*(H*(X); ) ~=H ( X) and H*(H (X); `) ~=H (( X)hT). 22 8 The derived functors of an exterior algebra In the rest of this paper we take p = 2. Let = (oe) 2 F be an exterior algebra on one generator of degree |oe| 2. Note that ~oe = 0 for dimensional reasons. We intend to compute the higher derived functors of the various functors we have been considering for this algebra. Proposition 8.1. There are isomorphisms __ __ H*( ; ) ~= ( ) [!] , H*( ; e) ~=e ( ) [e!]. The inner degrees are |fli(!)| = i(2|oe| - 1), |fli(e!)| = i(4|oe| - 1) and the grading of the homology groups are given by __ __ Hi( ; ) ~= ( ) fli(!) , Hi( ; e) ~=e ( ) fli(e!). Proof. The algebra is the pushout of F2 F2[y] ! F2[x] where y 7! x2. Put ~x = 0 and ~y = 0. By Proposition 6.3 we find __ Hi( ; E) ~=Tor E(F2[y])i(F2, E(F2[x])) for E = , e. __ The result follows by standard computations. |__| In order to compute derived functors of the other functors we need an explicit simplicial resolution of . By Theorem 6.4, Theorem 7.6, Proposition 7.3 and Proposition 7.7 we may use an almost free resolution of in sAlg and equip it with ~ = 0. Proposition 8.2. There is an almost free resolution Ro 2 sAlg of the algebra with Rn = F2[x, y1, y2, . .,.yn] for n 0. The structure maps di : Rn ! Rn-1 and si : Rn ! Rn+1 are given by si(x) = x ( yj , j i si(yj) = yj+1 , j > i di(x) = x 8 >>x2 , i = 0, j = 1 >< yj-1 , i < j, j > 1 di(yj) = >>yj , i j, j < n >: 0 , i = n, j = n The degrees of the generators are |x| = |oe| and |yi| = 2|oe| for all i. 23 Proof. We first give a description of the simplicial set 1o= Hom (-, [1]) suited for our purpose. Define the elements yj 2 1nfor n 0 and 0 j n + 1 by yj(i) = 0 if i < j and yj(i) = 1 if i j. We have 1n= {y0, . .,.yn+1}. The structure maps are as follows: ( ( yj-1 , i < j yj+1 , i < j diyj = and siyj = yj , i j yj , i j Let F2[-] denote the functor which takes a graded set to the polynomial algebra generated by that set. Let F2[ 1o, *] denote the pushout of F2 F2[a] ! F2[ 1o] where F2 and F2[a] are constant simplicial algebras. In degree n the maps are as follows: a 7! 0 2 F2 and a 7! yn+1 2 F2[ 1o]. Note that F2[ 1o] ' F2[*] by the simplicial contraction of 1o. The spectral sequence [Q ] II x6 Theorem 6 gives that ßi(F2[ 1o, *]) ~=F2 for i = 0 and 0 otherwise. Define Ro as the pushout of F2[x] F2[z] ! F2[ 1o, *] where in degree n the maps are z 7! x2 and z 7! y0. For this pushout Quillen's spectral sequence gives that ( , i = 0 ßi(Ro) ~=Tor F2[z]i(F2, F2[x]) = 0 , i > 0 Thus Ro has the right homotopy groups. Further Rn is as stated and the structure maps are as stated. Note that Ro is almost free. The degrees are __ correct since the structure maps must be degree preserving. |__| Lemma 8.3. Hn( ; ) has the following F2-basis: dy1 . .d.yn, xdy1 . .d.yn, dxdy1 . .d.yn, xdxdy1 . .d.yn. Proof. Using the formulas in Proposition 8.2 it is easy to check that the four given classes are in the kernel of di for all i. To check linear independency, we introduce two gradings of . Firstly, the wedge grading on (Rn) is defined as the number of wedge factors, ie. the number of d's in a homogeneous element. Secondly the polynomial grading is defined as follows: Give x grading 1, yj grading 2 and each dx or dyj grading 0 and extend multiplicatively. Note that the maps di preserve both gradings. We write q,t(Rn) for the elements in (Rn) of wedge degree q and polynomial degree t. Thus there is a direct sum decomposition M Hn( ; ) = Hn( ; q,t). q,t 0 24 The classes we consider sit in different bigradings, so we only have to check that they individually do not represent the trivial class. We have the following bases for n,0(Rn+1), n,0(Rn) and n,0(Rn-1) respectively: {dxdy1 . .c.dyj.d.y.n-1} [ {dy1 . .c.dyj.c.d.yk.d.y.n+1}, {dxdy1 . .c.dyj.d.y.n-1} [ {dy1 . .d.yn}, {dxdy1 . .d.yn-1}. We use the normalized complex consisting of \i>0ker(di) with differential d0 to compute the homology. For this normalized complex we have the respective bases ;, {dy1^. .^.dyn}, {dx^dy1^. .^.dyn-1}. Taking homology and using that n,0(Rn-2) = 0 we see that the classes dxdy1 . .d.yn-1 and dy1 . .d.yn do not represent zero. Similarly, xdxdy1 . .d.yn-1 and xdy1 . .d.yn do not represent zero. The __ result follows by shifting dimensions. |__| Lemma 8.4. Hn( ; H) has an F2-basis as follows: y1 . .y.ndy1 . .d.yn, x2y1 . .y.ndy1 . .d.yn, xdxy1 . .y.ndy1 . .d.yn, x3dxy1 . .y.ndy1 . .d.yn. __ Proof. This follows from Lemma 8.3 and the Cartier isomorphism. |__| Corollary 8.5. The long exact sequence associated to B ! Z ! H splits into short exact sequences when evaluated at : 0 ! H*( ; B) ! H*( ; Z) ! H*( ; H) ! 0. Proof. The generators from Lemma 8.4 are visibly in the image of the map __ Hn( ; Z) ! Hn( ; H). |__| Definition 8.6. Let fli,j2 (Rj) denote the following elements: fli,j= y1y2 . .y.idy1dy2 . .d.yj for i, j > 0 fl0,j= dy1dy2 . .d.yj for j > 0 and fl0,0= 1. These elements are actually in Z( ) if i j, and in B( ) in case i < j. They are even in the normalized chain complex, so they define classes in H*( ; Z) respectively H*( ; B). 25 Theorem 8.7. For n 0 the homology group H2n( ; B) has F2-basis {dxfl0,2n} [ {fl2i,2n, x2fl2i,2n, xdxfl2i,2n, x3dxfl2i,2n|0 i < n} and the homology group H2n+1( ; B) has F2-basis {fl0,2n+1, dxfl0,2n+1, xdxfl0,2n+1}[ {fl2i+1,2n+1, x2fl2i+1,2n+1, xdxfl2i+1,2n+1, x3dxfl2i+1,2n+1|0 i < n}. Similarly, H2n( ; Z) has F2-basis {dxfl0,2n} [ {fl2i,2n, x2fl2i,2n, xdxfl2i,2n, x3dxfl2i,2n|0 i n} and H2n+1( ; Z) has F2-basis {fl0,2n+1, dxfl0,2n+1, xdxfl0,2n+1}[ {fl2i+1,2n+1, x2fl2i+1,2n+1, xdxfl2i+1,2n+1, x3dxfl2i+1,2n+1|0 i n}. Proof. Recall that Hm ( ; ) ~= ( ) fl0,m and Hm ( ; H) ~= e( ) flm,m . From the splitting Hm ( ; Z) ~= Hm ( ; B) Hm ( ; H) together with the computation of Hm ( ; H) in Lemma 8.4 , it follows that the statements about Hm ( ; Z) and Hm ( ; B) are equivalent. Recall the long exact sequence Hm+1 ( ; B) --b-! Hm ( ; Z) --i*-! Hm ( ; ) --d*-! Hm ( ; B) (7) Claim: The image of d* is a one dimensional vector space, generated by a class, which in the normalized chain complex is represented by dxfl0,m. Note that three out of four of the generators of Hm ( ; ) are annihilated by d*. So the image of d* is spanned by the single class, represented in the normalized complex by dxfl0,m 2 B( ). This element actually represents a non zero homology class, since the map Hm ( ; B) ! Hm ( ; ) induced by inclusion, send it to the element represented by dxdy1 . .d.ym . But this is non-trivial according to Lemma 8.3, and the claim follows. We will now prove the theorem by induction on m. We first treat the case m = 0. Here we see that the map d0 : H0( ; ) ! H0( ; B) is surjective by the long exact sequence (7), and the statement follows directly from the claim we just proved. Assume that the theorem holds for m. The long exact sequence (7) gives us a short exact sequence 0 ! im(dm+1 ) ! Hm+1 ( ; B) !b ker(im ) ! 0 26 So, in order to prove the theorem, we have to show that b maps the classes given in the statement of the theorem with dxfl0,m+1 removed to a basis for the kernel of im . By induction we have a basis for Hm ( ; Z). From this we see that the kernel of the map i2n+1 : H2n+1( ; Z) ! H2n+1( ; ) has basis {fl2i+1,2n+1, x2fl2i+1,2n+1, xdxfl2i+1,2n+1, x3dxfl2i+1,2n+1|0 i n} and that the kernel of the map i2n : H2n( ; Z) ! H2n( ; ) has basis {x2fl0,2n, x3dxfl0,2n} [ {fl2i,2n, x2fl2i,2n, xdxfl2i,2n, x3dxfl2i,2n|1 i * * n}. It is convenient now to pass from the normalized to the unnormalized complex. The normalized complex is a subcomplex of the unnormalized one, and in the notation we do not distinguish between a class in the subcomplex and its image in the unnormalized one. Let us first consider the odd case, m = 2n + 1. ThePelement fl2i,2n+22 B(R2n+2) is a cycle with respect to the boundary @ = dk. We compute its image under the map b. Let ffr 2 (R2n+2) denote the following element: ffr = y1y2 . .y.2iyrdy1dy2 . .c.dyr.d.y.2n+2 , 2i + 1 r 2n + 2 P 2n+2 where the hat means that the factor is left out. Put fi = r=2i+2ffr. We have dffr = fl2i,2n+2for each r, so that also dfi = fl2i,2n+2. This means that b(fl2i,2n+2) is represented by 2n+2X @fi = (yr-1 + yr)fl2i,2n+1+ y2n+1fl2i,2n+1= fl2i+1,2n+1. r=2i+2 Since b is linear with respect to multiplication by x2, xdx, x3dx this gives the desired result for H2n+2( ; B). In the even case m = 2n, a similar argument shows that b(fl2i+1,2n+1) is represented by fl2i+2,2n. Checking with the lists of classes above, we see that we are left to prove that b(fl0,2n+1) = x2fl0,2n. The argument is very similar. Let ffr = yrdy1dy2 . .c.dyr.d.y.2n+1and fi = ff1 + ff2 + . .+.ff2n+1. Then dffr = fl0,2n+1so also dfir = fl0,2n+1. Thus b(fl0,2n+1) is represented by __ @fi = x2fl0,2n. |__| 27 Proposition 8.8. There are short exact sequences for i 0, t 1 as follows: 0 -- - ! Hi( ; u`) --- ! Hi( ; `) --- ! Hi( ; L) -- - ! 0 , 0 -- - ! Hi( ; ut+1`) --- ! Hi( ; ut`) --- ! Hi( ; ute ) --- ! 0. Proof. The first short exact sequence follows if we can prove that the con- necting homomorphism b : Hi+1( ; L) ! Hi( ; u`) is trivial. So consider the diagram 0 --- ! u`(Ri+1) --- ! `(Ri+1) --- ! L(Ri+1) --- ! 0 ? ? ? ? ? ? y y y 0 --- ! u`(Ri) --- ! `(Ri) --- ! L(Ri) --- ! 0. The element q(y1) . .q.(yj)ffi(yj+1) . .f.fi(yi+1) 2 `(Ri+1) maps to the ele- ment flj,i+12 L(Ri+1). By the relations ffi(a)2 = ffi(~a), q(a)2 = OE(~a) + ffi(a2~a) and ffi(a)q(b) = ffi(a~b) + ffi(ab)ffi(b) we see that this element ma* *ps down to zero in `(Ri). So the connecting homomorphism b is trivial. __ The proof for the second short exact sequence is similar. |__| Corollary 8.9. For each i 0 there are isomorphisms of F2-vector spaces M t Hi( ; `)~=Hi( ; L) u Hi( ; e) t 1 ~=Hi( ; B) F2[u] Hi( ; e) . Let F : F ! Alg be a functor. We_define_the total degree of a class in Hi(R; F )n to be n - i. For F = `, this corresponds through the spectral sequences of Theorem 7.2 to the grading of cohomology groups. We write PF (t) for the Poincar'e series corresponding to the total degree of H*( ; F )*. Theorem 8.10. Let s denote the degree of the class oe 2 . Then we have the following Poincar'e series: P__(t)= (1 + ts)(1 - ts-1)-1, Pe (t)= (1 + t2s)(1 - t2s-1)-1, PB(t) = ts-1(1 + ts-1 - t2s-1)(1 - t2s-1)-1(1 - t2s-2)-1, P`(t)= (1 + ts-1 - ts+1 + t2s-1)(1 - t2)-1(1 - t2s-2)-1. Proof. The first two formulas follow from Proposition_8.1: The total degree of fli(!) is |fli(!)| - i = (2s - 2)i and ( ) = (doe) so P__(t) = (1 + ts)(1 + ts-1)(1 - t2s-2)-1 = (1 + ts)(1 - ts-1)-1. 28 A similar argument gives Pe (t). To determine PB(t) we must count the classes given in Theorem 8.7, according to the total degree. We divide these classes into three groups. The first group are those of the form dxfl0,m. The Poincar'e series of the subspace generated by those classes is ts-1=(1 - t2s-2). The second group are those of the type fl0,2n+1or xdxfl0,2n+1. These have Poincar'e series t2s-2(1 + t2s-1)=(1 - t4s-4). The third group is the remaining classes. They span a free e ( )-module with basis X = {fl2i,2n, fl2i+1,2n+1|0 i < n, 0 n}. We introduce the following operation on the set X: T (fli,n) = fli+1,n+1. This operation has total degree 4s-2. All generators are obtained by applying T a non-negative number of times starting from one of the elements of Y = {fl0,2n|n 1}. The set Y has Poincar'e series t4s-4=(1-t4s-4). So, the set X has Poincar'e series t4s-4(1 - t4s-4)-1(1 - t4s-2)-1. We multiply this by (1 + t2s)(1 + t2s-1* *), make a small reduction, and obtain the Poincar'e series for the third group of classes: t4s-4(1 + t2s)(1 + t2s-1)-1(1 + t4s-4)-1. To get the Poincar'e series of B, we add the three series obtained so far. ts-1 t2s-2(1 + t2s-1) t4s-4(1 + t2s) PB(t) = _________+ _______________+ _____________________ 1 - t2s-2 1 - t4s-4 (1 - t2s-1)(1 - t4s-4) The stated formula for PB(t) follows after some reductions. Finally, Corollary 8.9 gives that P`(t) = PB(t) + (1 - t2)-1Pe (t) which __ leads to the stated formula after some reductions. |__| 9 The spectral sequences for spheres Let X be a pointed space, Y = X it's reduced suspension. We have es- tablished a spectral sequence converging to H*(ET xT Y ; F2) in general. But in this special case, we are fortunate to have a direct calculation of the homology H*(ET+ ^T Y ; F2). If the homology of X is of finite type, the (finite) dimensions of these homology and cohomology groups agree. So if we for the particular space X can check that the Poincar'e series of the homology of H*(ET xT Y ; F2) as computed in [CC ] agrees with the Poincar'e series of the E2 term of our spectral sequence, we know that our spectral sequence collapses. (See also [BM ] for an easier proof of the results in [CC ]). Let us now consider the special case of spheres X = Ss-1 and Y = Ss. The purpose of this section is to show that in this case the two Poincar'e series actually agree, forcing the spectral sequence to collapse. 29 Theorem 9.1. The Poincar'e series of H*(ET xT Ss; F2) is (1 + ts-1 - ts+1 + t2s-1)(1 - t2)-1(1 - t2s-2)-1. Proof. We compute a sequence of related Poincar'e series. First, let A = H~*(Ss-1) considered as a graded vector space. This has Poincar'e series ts-1. For each m 1, the cyclic group Cm acts on A m . This is a 1-dimensional vector space over F2. We now consider the homology groups H*(Cm ; A m ) We first look at homological dimension 0. H0(Cm , A m ) ~= A m , so it has Poincar'e series tm(s-1). In higher homological degrees, there are two cases. If m is odd, the groups all vanish, and we get a trivial Poincar'e series. If m is even, and i 1, Hi(Cm , A m ) ~=A m Since this single group has homological degree i, its Poincar'e series is ti+m(s-1). Now, recall from [CC ], proposition 9.3 that ~H*(ES1+^S1 Ss) ~= m 1 H*(Cm ; A m ) (Actually, we are correcting a misprint in [CC ] here. The homology groups on the right hand side of the formula should not be reduced). The Poincar'e series of the right hand side contains the sum of the contri- butionPof the homology in dimension zero. The Poincar'e series of this part is m 1 tm(s-1) = ts-1(1 - ts-1)-1. It also contains the sums of the contri- butions of the reduced group homologies. Since this is trivial if m is even, we can as well put m = 2n, and the Poincar'e series of the reduced part is X X ti+2n(s-1)= t2(s-1)+1(1 - t)-1(1 - t2(s-1))-1 i 1 n 1 Summing, we get that the Poincar'e series for H~*(ES1+^S1 ( X)) is (ts-1 - ts + t2s-2)(1 - t)-1(1 - t2s-2)-1 Finally, we note that there is a short exact sequence of homology groups: 0 ! ~H*(BT) ! ~H*(ET xT Ss) ! ~H*(ET+ ^T Ss) ! 0. This shows that the Poincar'e series of H*(ET xT Ss) is (ts-1 - ts + t2s-2)(1 - t)-1(1 - t2s-2)-1 + (1 - t2)-1 __ Bringing on common denominator and adding proves the theorem. |__| 30 Proposition 9.2. The Poincar'e series of H*( Ss; F2) is (1 + ts)(1 - ts-1) when s 2. Proof. The mod 2 cohomology ring of Ss is a special case of Theorem 2.2 of [KY ] except for the case s = 2. It is however shown (Remark 2.6) that the Eilenberg-Moore spectral sequence also collapses when s = 2 so we can compute the Poincar'e series from the E2-term. It has the following form (see the proof of Theorem 2.2): E*,*2~= (x) (__x) [!] where the respective bidegrees of x, __xand fli(!) are (0, s), (-1, s) and (-2i, 2is) such that the respective total degrees becomes s, s-1 and 2i(s-1). Thus the Poincar'e series is (1 + ts)(1 + ts-1)(1 - t2(s-1))-1 __ and the result follows by a small reduction. |__| Theorem 9.3. If we let X = Ss with s 2 and use F2-coefficients, then the spectral sequences of Theorem 7.2 collapses. Thus there are isomorphisms of graded F2-vector spaces: __* * s * s * * s H*(H*(Ss); ) ~=H ( S ) and H*(H (S ); `) ~=H (( S )hT). Proof. By Theorem 8.10, Theorem 9.1 and Proposition 9.2 the Poincar'e series of the E2-terms agree with the Poincar'e series of the targets. So the spectral __ sequences collapses. |__| 10 Appendix: On a filtration of the functor ` In this appendix we identify the graded object associated with the filtration `(A) u`(A) u2`(A) . . . in the case where p = 2 and A is a polynomial algebra. Recall that the functors L, e : F ! Alg are defined by L(A) = `(A)=(u) and e (A) = L(A)=Iffi(A) where Iffi(A) is the ideal (ffi(x)|x 2 A) L(A). We want to define a map `(A) ! e (A)[t] such that the elements OE(x), q(x) and u in the domain are send to the elements OE(x), q(x) and t2 in the target. Unfortunately, this cannot be done by a ring map. But if we pay the penalty of changing the multiplicative structure of the target, we can almost get such a map. 31 Definition 10.1. tw (A) is the free graded commutative algebra on gener- ators OE(x), q(x) for x 2 A and t, of degrees |OE(x)| = 2|x|, |q(x)| = 2|x| - 1 and |t| = 1, modulo the relations q(x + y) = q(x) + q(y), OE(x + y) = OE(x) + OE(y), q(xy) = OE(x)q(y) + OE(y)q(x), OE(xy) = OE(x)OE(y), q(x)2 = OE(~x) + tq(~x). Clearly, tw(A)=(t) ~=e (A). The ring tw(A) is just a twisted version of the polynomial ring over e (A) in t in the following case: Theorem 10.2. Assume that the underlying algebra of A is a polynomial algebra. Then the graded ring Gr*( tw (A)) corresponding to the filtration of tw (A) by powers of t equals e (A)[t]. Proof. As an intermediate step, let us consider the ring R(A) which is defined exactly like tw (A) except that we do not include the last relation q(x)2 = OE(~x) + tq(~x). If A is a polynomial algebra on generators {xi|i 2 I}, then R(A) is a polynomial ring on generators OE(xi) and q(xi). To obtain tw(A) from R(A), we have to add the relations q(p)2 = OE(~p)+tq(~p), where p is any polynomial in the generators xi. Actually, it is sufficient to do this for the generators themselves, as this relation for p1p2 follows from the relations for p1 and p2. Because, assume those are satisfied, then we calculate q(p1p2)2= OE(p1)2q(p2)2 + OE(p2)2q(p1)2 = OE(p1)2(OE(~p2) + tq(~p2)) + OE(p2)2(OE(~p1) + tq(~p1)) = OE(~(p1p2)) + tq(~(p1p2)). Thus we can write tw(A) as an algebra: F2[t, OE(xi), q(xi)|i 2 I]={q(xi)2 = OE(~xi) + tq(~xi)} From this it is clear, that tw (A) is a free F2[t, OE(xi)|i 2 I]-module, with generators {q(xi1) . .q.(xin)|ir 6= is forr 6= s, n 0}. (The empty product means 1 here.) It follows that Gr*( tw (A)) is a free module over Gr*(F2[t, OE(xi)|i 2 I]) with the same generators. So, to finish the proof, we only have to determine the multiplicative struc- ture of tw (A). The multiplicative relations are given by the relations. In 32 the graded ring they are q(xi)2 = OE(~xi). So, we have a presentation of the graded ring as F2[t, OE(xi), q(xi)|i 2 I]={q(xi)2 = OE(~xi)}. __ But this is exactly e (A)[t]. |__| Theorem 10.3. Let A be an object in F and i 1 an integer. Multiplication with ui defines a natural surjective F2-linear map ui`(A) ui : e(A) ! _________. ui+1`(A) If the underlying algebra of A is a polynomial algebra, then this map is an isomorphism and M Gr*(`(A)) ~=L(A) uj e(A). j 1 Proof. Multiplication with ui gives a surjective map `(A) ! ui`(A) and ui(Iffi(A)) = 0, ui(u`(A)) = ui+1`(A) so the map factors through e (A). We define a natural ring map v : `(A) ! tw(A) by the formulas v(OE(x)) = OE(x) + tq(x), v(q(x)) = q(x), v(u) = t2, v(ffi(x)) = 0. To see that v is well defined, we have to check that the relations in the definition of ` goes to 0. This is trivial for all relations except three which* * is verified as follows: v(OE(xy))= OE(xy) + tq(xy) = (OE(x) + tq(x))(OE(y) + tq(y)) + t2q(x)q(y) = v(OE(x)OE(y) + uq(x)q(y)), v(q(xy)) = q(xy) = OE(x)q(y) + q(x)OE(y) = (OE(x) + tq(x))q(y) + (OE(y) + tq(y))q(x) = v(OE(x)q(y) + q(x)OE(y)), v(q(x)2)= q(x)2 = v(OE(~x) + ffi(x2~x)). By the map v we get a commutative diagram as follows: e (A) -- - ! tw(A)=t2 tw (A) ? ? ? ? uiy t2iy ui`(A)=ui+1`(A) -- - ! t2i tw(A)=t2i+2 tw(A) When the underlying algebra of A is a polynomial algebra, then Theorem 10.2 gives that the top and the right vertical maps are injective. 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