The Eilenberg-Moore spectral sequence for K-theory; applications to p-compact homogeneous spaces A. Jeanneret & A. Osse* Abstract We construct the Eilenberg-Moore spectral sequence for some gen- eralized cohomology theories, along the lines of Smith and Hodgkin. We prove its multiplicativity and give some sufficient conditions for its convergence to the desired target. As applications, we compute the K-theory of various spaces associated to p-compact groups. AMS subject classification (1991): 55T20, 55N15, 55P35, 57T35. Keywords: Spectral sequences, K-theory, completed tensor product, p-compact groups. Introduction The topology of homogeneous spaces constitutes an important aspect of Lie theory. The celebrated works of Borel, Bott, Baum, Smith and others com- pletely describe their rational cohomology and provide important informa- tions on their integral cohomology. For the K-theory of homogeneous spaces, the Eilenberg-Moore type spectral sequence constructed by Hodgkin has been the most successful tool. Recently Dwyer and Wilkerson [7] have introduced a homotopical gener- alization of the Lie groups, namely the p-compact groups. In this homotopy Lie theory, the notions of Weyl group, maximal torus, homogeneous spaces are well defined, even if there is still no "differentiable structure" in sight. The present work originates in understanding the topology of the p- compact homogeneous spaces of Dwyer and Wilkerson. An important in- variant of these spaces is their p-adic cohomology. Its rational part can be ______________________________ *Supported by grant No 20-46899.96 of the Swiss National Fund for Scientific* * Research 1 computed by combining a classical Eilenberg-Moore spectral sequence ar- gument with the result of [7]. At the moment, the main indications about the torsion in p-adic cohomology are conjectural (see however the results of [16]). Because of the "lack of differentiability", the K-theoretical construc- tions of Hodgkin do not apply to the p-compact setting. To circumvent to this difficulty, we have to go back to the original definition of the spectral sequence, by basing all the constructions on the geometric cobar resolution. This has led us to rewrite, adapt and simplify many classical arguments. This approach turns out to be successful and allows us to deal with other generalized cohomology theories. To be more precise, let E*(- ) be a gener- alized multiplicative cohomology theory such that E*(pt) is a graded field. Examples of such theories are given by singular cohomology with coefficient in a field, complex mod p K-theory and Morava K-theories. With our as- sumption, E*(X) is a complete Hausdorff topological E*(pt)-algebra for any space X. Moreover, E*(- ) satisfies the K"unneth isomorphism (with com- pleted tensor products). These two properties are the crucial ingredients for the construction and the study of the Eilenberg-Moore spectral sequence for E*(- ). Our main result can be stated as follows: Main Theorem Let B be a connected space and E*(- ) as above. For any pull-back diagram X xB Y ____-X | | | | | |p ?| f ?| Y _______-B there exists a strongly convergent spectral sequence {E*;*r(X; Y ); dr}r2 sati* *s- fying the following properties: 1. The spectral sequence is multiplicative and compatible with the stable operations of E*(- ). 2. Ei;*2~=Tor-iE*(B)(E*(X) ; E*(Ya))s algebras, where Tor -iE*(B)(-; -)is the i-th derived functor of the completed tensor product. 3. If p : X -! B is a fibration and E*(B )is an exterior algebra on odd degree generators, then the spectral sequence converges to E*(X xB Y ). The third point of the Theorem desserves the following comment. As easily seen, if E*(B )is an exterior algebra, then E*(B )is connected; hence B is 1-connected. It is well known that this weaker property of B suffices to 2 ensure the convergence of the spectral sequence, when E*(- ) = H*(-; K), the singular cohomology with coefficient in a field K (see [18]) With respect to our original aim, we apply the main theorem to compute the K-theory of various p-compact homogeneous spaces. These computations lead to a new proof of the main result in [12]. We refer to Section 5 for more precise statements. The paper is organized as follows. The construction of the spectral se- quence is given in Sections 1 and 2. The ideas are standard but our methods are slightly different from those in [18] and [11]. The multiplicative structure is discussed in Section 3. The reader will notice that this is the most delicate part of the paper; we assure her/him that we did our best to present the ideas as clearly as possible. The convergence questions are treated in Sections 2 and 4. Proposition 2.2 of Section 2 is already stated in Hodgkin's paper, but the proof presented here is new and hopefully simpler. Section 4 is crucial for the application to the p-compact groups. In contrast to Hodgkin, we are able here to get rid of the "differentiability hypothesis". In addition, the geomet- rical nature of our constructions readily implies that the spectral sequence is compatible with the stable operations of E*(- ). We mention that the results of Section 1 to Section 4 are stated for sectionned spaces over B and imply the main theorem, via the basepoint adjunction trick discussed in Section 1. The applications are discussed in Section 5. We end with an appendix on profinite rings and modules. The theme of the appendix is well known and has been considered by some topologists (see [3] and [23]). We have decided to discuss it in some details because of its central role in our arguments and also because we didn't find any reference suitable to our needs. Aknowledgements. We would like to thanks U. Suter and U. W"urgler for helpful discussions. We also thank J.M. Boardman for sending us a copy of [2]. 1 The geometric cobar resolution For the main construction of this section, we need some recollections about fiberwise topology. Until further notice, B is a fixed pointed and connected space; T op=B will denote the category of spaces over B with fibre maps as morphisms. The base point inclusion {pt} ,! B, viewed as an element of T op=B, will simply be denoted pt. The categorical product in T op=B is the familiar fibred product. In the sequel we will work in the pointed version of this category, namely the category (T op=B)* of sectionned spaces over B. The objects are triples 3 p B -s! X -! B such that p O s = id and morphisms are commutative diagrams X1 s1 ae> | Z p1 aeae || ZZ" B |OE B Z Z || ae> s2 Z" ?|aeaep2 X2 When there is no danger of confusion, we simply write X (respectively p : X1 - ! X2 ) instead of B -s! X -! B (respectively the diagram above). As the reader may have noticed, we recover the categories T op of topological spaces and T op* of pointed topological spaces by setting B = pt. It is well known that all the usual homotopic theoretic constructions can be per- formed in the category (T op=B)*. For instance, we have fiberwise homotopy, fiberwiseWmapping cone CB (-), fiberwise suspension B (-), fiberwise wedge - B-, etc. We refer to [18] for the definitions. Two other constructions play a role in what follows: 1. If (X; x0) is a pointed space, (X; x0)B will denote the sectionned space over B pr2 B -s0!X x B -! B where pr2 is the projection on the second factor and s0(b) = (x0; b). p 2. Let X - ! B be a space over B and A a closed subspace of X. The fiberwise collapse of X with respect to A is the sectionned space p B -s! X=B A -! B defined by: ` o X=B A = (X B)=a ~ p(a) 8a 2 A o p(x) = p(x); 8x 2 X and p(b) = b 8b 2 B o s(b) = b; 8b 2 B. In order to`simplify the notation, we write X=A for X=B A and X+ for X=B/O = X B (base point adjunction). We mention in passing that X=A is fiberwise homotopy equivalent to the fiberwise cone of the obvious inclusion A+ ,! X+ . We turn to an important fact, also treated in [18] and [11]: for any map : X1 - ! X2 in (T op=B)* there is a Puppe sequence of maps in (T op=B)* i() j() B() X1 -! X2 -! CB () -! B X1 - ! B X2 -! : : : (1) 4 with the following properties: 1. For any Z in (T op=B)*, the Puppe sequence of the map ^B id is obtained from (1) by smashing with Z. 2. If s1; s2; s respectively denote the sections of X1; X2 and CB () then i() X1=s1(B) -! X2=s2(B) -! CB ()=s(B) is a cofibration sequence in the category T op*. Definition 1.1 A cohomology theory on (T op=B)* consists of a sequence of contravariant functors {ehi: (T op=B)* -! Ab }i2ZZ(Ab is the category of abelian groups) and a sequence of natural transformations {ffii} such that: 1. ffii: ehi(X1)-! ehi+1(CB ())is defined for any cofibration sequence i() X1 -! X2 -! CB () and the following sequence is exact: i-1 i i()* i * i : :-:! ehi-1(X1) ffi-!eh(CB ()) -! eh(X2) -! eh(X1) -! : : : 2. (Dold's cylinder axiom) For any space X and any : X x [0; 1]-! B; we have eh*(X x [0; 1]=X) = 0, where the inclusion of X into X x [0; 1] is given by x 7! (x; 0). 3. (Strong additivity) For any family {Xfi}fi2Jof sectionned spaces over B, we have _ Y eh*( Xfi) ~= eh*(Xfi) : B It can be checked that these axioms imply that a cohomology theory eh*(-) on (T op=B)* is homotopy invariant and possesses a natural suspension iso- morphism, that is eh*+1(B X) ~=eh*(X) for any X in (T op=B)*. Before giving examples, we would like to discuss product structures. A cohomology theory eh*(-) on (T op=B)* is called multiplicative if it is equipped with a natural pairing of graded abelian groups : eh*(X) eh*(Y )-! eh*(X ^B Y ); X; Y 2 (T op=B)* which is associative and has a unit 1 2 eh0((S0; *)B ). The sectionned space (S0; *)B plays the role of a point in our category, since (S0; *)B ^B X = X for 5 any X in (T op=B)*. Consequently eh*((S0; *)B ) is a ring, called the coefficie* *nt ring of eh*(-), and eh*(X) is a two sided module over it. In [11, Lemma 4.1] it is shown that the pairing factors through: : eh*(X) h eh*(Y )-! eh*(X ^B Y ) ; X; Y 2 (T op=B)* where h is a short hand for eh*((S0; *)B ). Example. Let eE*(- )be a multiplicative cohomology theory on the category of pointed spaces and E*(- ) the associated unreduced theory. Then eE*B(-:)(T op=B)* -! Ab ; X 7! E*(X=s(B) ) defines a multiplicative cohomology theory on (T op=B)*. The coefficient ring is eE*B((S0; *)B)~=E*(B), so that any eE*B(X)is an E*(B) -module. More generally, for any fibration X -! B the functor (T op=B)* -! Ab; Y 7! eE*B(X+ ^B Y ) is also a multiplicative cohomology theory on (T op=B)*, whose coefficient ring is E*(X) . The interested reader might consult the details in [5, Section 3.4]. From now on, eE*(- )will denote a multiplicative cohomology theory on the category T op* and E*(- ) the associated unreduced theory. We will al- ways assume that E*(pt) is a graded field; that is, its non zero homogeneous elements are invertible. This assumption implies that Ee*(- )satisfies the K"unneth isomorphism (see [3]). After all these preliminaries, we are now ready to present the geometric cobar resolution. To this end, we suppose that we are given a space X in (T op=B)*. From this datum, we perform the following construction: 1. First we set X0 = X. p 2. For each integer i 0 we suppose that B - s! Xi -! B has been constructed and we successively define: o eXi= (Xi=s(B))B o OEi: Xi -! eXi, OEi(x) = ([x]; p(x)) o Xi-1 = CB (OEi). The most important properties of this construction are given in 6 Lemma 1.2 With the notations above and for any integer i 0, the fol- lowing statements are true: 1. For any space Y 2 (T op=B)*, the spaces (Xei^B Y )=s(B) and Xi=s(B)^ Y=s(B) are homotopy equivalent in T op*. 2. eE*B(Xei)~=eE*B(Xib)E*(pt)E*(B) as E*(B) -modules, where the E*(B) - action on the right hand side is given by right multiplication. 3. This induced morphism OE*i: eE*B(Xei)-! eE*B(Xii)s surjective. Proof. The first part can be safely left to the reader. For the second part, take Y = (S0; *)B in the first part to obtain Xei=s(B) = Xi=s(B) ^ B+ . It suffices then to apply the K"unneth isomorphism. Concerning the third part, let i: Xi=s(B) ^ B+ - ! Xi=s(B) denote the collapsing of B to a point. Then i O OEi = id, where OEi: Xi=s(B) -! eXi=s(B) is induced by OEi. Consequently *iis a right in- verse for OE*iand we are done. Thanks to this Lemma, the long exact sequences of the OEi's break into short exact sequences: 0 -! eE*B(Xi-1-)! eE*B(Xei)-! eE*B(Xi-)! 0 : As usual in homological algebra, we splice all these sequences together as follows: 0 0 "ZZ ae Z =aeae eE*B(X-1 ) ae "Z =aeae ZZ 0 oe__ eE*B(X)oe__eE*B(Xe0)oe_____ eE*B(Xe-1)oe______eE*B(Xe-2)oe___ : : : "ZZ ae Z =aeae eE*B(X-2 ) ae "Z =aeae ZZ 0 0 As a result, we obtain a long exact sequence of E*(B) -modules: 0 - Ee*B(X)- Ee*B(Xe0)- Ee*B(Xe-1)- : : : (2) 7 Fix i 0 and write eE*B(Xi )= eE*(Xi=s(B) )= lim -Viff, where each Viffis a finite dimensional E*(pt)-module (see Example 1 in the Appendix). By Lemma 1.2 and Theorem 6.3.ii), we have Ee*B(Xei)~=( lim V ff) b * E*(B) ~= lim (V ffb * E*(B) ) : - i E (pt) - i E (pt) Since each ViffbE*(pt)E*(B) is obviously a free E*(B) -module, we can in- voke Theorem 6.1.ii) to conclude that eE*B(Xei)is a projective E*(B) -module. Consequently the sequence (2) is a projective resolution of the E*(B) -module eE*B(X). 2 The spectral sequence The aim of this section is to construct the spectral sequence announced in the introduction, to identify its E2-term and to discuss its convergence. This spectral sequence will be defined by means of derived exact couples. In order to simplify the presentation, we will always replace finite sequences of maps by inclusions in (T op=B)*, via the fibrewise mapping cylinder construction. With this convention, the cofibre of a map f : A -! C will simply be written C=A. Let X, Y be spaces in (T op=B)* and let {Xi -! eXi-! Xi-1}i0 be the geometric cobar resolution of X. Set Wi(X; Y ) = Wi = Xi^B Y ; for all i 0: For a fixed integer i 0, we iterate the Puppe construction of the OEj's to obtain the sequence Xi-1 -! B Xi -! -i+1BX0. The latter induces a cofibration fii-i+1 ffi-i+1 B Wi=Wi-1 -! B W0=Wi-1 -! B W0=B Wi : (3) To introduce the spectral sequence we define, for i 0 Di;j1= eEj+1B(-i+1BW0=Wi-1 ) Ei;j1= eEj+1B(B Wi=Wi-1 ;) and for i > 0 Di;j1= Ei;j1= 0 : The cofibration (3) induces a long exact sequence in eE*B-cohomology which can be written as ff*ii;j fi*ii;j fl*ii+1;j : :-:! Di+1;j-11-! D1 -! E1 -! D1 -! : : : 8 where fl*iis induced by the Puppe map fli: -i+1BW0=B Wi -! B (B Wi=Wi-1) : We splice all these exact sequences to form the following unraveled exact couple (in the sense of Boardman [2]) . .-.! Di+1;*1__________-Di;*1__________-Di-1;*1___- : : : I@@ I@ @ @@ Ei;*1 Ei-11 Observe that the bidegree of ff*i(respectively fi*i, fl*i) is (-1; 1) (respe* *ct- ively (0; 0), (1; 0)). By standard techniques this yields a spectral sequence {E*;*r(X; Y ); dr}r1 (or simply {E*;*r; dr}r1 when X and Y are understood), with differentials dr : Ei;jr-! Ei+r;j+1-rr. As usual, for any r 2, the cycles and boundaries are defined as Z*;*r= Ker(dr-1); B*;*r= Im(dr-1) : Since E*;*r= Z*;*r=B*;*r, it is important to have a geometric description of the groups Z*;*rand B*;*r. For this purpose, we write o for the natural inclusion (Wi-1; B Wi) (Wi-1; rBWi+r-1) o ffi = ffi1 O oe, where ffi1 is the coboundary of the triple (Wi-r; r-1BWi-1; rBWi) and oe the suspension isomorphism eEj+rB(rBWi=r-1BWi-1 )~=eEj+1B(B Wi=Wi-1 : ) Then we have Zi;jr~=Im{* : eEj+1B(rBWi+r-1=Wi-1 -)! eEj+1B(B Wi=Wi-1 } ) Bi;jr~=Im{ffi : eEj+r-1B(r-1BWi-1=Wi-r -)! eEj+1B(B Wi=Wi-1} :) With these identifications the differentials dr are induced by the composite eEj+1B(rBWi+r-1=Wi-1 )__ffi2-eEj+2B(-i+1BW0=rBWi+r-1 ) | * |fii+r |? Eej+2B(r+1BWi+r=rBWi+r-1 ) ~=||oe |? eEj-r+2B(B Wi+r=Wi+r-1 ;) 9 where ffi2 is the coboundary of the triple (Wi+1; rBWi+r+1; -i+1BW0) and fi*i+r is as above. The proof of these facts is standard; the interested reader might want to perform it by chasing on diagrams like the one on the top of page 661 in [22]. Everything is now in place for the identification of the E2-term of that spectral sequence. This is based on the following Lemma 2.1 For any i 0 and any space Y in (T op=B)*, the product in eE*B-cohomology induces an isomorphism Ee*B(Xei)b* eE*(Y )~=eE*(Xe ^ Y ): E (B) B B i B Proof. Consider the diagram eE*B(Xei)b* eE*(Y )____________________-eE*(Xe ^ Y ) E (B) B B i B | | | | |~= |~= | | | | ?| ?| (Ee*(Xi=s(B) )bE*(pt)E*(B) ) bE*(B) eE*(Y=s(B) _)_-eE*(Xi=s(B) ^ Y=s(B) ) We proceed as in [11, page 49] to show that it commutes. By Lemma 1.2, the two vertical maps are isomorphism. The lower horizontal map is also an isomorphism (this is the ordinary K"unneth isomorphism in E*-cohomology). This complete the proof of the lemma. Let us now deal with the differential d1 : Ei-1;*1-! Ei;*1. By definition it is the composite fi*iOfl*i-1. As easily checked, the following diagram commutes -i+2BW0 6| | | 2BWi=B Wi-1 ________-2BWi-1 ________-B (B Wi-1=Wi-2) @@ fii 6|| @R | fli-1 -i+2BW0=B Wi-1 The horizontal composition is equivalent to OEi+1^Y2 OEi^Y2 B eXi+1^B Y - ! B eXi^B Y -! B eXi^B Y : 10 This observation and Lemma 2.1 imply that the complex 0 - E0;*1d1-E-1;*1d1- E-2;*1d1- : : : is isomorphic to the one obtained by applying the functor - bE*(B) eE*B(Y ) to the complex 0 - Ee*B(Xe0)- Ee*B(Xe-1)- : : : We have thus shown that Ei;*2~=Tor-iE*(B)(Ee*B(X); eE*B(Y)); where Tor -iE*(B)(-; -)is the i-th derived functor of the completed tensor product (see the Appendix). We now turn to the convergence questions. Let us introduce the graded group H*(X; Y ) by taking Hr(X; Y ) as the direct limit of the sequence D0;r1-! D-1;r+11-! : :-:! Di;r-i1-! : : : The group H*(X; Y ) is filtered (as graded group) by FiHr(X; Y ) = Im{Di;r-i1-! Hr(X; Y )} : According to Boardman ([2, Section 6]), this filtration is Hausdorff and com- plete and the spectral sequence constructed in the previous section converges to H*(X; Y ); that is, Ei;r-i1~=FiHr(X; Y )=Fi+1Hr(X; Y ) ; for alli : The expected target group is eE*B(W0 )= eE*B(X ^B Y .)In this section we will give conditions under which H*(X; Y ) is isomorphic to eE*B(X ^B Y .) For each i 0, we consider the exact triangle induced in E*-cohomology by the cofibration Wi-1 -! -i+1BW0 -! -i+1BW0=Wi-1. The direct limit of these triangles is the exact triangle * H*(X; Y ) ________-eEB(W0 ) I@@ (4) @ G*(X; Y ) where Gr(X; Y ) is the direct limit of the sequence eEr+1B(W-1 -)! eEr+2B(W-2 -)! : :-:! eEr-i+1B(Wi-1-)! : : : 11 If the group G*(X; Y )vanishes, then the map of the triangle (4) will be an isomorphism so that our spectral sequence will have the expected abutment. The following proposition describes the main property of this obstruction group and gives a sufficient condition for the convergence to eE*B(X ^B Y .) Proposition 2.2 Let X be an element of (T op=B)* such that the projec- tion p : X -! B is a fibration. The functor Y 7! G*(X; Y ) is a cohomology theory on (T op=B)*. Consequently, if G*(X; pt+) = 0, then G*(X; Y )= 0 for all spaces Y in (T op=B)*. Proof. The first axiom of Definition 1.1 is easily checked using the exactness of the direct limits and the properties of the smash product and cofibrations in the category (T op=B)*. The cylinder axiom requires the extra assumption on the projection p. By Proposition 4.8 of [18] and Section 3.4 of [5], the functors Y 7! Ee*B(Xi^B Y )are cohomology theories for each i 0. We invoke one more time the exactness of the direct limits to conclude. We are left to show that G*(X; Y ) is strongly additive. Let {Yfi}fi2Jbe a family of spacesWin (T op=B)*. We want to prove that the natural injections Yfi,! B Yfiinduce an isomorphism W Y G*(X; BYfi) ~= G*(X; Yfi): We start with three observations: W W 1. W0(X; BYfi) is homotopy equivalent to B W0(X; Yfi). W Q 2. eE*B( B W0(X; Yfi))is naturally isomorphic to eE*B(W0(X; Yfi)). 3. The following triangle is exact (the direct product preserves exactness) Y Y H*(X; Yfi)___________- eE*B(W0(X; Yfi)) I@@ @Y G*(X; Yfi) Because of these observations, it is sufficient to prove that W Y H*(X; BYfi) ~= H*(X; Yfi): From the definition of the spectralQsequence andQexactness of the direct product, one checks easily that { E*;*r(X; Yfi);W dr}r1 is a spectral se- quence. Moreover, the natural injections Yfi,! B Yfiinduce a morphism of spectral sequences W Q r : E*;*r(X; BYfi)-! E*;*r(X; Yfi): (5) 12 As the completed tensor product commutes with the direct products, the morphism W Q * 2 : Tor*E*(B)(Ee*B(X); eE*B( B Yfi))-! Tor E*(B)(Ee*B(X); eE*B(Yfi)) is an isomorphism. ThusQthe spectral sequencesQof (5) are isomorphic.Q We filter the group H*(X; Yfi) by setting Fi H*(X; Yfi)= FiH*(X; Yfi). This filtrationWis Hausdorff and complete. We observe that the natural injec- tions Yfi,! B Yfiinduce a continuous homomorphism W Q 0 : H*(X; BYfi) -! H*(X; Yfi) : (6) Its associated graded homomorphism is nothing but the isomorphism W Q 1 : E*;*1(X; BYfi)-! E*;*1(X; Yfi): As the two groups involved in (6) are Hausdorff and complete, 0 has to be an isomorphism. The last assertion follows from the comparison theorem 4.1 of [5]. 3 The multiplicative structure The discussion of the multiplicative structure of the Eilenberg-Moore spectral sequence requires more general definitions than those given in Section 2. Since this material will be used only in the present section, we have chosen to introduce it here. Definition 3.1 Let m 1 and X 2 (T op=B)*. A negative m-filtration of X is a sequence U* of spaces {Ui}i0 and maps i : Ui-1 -! mBUi in (T op=B)*, with U0 = X. A morphism f* : U* -! V* of negative m-filtrations is a sequence of maps fi: Ui -! Vi making the obvious diagrams commutat- ive. Examples. 1. For X in (T op=B)* and m 1 the geometric cobar resolution of degree m - 1 of X is inductively defined by setting X0 = X and for i 0, o eXi= (Xi=s(B))B o OEi: Xi -! eXi, OEi(x) = ([x]; p(x)) o Xi-1 = CB (m-1BOEi). 13 We set Xi(m) := Xi and take i : Xi-1(m) -! mBXi(m) to be the next map in the Puppe sequence of the cofibration m-1BXi(m) -! m-1BeXi(m) -! Xi-1(m) : The resulting negative m-filtration will be denoted X*(m) and called the cobar m-filtration. When m = 1, we recover the 1-filtration associ- ated to the geometric cobar resolution of Section 1; we will then simply write X* := X*(1). 2. Let U* be a negative m-filtration of X 2 (T op=B)* and let Y 2 (T op=B)*. One constructs a negative m-filtration U* ^ Y by smash- ing all the constituents of U* by Y . In particular, we suspend negative filtrations by smashing them with Y = (Sk; *)B . We recall here a construction of Hodgkin, since it will play a central role in our discussion. It might be illuminating to view this construction as the geometric counterpart of the tensor product of chain complexes in homological algebra. To get into work, we fix X 2 (T op=B)* and let U* be a negative m-filtration of U0 = X. For any integer i 0 we consider the sequence i+1m i+2 0 -im Ui -! B Ui+1 -! : :-:! B U0 : (7) We may and will assume (in accordance with our convention) that all the maps inVthis sequence are inclusions. Let Zi be the subspace in (T op=B)* of -imBU0 B -imBU0 defined by [ Zi = -kmBUi-k ^B -lmBUi-l: k+l=i By comparing the sequence (7) for the index i and i-1, Hodgkin constructed a natural map Oi : Zi-1 -! 2mBZi and showed (see [11, page 24]) that _ -(i-k)m -(i-l)m 2mBZi=Zi-1 ' B (mBUk=Uk-1) ^B B (mBUl=Ul-1) : (8) k+l=i As in [11], the negative 2m-filtration {Zi; Oi}i0 will be noted U* U*. We continue our recollection of Hodgkin's work, by explaining how neg- ative filtrations give rise to spectral sequences. Let U* be a negative m- filtration and eE*B(- )a cohomology theory on (T op=B)*. We fix an integer 14 i 0 and extract the part of the sequence (7) given by the inclusions Ui-1 ,! mBUi ,! (-i+1)mBU0. This sequence induces a cofibration mBUi=Ui-1 -! (-i+1)mBU0=Ui-1 -! (-i+1)mBU0=mBUi : With respect to this cofibration we define Di;j1= eEj+1+(-i+1)(m-1)B((-i+1)mBU0=Ui-1 ) Ei;j1= eEj+1+(-i+1)(m-1)B(mBUi=Ui-1: ) We extend to all integers by setting, for i > 0 Di;j1= Ei;j1= 0 : As in Section 2 we obtain an unravelled exact couple; the associated spec- tral sequence is written {Ei;jr(U*); dr}r1 . Here also, the result of Board- man [2] implies that the spectral sequence strongly converges to H*(U* ) := lim-!(D0;*1! D-1;*+11! : :):. The latter is related to the desired abutment via a natural map : H*(U* ) -! eE*B(U0 ). Lemma 3.2 Let X; Y 2 (T op=B)* and m 1. Write X*(m) (respectively X*) for the cobar m-filtration (respectively 1-filtration) of X. Then there is a natural morphism of spectral sequences E*;*r(X*(m) ^ Y ) -! E*;*r(X* ^ Y ) which is an isomorphism for r 2: Proof. We construct inductively maps OEi: Xi(m) -! -i(m-1)BXi by taking OE0 = id and requiring the commutativity of the diagram m-1BXi(m) ______-m-1BeXi(m) ________-Xi-1(m) | | | OEi| eOEi| OEi-1| ?| ?| ?| (-i+1)(m-1)BXi___-(-i+1)(m-1)BeXi___-(-i+1)(m-1)BXi-1 where eOEiis the composite m-1BeXi(m) -! m-1B(-i(m-1)BXi=s(B) x B) -! (-i+1)(m-1)BeXi: 15 Then the map OEi^ id: Xi(m) ^B Y - ! -i(m-1)BXi^B Y induces the de- sired homomorphism of spectral sequences. As in Section 1, one constructs a projective E*(B) -resolution 0 - Ee*B(X)- Ee*B(Xe0(m) )- Ee*B(Xe-1(m) )- : : : The comparison theorem of projective resolutions now implies that OEi^ id induces an isomorphism of E2-terms and the lemma follows. This is the right place to start the discussion of the multiplicative prop- erties of the spectral sequence. Proposition 3.3 Let X; Y 2 (T op=B)* and set W* = X* ^ Y , where X* is the cobar 1-filtration of X. For 1 r 1, there exist associative pairings r : Ei;jr(W*) Ep;qr(W*)- ! Ei+p;j+qr(W* W*) ; a b 7! a . b satisfying the following properties: 1. 1 is induced by the multiplication of eE*B(- ). 2. r+1 is induced by r, via the isomorphism Er+1 ~=H(Er). 3. For all a 2 Ei;jr(W*) and b 2 Ep;qr(W*), we have dr(a . b) = dr(a) . b + (-1)j+1a . dr(b) : Proof. We begin with some notation. For any negative m-filtration U*, we set: o Ai;jr(U*) = eEj+1+(-i+1)(m-1)B((rmBUi+r-1=Ui-1).) o For s r, ffr;s: Ai;jr(U*)-! Ai;js(U*)is the morphism induced by the inclusion (Ui-1; smBUi+s-1) ,! (Ui-1; rmBUi+r-1): o i;jr: Ai;jr(U*)-! Ai+r;j-r+1r(U*)is the coboundary operator of the triple (Ui-1; rmBUi+r-1; 2rmBUi+2r-1). Let us observe that Ai;j1(U*) = Ei;j1(U*), the first term of the spectral seque* *nce associated to U*. By proceding as in the case m = 1 (see Section 2), we see that Zi;jr~=Im{ffr;1: Ai;jr(U*)-! Ai;j1(U*)} Bi;jr~=Im{ffi : Ai-r+1;j+r-2r-1(U*)-! Ai;j1(U*)} ; 16 where ffi is defined as in Section 2. We set = j + 1 + (-i + 1)(m - 1) and consider the commutative diagram (obtained by playing around with adequate triples): eE+1B(((-i+1)mBU0=rmBUi+r-1) ) ffaei> | Z fi*i+r aeae | ZZ" eEB((rmBUi+r-1=Ui-1) ) | eE+1B((mBUi+r=Ui+r-1) ) | ZZ | ffae> Z" ?| aeae eE+1B((2rmBUi+2r-1=rmBUi+r-1) ) As in Section 2 and with the identifications above, the differential dr is equal fi*i+rO ffi. Consequently, dr is induced by the morphism *;*r. We now go back to our data and define pairings OEr: Ai;jr(W*) Ap;qr(W*)- ! Ai+p;j+qr(W* W*) in the following manner. First we use the product of eE*B(- )and the suspen- sion isomorphism to send Ai;jr Ap;qrinto eE*B(-p+1B(rBWi+r-1=Wi-1) ^B -i+1B(rBWp+r-1=Wp-1) ): Then we proceed as in [11, page 30] to send the latter into Ai+p;j+qr(W*W*). We observe that the arguments in Hilfsatz 13 and 14 of [13] apply verbatim to our situation. To be entirely honest, we should mention that Kulze's proofs are based on two hypothesis (the axioms M1 and M2 (page 290) of [13]). Fortunately, these hypotheses are satisfied in our case (see Theorem 9.10 (page 238) of [1]). As in the classical case, the next step consists in comparing the spectral sequences of the negative filtrations W* W* and W*, via the diagonal map. Unfortunately we have not been able to proceed directly. However, as we will see in a moment the suspensions of these filtrations can be compared. This will be sufficient for our purpose. The reason is the following: For any negative m-filtration U*, the suspension isomorphism induces an isomorphism of spectral sequences (This is easily checked at the level of exact couples) ~= i;j+1 oe : Ei;jr(U*) -! Er (U*): (9) Lemma 3.4 Let U* be a negative m-filtration of U0 = V . As above, we write V*(m) for the cobar m-filtration of V . There exists a negative m- filtration Z* of V and two morphisms of negative filtrations g* f* V*(m) - Z* -! U* such that f0 = g0 = id: 17 Proof. To begin with we set Z0 = V and f0 = g0 = id. Given fi: Zi -! B Ui and gi: Zi -! B Vi(m) , let U0i= mBUi=Ui-1; Vi0= mBeVi(m) and Z0i= m-1BeZi=s(B) x (U0ixB Vi0) : We define Zi-1 as the fiberwise cone of the obvious map m-1BZi -! Z0iand we construct fi-1 and gi-1 by requiring that the following diagram commutes: mBUi ______-U0i______-B Ui-1 6| 6|0 6|| |fi |fi |fi-1 | | | m-1BZi _____-Z0i_______-Zi-1 | | 0 | |gi |gi |gi-1 ?| ?| ?| mBVi(m) ____-Vi0____-B Vi-1(m) Here f0iand g0iare the obvious projections. Theorem 3.5 Let X; Y 2 (T op=B)* and set W* = X* ^ Y , where X* is the cobar 1-filtration of X. There exist associative pairings, for 2 r 1, r : Ei;jr(W*) Ep;qr(W*)- ! Ei+p;j+qr(W*); a b 7! ab satisfying the following properties: 1. If E*;*2(W*) is identified with Tor *E*(B)(Ee*B(X); eE*B(Y)), then 2 beco* *mes the usual internal product of Tor *E*(B)(-; -). 2. r+1 is induced by r, via the isomorphism Er+1 ~=H(Er). 3. For all a 2 Ei;jr(W*), b 2 Ep;qr(W*) we have dr(ab) = dr(a)b + (-1)j+1adr(b) : 4. Let Hr(X; Y ) be the limit of the spectral sequence {Ei;jr(W*); dr}r1 . There is a product : H*(X; Y ) H*(X; Y ) -! H*(X; Y ) which in- duces 1 . In addition, the natural homomorphism : H*(X; Y ) -! eE*B(X ^B Y ) respects the products. 18 Proof. For the construction of the r's, we already have the composite r *;* E*;*r(W*) E*;*r(W*) ___________-Er (W* W*) ~=||or ?| E*;*r((X* X*) ^ (Y ^B Y )) (10) ~=||oer ?| E*;*r(B (X* X*) ^ (Y ^B Y )) where r has been defined in Proposition 3.3, or is induced by the twists X ^B Xj ' Xj ^B X and oer is the suspension isomorphism discussed above (see (9)). Next we apply Lemma 3.4 with U* := X* X* and V := X ^B X. This yields a negative 2-filtration Z* and two morphisms of spectral sequences f* *;* E*;*r(B U* ^ (Y ^B Y )) ______-Er (Z* ^ (Y ^B Y )) 6| g* | (11) | E*;*r(B V*(2) ^ (Y ^B Y )) : We claim that g* is an isomorphism for any r 2. This can be checked as in Lemma 3.2, using property (8) and the construction of Z* (see Lemma 3.4). To obtain the pairing r, we compose the diagrams (10) and (11) with the sequence oe-1 *;* E*;*r(B V*(2) ^ (Y ^B Y )) ____-Er (V*(2) ^ (Y ^B Y )) * || ?| E*;*r(X*(2) ^ Y ) (12) ~=|| ?| E*;*r(W*) where oe-1 is the inverse of the suspension isomorphism (9), * is induced by the diagonals X - ! X ^B X and Y -! Y ^B Y and the last arrow is the isomorphism of Lemma 3.2. Even though suspensions appear in the construction, we note that the r's are bigraded morphisms. We now go through the claimed properties of the pairings. The first one follows from the definition of the internal product of Tor *E*(B)(-; -)(see [21, page 65]). The next two properties are consequences of Proposition 3.3. 19 Let us deal with the multiplicative structure of H*(X; Y ) = H*(W* ) . First we proceed as in Proposition 3.3 to define morphisms Di;j1(W*) Dp;q1(W*) -! Di+p;j+q1(W* W*): The direct limit of these morphisms yields a pairing Hr(W* ) Hs(W* ) -! Hr+s(W* W* ): To obtain the desired product on H*(W* ) , we compose this pairing with a sequence of morphisms following the same pattern as in diagrams (10), (11) and (12). Of course one needs to check that g* : H*(B V*(2) ^ (Y ^B Y )) ) -! H*(Z* ^ (Y ^B Y ))) is an isomorphism, but this is true because the corresponding spectral se- quences are isomorphic (see the claim after diagram (11)). The naturality of these constructions implies that is multiplicative and induces 1 . 4 An example of convergence Let B be a pointed space and p : EB - ! B be the path space fibration. The adjoint of the identity of B will be denoted e : B -! B . As- sume that E*(B )~= (1; : :;:n) with i 2 Eodd(B) for i = 1; : :;:n. An easy calculation with the Rothenberg-Steenrod spectral sequence implies that E*(B) ~=E* [[ae1; : :;:aen]] where aei 2 Eeven(B) is chosen so that e*(aei) = * *oei for i = 1; : :;:n. Theorem 4.1 Let B be a connected and pointed space, p : EB - ! B the path space fibration and assume that E*(B ) ~= (1; : :;:n) with i 2 Eodd(B) . Then the Eilenberg-Moore spectral sequence of the pull-back dia- gram B ____-EB | | | | | | ?| ?| pt _____-B converges strongly to E*(B ). 20 Following the notation of Section 1, we write {Xi -! Xei- ! Xi-1}i0 for the cobar resolution of X0 = EB+ and we set Wi := Xi^B pt+ ; (i 0) : The first steps of this construction are summarized in the following com- mutative diagram (where the vertical maps are induced by the inclusion pt+ ,! B+ = (S0; *)B ): i0 W0 ____-EB+ ___-W-1 ___-B W0 | | | | j0|| | j1|| ||j0 ?| ?| ?| ?| X0 _____-eX0____-X-1 ___-B X0 : OE0 By definition eE*B(W0 )~=E*(B )and the contractibility of EB implies that i*0 * the composite Ee*(B ) ,! E*(B ) ~= eE*B(B W0 )-! eEB(W-1 )is an isomorphism. By abuse of notation, we will identify the generators oei 2 eE*(B )to their images in eE*B(W-1. )A similar argument shows that eE*B(X-1 ) ~= eE*(B); here also we will identify the aei's with their images in eE*B(X-1.) Lemma 4.2 With the notation and identifications above, the morphism induced in Ee*B-cohomology by the map j1 : W-1 -! X-1 satisfies: j*1(aei) = -oei, for i = 1; : :;:n. Proof. Let ' : B -! B be the adjoint of the map inv : B -! B which sends a loop ff to its inverse ff-1. We leave as an exercise to check that, in E*-cohomology, '*(aei) = -oei, for i = 1; : :;:n. We will construct two maps (in T op*) F : X-1=s(B) - ! B and f : W-1=s(B) - ! B making the following diagram commutative: f W-1 ___-W-1=s(B) ___-B | | j-1|| || ||' (13) ?| ?| |? X-1 ___-X-1=s(B) ____-B : F The space X-1 is by definition the cofiber of OE0: X0 = EB+ - ! eX0= X0=s(B) x B : 21 ` More precisely, X-1 = CB (X0) Xe0=(ff; 0) ~ OE0(ff), where CB (X0) denotes the reduced cone over B of X0 (see [18] for its definition). We construct a map X-1 -! B by sending ae ff(1 - t) ifff 2 EB (ff; t) 7! ff ifff 2 B ([x]; b)7! b if([x]; b) 2 eX0: We leave as an exercise to check that this map is well defined and induces F : X-1=s(B) - ! B . It is also easily checked that the composite a F (EB x B) pt = eX0=s(B) -! X-1=s(B) -! B is the projection onto the second factor. This shows that F *(aei) = aei for i = 1; : :;:n. Similarly W-1`is the cofiber of the map W0 = B + - ! EB+ . Thus W-1 = CB (W0) EB+ =(ff; 0) ~ ff. We construct a map W-1 - ! B by sending ae [(ff; t)] ifff 2 B (ff; t)7! [(ffl0; 0)]ifff 2 B ff 7! [(ffl0; 0)]ifff 2 EB+ ; where ffl0 2 EB+ stands for the trivial loop. It is easily checked that this map is well defined and induces f : W-1=s(B) - ! B . We also leave as an exercice that the following diagram commutes up to homotopy: i0 W-1=s(B) _____-B W0=s(B) | | f|| |= ?| a ?| B oe______(B pt) pr This implies that f*(oei) = oei for i = 1; : :;:n. The commutativity of the diagram (13) is straightforward and implies the lemma. To proceed further we need to consider one more step in the cobar res- olution. More precisely, we will study the inclusions W-1 ,! B W0 and W-2 ,! B W-1. For simplicity, we will use the following identifications eE*B(B W-1=W-2 ~)=eE*B(B eX-1^B pt+ )~=eE*(B ~)=eE*-1(B): The second isomorphism follows from Lemma 1.2 and the others are obvious. 22 Lemma 4.3 Let q1 be the projection B W-1 -! B W-1=W-2 . With the identifications above, we have q*1(aei) = -oei ; fori = 1; : :;:n. Proof. We consider the following commutative diagram in (T op=B)*, ob- tained from the natural map X-2 ,! B X-1: q1 B W-1 ___-B W-1=W-2 | j1|| ||j2 ?| q2 ?| B X-1 ___-B X-1=X-2 : With our usual identifications, we have eE*B(B X-1=X-2 )~=eE*B(B eX-1~)=eE*-1B(Xe-1)~=eE*(B) E*(B) : N The two homomorphisms j*2and q*2send aei 1 to aei. We note thatNthese homomorphisms behave differently, for example on the elements aei aej. Lemma 4.2 and the commutative diagram above now implies the assertion. Proof (of Theorem 4.1). We will show, as claimed, that the map : H*(EB+ ; pt+)- ! eE*B(W0 ) is a ring isomorphism. In our situation, the E2-term of the Eilenberg-Moore spectral sequence is isomorphic to Tor *E*(B)(E*(pt); E*(pt))~=(y1; : :;:yn) ; the argument is standard and involves the Koszul resolution (see [21]). For dimensional reasons, the generators yi are permanent cycles. The multi- plicative properties of the spectral sequence imply that it collapses; that is, E2 ~=E1 . Hence, H*(EB+ ; pt+)is a free E*(pt)-module of rank 2n. We next consider the diagram * i* eE*B(W0 )__i0-eE*+1B(W-1__)___________1-eE*+2B(W-2__)_- : : : I@@q*1 @ eE*B(B W-1=W-2 ) Since Im(q*1) = Ker(i*1), Lemma 4.3 implies that the image of the generators i 2 eE*B(W0 )~=E*(B )are zero in the obstruction group G*(EB+ ; pt+)= lim-!eE*B(Wi.)Therefore the generators i lie in the image of , showing the surjectivity of the latter. A dimension count now implies the claim. 23 Theorem 4.4 Let B be a connected and pointed space such that E*(B )~= (1; : :;:n), with i 2 Eodd(B) . Let X, Y in (T op=B)* and assume that the projection p : X -! B is a fibration. Then the Eilenberg-Moore spectral sequence {E*;*r(X; Y ); dr}r1 converges strongly to eE*B(X ^B Y .) Proof. Recall that G*(- ; -) denotes the obstruction to the "good beha- viour" of the Eilenberg-Moore spectral sequence (see Section 2). By The- orem 4.1, we have G*(EB+ ; pt+) = 0, where p : EB - ! B is the path space fibration. It follows from Proposition 2.2 that G*(EB+ ; X) is also trivial. The argument of Hodgkin (see pages 40-41 of [11]) shows that 0 = G*(EB+ ; X) = G*(X; EB+ ). Since p : X -! B is a fibration, its ho- motopy fiber is homotopy equivalent to the fiber over the base point; hence G*(X; EB+ ) = G*(X; pt+) = 0. We invoke one more time Proposition 2.2 to obtain G*(X; Y )= 0 and this concludes the proof. Remark. If E*(B )is not an exterior algebra, the spectral sequence may not converge to the desired target. For instance, let E*(- ) = K*(- ; IF2)be the mod 2 complex K-theory and B = BSO(3), the classifying space of the Lie group SO(3). As well known, we have K*(B; IF2) ~= IF2[[ae]] and K*(B ; IF2)~=IF2[]=(4) : For the path space fibration B -! EB -! B, The limit of the Eilenberg- Moore spectral sequence is H*(EB+ ; pt+)~= (y), which is obvously different from K*(B ; IF2). 5 Applications In this section, we will use our main theorem to describe the K-theory of certain spaces associated to p-compact groups. The basic references for the theory of p-compact groups are [7] and [15]; we refer to these papers for the relevant facts about these objects. Throughout this section, p is a fixed prime and R denotes either the field IFp of order p or the ring ZZ^pof p-adic integers. The complex K-theory with coefficient R will be denoted K*(-; R). Our first result provides a large class of p-compact groups satisfying the hypothesis of the third point in the main theorem. The result might be well known to the experts, but we haven't found any explicit reference. Theorem 5.1 Let X be a connected p-compact group. Then K*(X; R) is an exterior algebra on odd degrees generators if and only if ss1(X) is torsion free. 24 Proof. A Bockstein spectral sequence argument shows that K*(X; IFp) is an exterior algebra on odd degrees generators if and only K*(X; ZZ^p) is an exterior algebra, if and only if K*(X; ZZ^p)is torsion free. This reduces our problem to the case R = ZZ^p. Observe that ss1(X) is a finitely generated ZZ^p-module, hence ss1(X) ~= ZZ^pr ss where ss is a finite abelian p-group. We combine the arguments of Theorem 4.3 (page 63) in [14] and corollary 3.3 of [15] to show that X is homotopy equivalent (only as a space) to Y x (S1^p)r, where Y is a connected p-compact group with ss1(Y ) ~=ss. If ss1(X) is torsion free, then Y is a 1-connected p-compact group and we can invoke a theorem of Kane and Lin (see [12, Theorem 1.2]) to conclude that K*(X; ZZ^p) is an exterior algebra. The argument for the converse is implicit in [10]. Suppose that ss is non trivial and choose a cyclic subgroup i : ZZ=p ,! ss. Let Y < 1 > be the universal cover of Y and M(p) = S1 [p e2 be the 2-skeleton of BZZ=p. Consider the pullback diagram Y < 1 > = Y < 1 > = Y < 1 > | | | | | | | | | ?| ?| ?| Y (p) ______-Y 0________-Y | | eq|| || ||q ?| j ?| Bi ?| M(p) ____-BZZ=p _____-Bss : Since Y < 1 > is 2-connected, obstruction theory tells us that the fibration eqis trivial, hence eqinduces an injection in K-theory. As well known, the map Bi induces a surjection in K-theory; an easy computation shows that the same is true for the map j. All these facts imply that there exists a class 6= 0 in the image of q* : K*(Bss; ZZ^p)-! K*(Y ; ZZ^p). A Chern character argument shows that is a non trivial torsion class in K*(Y ; ZZ^p)and this implies that K*(X; ZZ^p)has non trivial torsion. Proposition 5.2 Let X be a connected p-compact group such that ss1(X) is torsion free and let i : T - ! X be a maximal torus. The ring homomorphism Bi* : K*(BX; R) -! K*(BT ; R) is injective and makes K*(BT ; R) into a free and finitely generated K*(BX; R)- module. Proof. The arguments of the proof of Theorem 2.7 in [12] are valid in this more general situation. The injectivity is due to the equality of the Krull dimensions. 25 Everything is now in place for the main result of this section. Theorem 5.3 Let X be a connected p-compact group such that ss1(X) is torsion free. Let i : T - ! X be a maximal torus and X=T the associated homogeneous space; that is, X=T is the homotopy fibre of Bi : BT -! BX : Then the inclusion X=T ,! BT induces an isomorphism K*(X=T ; R) ~=K*(BT ; R) K*(BX;R) K*(pt; R) ; where the K*(BX; R)-module structure on K*(BT ; R) (respectively K*(pt; R)) is given by the induced map Bi* (respectively the augmentation map). Proof. Let us first deal with the case R = IFp. We may and we will assume that the map Bi : BT -! BX is a fibration. Thanks to Theorem 5.1, we can apply our main theorem to the pullback diagram X=T ___-BT | | | |Bi | | ?| ?| pt ____-BX : Consequently, there is a strongly convergent Eilenberg-Moore spectral se- quence Ei;*2= T oriK*(BX ; IF (K*(BT ; IFp); K*(pt; IFp)) ) K*(X=T ; IFp): p) Proposition 5.2 above implies that the spectral sequence is trivial, i.e. E*;*2= E0;*2= E*;*1and the claim follows. The case R = ZZ^pis a consequence of the preceding one, the universal coefficients theorem and Nakayama's lemma. For a general p-compact group and even if the Eilenberg-Moore spectral sequence does not behave as expected, we still have the following qualitative result. Corollary 5.4 Let X be a connected p-compact group, i : T - ! X a maximal torus and W the corresponding Weyl group. Then K1(X=T ; R) = 0 and K0(X=T ; R) is a free R-module of rank |W |. Proof. Let X < 1 > be the universal cover of X. As we have seen above, X < 1 > is a p-compact group. In the proof of corollary 5.6 of [15], it is shown that X=T is homotopy equivalent to X < 1 > =S, where S - ! X < 1 > is maximal torus for X < 1 >. Since the latter is 1-connected, Theorem 5.3 26 applies. We invoke Proposition 9.5 of [7] to obtain the assertion about the rank. One of the main conjecture in the theory of p-compact groups states that H*(X=T ; ZZ^p)is torsion free and concentrated in even degrees. The inter- ested reader is refered to [16] for some partial results about this conjecture. The corollary above can be viewed as a positive solution of its K-theoretical version. As a second application, we will now give a slightly different proof of the main result in [12]. With this new method, we obtain an analoguous result for mod p K-theory. Even in the case of Lie groups, we are not aware of this mod p K-theory statement in the litterature. Corollary 5.5 Let X be a connected p-compact group, i : T - ! X a maximal torus and W the corresponding Weyl group. The map Bi induces a ring isomorphism K*(BX; R) ~=K*(BT ; R)W : Proof. By proceeding as in Section 3 of [12], we may and we will assume that X is 1-connected. Theorem 5.1 and a Rothenberg-Steenrod spectral sequence argument imply that K*(BX; R) ~= R[[ae1; : :;:aen]] with aei 2 K0(BX; R) and n equal to the rank of X. Similarly, K*(BT ; R) ~= R[[o1; : :;:on]] with oi 2 K0(BT ; R). For simplicity we set SX = R[[ae1; : :;:aen]]; ST = R[[o1; : :;:on]] and we will identify SX with its image in ST (this is justified because Bi* is injective by Proposition 5.2). By construction SX is contained in the ring of invariants SWT. To show the equality, we consider the following diagram ST ____-F rac(ST ) 6| 6| | | [| [| SWT ___-F rac(SWT) 6| 6| | | [| [| SX ___-F rac(SX ) where F rac(-) stands for the fractions field. By Proposition 5.2 and Corol- lary 5.4, ST is a free SX -module of rank |W |; it follows that F rac(SX ) ,! F rac(ST ) is a field extension of degree |W |. By Galois theory, F rac(SWT) = 27 F rac(ST )W ,! F rac(ST ) is a field extension of degree |W |. As consequence, the fields F rac(SX ) and F rac(SWT) coincide. Since ST is integrally closed (it is a power series ring over R) and the ring extension SX ,! ST is integral, we obtain that SX = SWT. Let us close this section by describing how our results extend to Morava K-theories. For n 1, K(n)*(-) denotes the n-th Morava K-theory, its coefficient ring is the graded field IFp[vn; v-1n] with |vn| = 2(pn - 1). Let X be a connected p-compact group, i : T - ! X a maximal torus and W the corresponding Weyl group. If K(n)*(X) is an exterior algebra on odd degrees generators, then the preceding arguments apply and we have: N 1. K(n)*(X=T ) ~=K(n)*(BT ) K(n)*(BX)K(n)*(pt). 2. K(n)*(BX) ~=K(n)*(BT )W . These statements naturally give rise to the following question: For each integer n 1, find all the p-compact groups X such that K(n)*(X) is an exterior algebra. As well-known, K(n)*(X) is an exterior algebra when H*(X; ZZ^p)is torsion free. Hence we recover Theorem 3.1 of [20]. In contrast to the paper just quoted, we can treat many spaces with torsion. For instance, our Theorem 5.1 provides a complete answer to the question when n = 1. More interestingly, Maria Santos (private communication) has observed that K(2)*(DI(4)) is an exterior algebra; here DI(4) is the exotic 2-compact group constructed by Dwyer and Wilkerson [6]. Are there any other examples of this type? 6 Appendix Let R be a graded ring with 1 and denote by Mod(R) the category of graded R-modules, where the morphisms are R-modules homomorphisms of degree 0. This category is abelian and possesses arbitrary direct and inverse limits (perform all the relevant contructions degreewise). If R is also commutative, then the graded tensor product yields a biadditive functor which is associat- ive, commutative and has R as a unit (up to coherence). Moreover, for any N 2 Mod(R), the functor - R N : Mod(R) -! Mod(R) ; M 7! M R N is right exact. In our applications, we will be dealing with particular graded rings and special subcategories of their modules categories. In the sequel, all inverse systems and limits are taken over direct sets. 28 A profinite graded ring is an inverse limit of graded rings of finite length (i.e noetheriean and artinian). We emphasize that, according to this defini- tion, every graded field is profinite; recall that a graded field is a graded r* *ing whose non zero homogeneous elements are all invertible. If R = lim -Rffis a profinite ring and ssff: R -! Rff are the canonical projections, the family of graded ideals {Ker(ssff)} equipp R with a topology which is complete, Hausdorff and compatible with the graded ring structure. Example 1 Let E*(- ) be a multiplicative (unreduced) cohomology theory such that E*(pt) is a graded field. In other words, all graded E*(pt)-modules are free. Given a CW-complex X, let {Xff} be the direct system of all finite CW-subcomplexes of X. Then we have ([3, Theorem 4.14]) E*(X) ~= lim -E*(Xff); the corresponding topology will be called the profinite topology of E*(X) . Since the E*(Xff) are finitely generated free E*(pt)-modules, they are rings of finite length; hence E*(X) is a profinite graded ring. Let R be a profinite graded ring. We consider the full subcategory F(R) of Mod(R) consisting of the objects M which are discrete topological R- modules and have finite length. Since the discrete topology is the only linear topology that an R-module of finite length can carry, the morphisms of F(R) are automatically continuous. A profinite R-module is an inverse limit of of objects in F(R). Thus it car- ries a natural topology which makes it into a complete Hausdorff topological R-modules. Let Modprof(R) be the subcategory of Mod(R) whose objects are profinite R-modules and whose morphisms are continuous R-module ho- momorphisms of degree 0. Example 2 E*(- ) is as in example 1 above. Let f : X -! B be a map of CW-complexes. By the CW-approximation theorem, the induced map E*(f) : E*(B) - ! E*(X) is a continuous ring homomorphism (with respect to the profinite topologies). It follows that E*(f) induces a profinite E*(B) - module structure on E*(X) . p Example 3 E*(- )is as above and B -s! X -! B is a sectionned space over B. Fix a finite subcomplex Xffof X, set Bff= s-1(s(B) \ Xff) and denote the image of Xffin X=s(B) by (X=s(B))ff. In the commutative diagram 29 s B _____- X ______-X=s(B) 6|| |6| 6| | | | [| s [| [| Bff ____-Xff ____-(X=s(B))ff the rows are cofibrations. Since p O s = id, the long exact sequences in E*-cohomology of these cofibrations reduce to the following commutative diagrams with exact rows: s* * 0 ______-eE*(X=s(B) )____-E*(X) ____-E (B) _____-0 | | | | | | (14) |? ?| * ?| s * 0 _____-eE*((X=s(B))ff )__-E*(Xff)___-E (Bff)____-0 : It follows that E*(X) ~= eE*(X=s(B) ) E*(B) as profinite E*(B) -modules, showing that eE*B(X):= eE*(X=s(B) )is a profinite E*(B) -module with respect to the topology induced by E*(X) . Using diagram (14), one checks that this topology is the same as the profinite topology. Hence we have shown that eE*B(X)is always a profinite E*(B) -module with respect to the profinite topology. Theorem 6.1 Let R be a profinite graded ring. i) The category Modprof(R) of profinite R-modules is abelian. It has enough projective objects and exact inverse limits. ii) Every inverse limit of projective objects of Modprof(R) is projective. The profinite R-module R is projective. Proof. As easily checked F(R) is abelian, artinian (i.e, every descending chain of subobjects of any object of F(R) stabilizes) and equivalent to a small category. Let P ro(F(R)) be the category of inverse systems in F(R) (see [19, page 21] for the definition). By [17] and [9, page 356]), P ro(F(R)) is an abelian category with enough projective objects and exact inverse lim- its. Moreover every inverse system is isomorphic to a strict one (i.e, whose transition morphisms are epimorphisms). Let us now consider the functor : P ro(F(R)) - ! Modprof(R) ; (Mff)ff2I7- ! lim -Mff: 30 The argument of Section 2.6 in [19] are easily adapted to our situation to show that is an equivalence of categories. Consequently Modprof(R) enjoys all the properties of P ro(F(R)) mentionned above, and we are done with the first point of the theorem. For the second part, we note that both P ro(F(R)) and Modprof(R) are proartinian in the sense of [4, page 563]. The first assertion follows from Corollaire 3.4 (page 567) in [4]. Finally R is projective because every morph- ism from R into a profinite R-module M is of the form r 7! r . m for some m 2 M. For the rest of this section, R denotes a profinite graded ring which is commutative. Our next aim is to study the topological tensor product in Modprof(R) . We start with Lemma 6.2 If M and M0 are in F(R), so is M R M0. Proof. Since M is finitely generated, its annihilator Ann(M) = {r 2 R; r . m = 0; 8m 2 M} is an open ideal in R (it is the intersection of the annihilators of the members of a finite generating set of M). Similarly Ann(M0) is an open ideal of R. We write R = lim -Rff, with each projection ssff: R -! Rff surjective (this is always possible).Then there exists ff such that Ker(ssff) Ann(M) \ Ann(M0). Consequently M, M0 and M R M0 can be regarded as Rff-modules. These new structures are strongly related to the former since any subset of M (respectively M0, M R M0) is a R- submodule if and only if it is a Rff-submodule. We observe now that Rffis a ring of finite length and M R M0 is a finitely generated Rff-module; this implies that M R M0 is a R-module of finite length, that is, an object of F(R). Let M = lim -Mffand N = lim -Nfibe two modules in Modprof(R) . Their completed tensor product is defined as M bR N = lim -(MffR Nfi) : The preceding lemma insures that M bR N lies in Modprof(R) . It can be shown that M bR N is the completion of the ordinary tensor product M N with respect to a certain filtration (see [3, page 603]). This last assertion is very useful in proving that M bR N = M R N when N is a finitely generated R-module. 31 Theorem 6.3 Let N be a profinite R-module. i) The functor - bRN : Modprof(R) -! 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