THE WEISS DERIVATIVES OF BO (-) AND BU (-) GREG ARONE March 7, 1999 To Mark Mahowald with deepest gratitude and admiration on the occasion of his birthday Contents 0. Introduction 2 1. The derivatives of BO(-) and BU(-) 8 2. A partially equivariant description of the homotopy type of the derivatives 9 3. Cohomology of the derivatives and their homotopy orbits 10 4. Homology approximations of the layers 14 5. Proof of Theorem 1.1 20 References 32 Abstract. We study Michael Weiss' "orthogonal tower" of the functors V 7! BO (V ) and V 7! BU(V ). We describe the Weiss derivatives of these functors and calculate the homology of the layers. The orthogonal tower studied here is related to the Goodwillie tower of the identity functor * *in various ways. We find that many of the results of Arone-Mahowald and Arone-Dwyer on the Goodwillie derivatives of the identity have interesti* *ng analogues in the context of the Weiss tower, but there are new things to be learned from it. ____________ 1991 Mathematics Subject Classification. 55P99. Key words and phrases. Orthogonal calculus. Partially supported by the NSF 1 2 GREG ARONE 0. Introduction Let F be R or C. Let U be an infinite-dimensional vector space over F with a positive-definite inner product. Let J be the category of finite-dimensional ve* *ctor subspaces of U with morphisms being linear maps respecting the inner product. M. Weiss provided [W95 ] a general framework for studying continuous functors from J to Spaces* (the category of spaces with a non-degenerate basepoint). One of the main constructions of Weiss associates a "Taylor tower" of sorts to a given continuous functor F : J !Spaces*. In more detail, the Taylor tower of F is a tower of fibrations of functors . .!.TnF ! Tn-1F ! . .!.T0F where TnF is, in a certain reasonable sense, a polynomial functor of degree n. * *More- over, for each n there is a natural transformation F ! TnF , which is in a suit* *able sense the best possible approximation of F by a polynomial functor of degree n. So, TnF is the n-th Taylor polynomial of F . T0F is the constant approximation of F , and the expansion is at infinity, so in fact T0F (V ) = hocolimk!1 F (Rk* *). For n > 0, the functor hofiber{TnF ! Tn-1F } should be thought of as the difference between the n-th and the n - 1-th Taylor approximations, in other words it is the n-th homogeneous term in the Taylor expansion of F . An important result of [W95 ] describes the general form of homogeneous functors. It turns out that ho* *mo- geneous functors of degree n are classified (up to natural homotopy equivalence) by spectra with an action of Aut (n) := Aut(Fn). Here we are using notation that allows us to include both real and complex vector spaces into the discussion. T* *hus Aut(n) is O (n) or U (n) depending on whether we are over the reals or over the complex numbers. A homogeneous functor of degree n has, up to natural homotopy equivalence, the form V 7! 1 CCCn^ SnVh Aut(n) where CCCnis some spectrum with an action of Aut (n) and nV stands for V Fn. Notice that SnV has a natural action of Aut(n). A subscript of the form hG means the homotopy orbits, in the based sense, with respect to an action of the group* * G. When the homogeneous functor is fiber{TnF ! Tn-1F } for some functor F , we say that CCCn, as a spectrum with an action of Aut (n), is the n-th derivative * *of F . It should be mentioned here that Weiss' theory has an obvious precursor in Good- willie's calculus of homotopy functors [G90 , G92 , G96 ]. In this paper we wil* *l use the term "Taylor tower" to mean either the Weiss tower of a functor of J , or t* *he Goodwillie tower of a homotopy functor of spaces. It should be clear from the context which one is meant. Here are a few examples of functors to which one can profitably apply Weiss' th* *eory. The examples are listed, roughly speaking, in order of increasing difficulty: * *(1) V (SV ^ X), (2) BAut (V ), (3) Emb(M x V; N x V ), (4) BTOP(V ). Here V is a generic object of J and the functors are functors of V . In the first examp* *le X is a fixed based space, SV is the one-point compactification of V and V Y WEISS DERIVATIVES 3 denotes (as usual) the space of continuous based maps from SV to Y . In the second example BAut (V ) is the classifying space of the orthogonal or unitary * *group of automorphisms of V . In the third example M and N are fixed (topological, smooth, etc.) manifolds with the dimension of M smaller than the dimension of N , and Emb(-; -) stands for the space of (topological, smooth, etc.) embedding* *s. In the last example Top(V ) is the group of homeomorphisms from V to itself. The first one of the examples above is purely homotopy-theoretic. It turns out that its Weiss tower is essentially equivalent to the Goodwillie tower of the i* *dentity functor, which was studied extensively by B. Johnson and by the author (with Mahowald) [J95, AM99 ]. The last two examples are much more geometric, and it is largely this kind of example that motivated Goodwillie's philosophy of calcu* *lus in the first place. The Taylor towers of these functors are not yet well unders* *tood. In this paper we study the Weiss tower of the functor V 7! BAut (V ). It is, * *in several ways, an intermediate between the purely homotopy-theoretic example and the geometric examples mentioned above. It is our hope (and part of our motivat* *ion) that understanding this example will bring us a little step closer to understan* *ding the more difficult geometric examples such as mentioned above. The 0-th term of our tower is the constant functor V 7! BAut := colimnBAut (Fn). In other words, it is BO or BU . Thus the Weiss tower is a tower of fibrations starting with BO (or BU ) and converging to BO (V ) (or BU (V )). Notice that the homotopy groups of the 0-th term are known, so the 0-th term of the tower is "better understood" than the inverse limit of the tower. For n > 0, the n-th layer (i.e., the homotopy fiber of the map Tn BAut(V ) ! Tn-1BAut (V )) is, up * *to a natural weak equivalence, of the form V 7! 1 (MFn^ SnV )h Aut(n)where MFnis some spectrum with an action of Aut(n). The spectrum MFnis the n-th derivative of the functor V 7! BAut (V ). Our main goal in this paper is to describe the s* *pectra MFn(together with the action of Aut (n)) for the real and complex cases, and to analyze the cohomology of the homotopy orbit spectra (MFn^ SnV )h Aut(n). We wi* *ll see that there are interesting connections between the Weiss derivatives of BAu* *t (V ) and the Goodwillie derivatives of the identity. For instance, in the complex ca* *se one can describe the transition between the results in [AM99 , AD99 ] on the Goodw* *illie tower of the identity and our present results on the Weiss tower of BU (-) in t* *he following table ______________________________________________________________ |__Goodwillie_tower_of_the_identity__|Weiss_tower_of_BU(V_)____ | |_______symmetric_group_n_______|_____unitary_group_U(n)______ | | poset of partitions of a finite set |poset of direct-sum | |_______________________________|decompositions_of_a_vector_space_| | the transitive elementarykabelian | Oliver'sUp-stubbornk | |_____subgroup_(Z=pZ)__of_pk____|____subgroup_pk_of_U(p_)_____ | | Tits building for GL k(Fp) | Tits building for the | |_______________________________|__symplectic_group_Sp2k(Fp)__ | 4 GREG ARONE We will now outline the method by which we are going to obtain our description * *of the derivatives of BAut (-). To begin with, the derivatives of the functor V 7! BAut (V ) are the same as of the functor V 7! fiber(BAut (V ) ! BAut ). Indeed, taking this homotopy fiber amounts to "substracting a constant" from our functor, and thus does not change the derivatives. The homotopy fiber is, in turn, equivalent to Aut=Aut (V ) ' colimn!1Mor(Fn; Fn V ) where by Mor we mean morphisms in J . Notice that the colimit can be replaced by homotopy colimit. Thus we have to understand the derivatives of the functor V 7! Mor(W; W V ), where W is a fixed object of J and then take the (homotopy) colimit as W goes to infinity. From here on, the idea is very similar to the one in [AK98 ]. There the authors wanted to describe the Goodwillie derivatives of the identity functor, and the method was to approximate the identity functor by its Q-completion (Q is short- hand for 1 1 ), as given by the Bousfield-Kan cosimplicial space [k] 7! Qk+1(X), and then use the fact that the Goodwillie derivatives of Qk+1(-) are relatively easy to describe using the stable splitting of Q(X) (the Snaith splitting). In our present case, we start with observing that the Weiss tower of the functor V 7! Q(Mor (W; W V )) is known, in a guise. It is given by H. Miller's stable splitting of Stiefel manifolds ([Mi85 ] - see also [C86 ] for a good survey of * *these ideas). By [op. cit., theorem 1.16], there is a homotopy equivalence Y i j Q(Mor (W; W V )) ' Q SAdn ^ Mor(Fn; W )+ ^ SnVAut(n) n1 where Ad n is the adjoint representation of Aut (n). Notice that the terms in t* *he product on the right side are homogeneous functors of V , and it is quite clear* * that the right hand side describes, up to a natural homotopy equivalence, the Weiss tower of the functor on the left side. Iin particular, the Weiss tower of this * *functor splits. Taking (homotopy) colimit as the dimension of W goes to infinity one fi* *nds that Y i j Q(Aut =Aut (V )) ' Q SAdn ^ SnVh Aut(n) n1 Here we use that colimW!1 Mor(Fn; W ) is a contractible, Aut (n)-free, space. Our next task is to pass from understanding the Taylor tower of Q(Mor (W; W V )) to understanding the Taylor tower of Mor (W; W V ). The trick is to consider t* *he Bousfield-Kan cosimplicial space [k] 7! Qk+1(Mor (W; W V )), whose total space is equivalent to Mor (W; W V ), and to observe that once we know the Weiss tower of the functor Q(Mor (W; W V )), it is easy to calculate the Weiss tower of Qk(Mor (W; W V )), using Miller's and Snaith's splittings. Indeed, after do* *ing some combinatorics (and Spanier-Whitehead duality), one finds that for k > 1 Y i j Qk(colimMor (W; W V )) ' 1 Map * Nk-1Ln+ ; 1 SAdn ^ SnV h Aut(n) n1 WEISS DERIVATIVES 5 where Nk-1Ln is the space of k - 1 chains in the category of direct-sum decom- positions of Fn. This is the (topological) category of decompositions of Fn as* * a direct sum of mutually orthogonal subspaces with morphisms being refinements of decompositions (Precise definitions will be given in section 1). Further inves* *tiga- tion shows that the coface maps in the Bousfield-Kan cosimplicial space are dua* *l, at least up to homotopy, to the face maps in the simplicial space NoLn, and this suggests the following theorem Theorem 1.1 Let n 1. The n-th derivative of the functor V 7! BAut (V ) is the spectrum Map * Ln; 1 SAdn where Ln is the unreduced suspension of the geometric realization of the catego* *ry of non-trivial direct-sum decompositions of Fn. We give the definitions required for theorem 1.1 in section 1. The proof of the theorem is relegated to section 5. The detailed proof is a bit long and techni* *cal. This is so because (just as in [AK98 ]), there are some technical difficulties * *with the plan of proving theorem 1.1 that we just outlined. The main problem is that the known models for Miller's and Snaith's splittings are not functorial enough for* * our purposes. More precisely, they do not interact well enough with the coface maps in the Bousfield-Kan cosimplicial space. In section 5 we will describe a new mo* *del for the Weiss tower of QkMor (W; W V ), one that has the required naturality properties. Again, our construction is very similar to the one in [AK98 ]. Its * *main features are that it is constructed in terms of homotopy inverse limits (fixed * *point spaces) rather than homotopy direct limits (orbit spaces) and that it only prov* *ides a tower of fibrations, which does not obviously split. Of course we will know tha* *t our tower of fibrations splits, for example because of [Mi85 ], but it will not be * *obvious from the construction. Another potential problem is convergence. Weiss' derivatives commute up to natu* *ral homotopy equivalence with finite homotopy limits of functors, but do not, in ge* *neral, commute with infinite homotopy limits. Therefore, they do not necessarily commu* *te with taking the total space of a cosimplicial space. However, in our case it is* * not hard to show that the Weiss derivatives behave well with respect to the Bousfield-Kan cosimplicial space [k] 7! Qk+1(Mor (W; W V )). This so basically because in the tower of fibrations associated with the Bousfield-Kan cosimplicial space, the d* *egree of the bottom non-trivial layer of the Weiss tower of the fibers starts at degr* *ee that goes to infinity as one goes up in the Bousfield-Kan tower. The argument is eas* *y, and is more or less given in the appendix of [AK98 ]. We will not say anything * *more on this issue. Obviously, the role of the poset of direct-sum decompositions of Fn in the Weiss tower of BAut (V ) is analogous to the role of the poset of partitions of a fin* *ite set in the Goodwillie tower of the identity. There are differences, however. One impor* *tant difference is that unlike the poset of partitions of a finite set, the poset of* * direct-sum 6 GREG ARONE decompositions is not a lattice. It does not have a greatest lower bound operat* *ion. It does have a well-defined least upper bound operation and in particular it ha* *s a final object (the decomposition with one component), but no initial object. Ind* *eed, an initial object would be a decomposition of Fn into a direct sum of lines, but such a decomposition is not unique. It follows that there is no reason to expe* *ct Ln to have an easily describable homotopy type, even non-equivariantly. This is in contrast with Kn, the realisation of the poset of partitions of a finite set* *, which non-equivariantly is equivalent to a wedge of (n-1)! spheres of the same dimens* *ion. There is, however, an efficient way to study the non-equivariant homotopy type * *of Ln, by comparing the Weiss tower with the Goodwillie tower of the identity. This idea is due to Goodwillie, and we are very grateful to him for allowing us to m* *ake use of it here. Basically, one considers the fibration sequence SV ! BO (V ) ! BO (V R) in the real case, or the fibration sequence SV ! BU (V ) ! BU (V C) in the complex case, and uses the fact that the Weiss derivatives of the fiber * *are essentially given by the Goodwillie derivatives of the identity. In section 2 w* *e follow through on this idea, and obtain the following theorem Theorem 2.1 There exists an O (n - 1)-equivariant equivalence R 1 1 0 Map *(LRn; 1 SAdn) ' Map *(S ^ Kn; S ) ^n O(n - 1)+ Similarly, in the complex case there exists a U (n - 1)-equivariant weak equiva* *lence C 1 1 n Map *(LCn; 1 SAdn) ' Map *(S ^ Kn; S ) ^n U(n - 1)+ Here n is considered a subgroup of O (n - 1) and U (n - 1) via the reduced regu* *lar representation. Kn stands for unreduced suspension of the geometric realization* * of the category of non-trivial partitions of a set with n elements. Now we can use the results of [AM99 , A98] to study the non-equivariant homoto* *py type of the Weiss derivatives. The spectra C 1 1 n Map *(LCn; 1 SAdn) ' Map *(S ^ Kn; S ) ^n U(n - 1)+ where studied in detail in [A98 ], and the real version can be analyzed similar* *ly. It turns out that in the complex case the n-th Weiss derivative vanishes unless* * n is a power of a prime. If n = pk then the n-th Weiss derivative turns out to be essentially the finite spectrum with Ak-1-free cohomology constructed by Mitche* *ll in [Mt85 ]. In particular, if n = p =prime, then the n-th derivative is the mod p Moore spectrum (with an interesting action of U (p). In the real case the n-th derivative vanishes unless n is a power of a prime or twice a power of a prime. When n = pk or n = 2pk, one can often re-express the n-th derivative in terms of a considerably smaller space. For instance, in the complex case, if n = pk one * *can WEISS DERIVATIVES 7 show that one can replace the action of pk on Kpk with the action of the affine group GL k(Fp) n Fkpon the Tits building for GL k(Fp) (section 3). In the cases when the derivatives do not vanish, we describe their cohomology groups, and sh* *ow that they have interesting freeness properties over the Steenrod algebra. In fa* *ct, as was observed in [A98 ], when the derivatives do not vanish, they are closely re* *lated to the finite spectra with Ak-free cohomology constructed by Mitchell in [Mt85 * *]. These freeness properties can be interpreted in terms of vk-periodic homotopy in the spirit of [AM99 ] (see theorems 3.1 and 3.2 for detailed statements). In view of the vanishing results of Theorem 3.1, and in view of the general sim* *ilarity of the Weiss tower to the Goodwillie tower of the identity, it is natural to lo* *ok for analogues in the present context of the results of [AD99 ] on approximating the partition poset with the Tits building. In [op. cit] the (p-completion of* * the) homotopy orbit spectrum k Map *(Kpk; 1 X^p )hpk was studied (in the case of X a sphere) by finding a mod p homology approximati* *on of Kpk and identifying it as the Tits building for GL k(Fp) (induced up from the group GL k(Fp) n Fkpto the symmetric group pk). In the present paper we would F k like to analyze the homotopy orbit spectrum Map *(LFpk; 1 SAdpk^Sp V)h Aut(pk)by finding a mod p homology approximation for LFpk. We will only do it in the comp* *lex case, which is a little simpler from this point of view. In section 4 we show t* *hat in the case n = pk, p a prime, there is an approximation for LCpkthat is very simi* *lar to the approximation of Kpk found in [AD99 ]. The role of the transitive elemen* *tary Abelian subgroup of pk is now played by the p-stubborn subgroup Upk U(pk) discovered by Oliver in [O94 ]. This is a version of the discrete Heisenberg gr* *oup. It is a central extension of the circle group by (Z=pZ)2k. One can also describ* *e the subgroup Upk U (pk) as the standard projective representation of (Z=pZ)2k. It is essentially the only faithful irreducible projective complex representation * *of an elementary abelian group. The Weyl group of Upkin U (pk) is the symplectic group Sp2k(Fp). Therefore, it is not surprising that the role that the group GL k(Fp) played in [AD99 ] is now played by the symplectic group Sp2k(Fp), and the Tits building for GL k(Fp) gets replaced by the Tits building for Sp2k(Fp). To summarize, let TSp (2k) be the T* *its building for Sp 2k(Fp) (precise definitions are given in section 4). Let Ak be* * the group N U(pk)Upk, the normalizer of Upkin U(pk). Extend the action of Sp2k(Fp) * *on TSp (2k) to an action of Ak by letting Upkact trivially. We show in section 4 t* *hat Theorem 0.1. There exists a U (pk)-equivariant map U (pk)+ ^Ak TSp(2k) ! Lpk, which is a mod p homology equivalence. This is a special case of theorem 4.6 below. It follows that the map alluded to* * in the theorem induces a mod p homotopy equivalence on homotopy orbit spectra kV 1 Ad pkV Map *(Lpk; 1 SAdpk^ Sp )hu(pk)! Map *(TSp (2k); S pk^ S )hAk 8 GREG ARONE This is our analogue of the main result of [AD99 ]. It implies, in particular,* * that the pk-thilayer of thejWeiss tower of BU(V ) is a wedge summand of the spec- kV trum 1-k SAdpk^ Sp hU . Precisely, it is the wedge summand split off by the pk symplectic Steinberg idempotent. 1. The derivatives of BO(-) and BU(-) In this section we describe the Weiss derivatives of the functor V 7! BAut (V )* *. Con- sider the vector space Fn with the standard inner product. Let LFnbe the catego* *ry of direct-sum decompositions of Fn. We will sometimes omit the superscript F fr* *om the notation when F is irrelevant or is clear from the context. Thus an object * *of Ln is an unordered collection of mutually orthogonal subspaces of Fn whose dire* *ct sum is all of Fn. We call these subspaces the components of the decomposition. There is a (unique) morphism of decompositions 1 ! 2 if every component of 1 is a subspace of some component of 2. This category is a poset and the role it plays in the Weiss tower of BAut (V ) is analogous to the role the poset of par* *titions of a finite set plays in the Goodwillie tower of the identity. Notice, however,* * that unlike the poset of partitions of a finite set, the category Ln is not a lattic* *e. It does not have a greatest lower bound operation. It does have a well-defined least up* *per bound operation and in particular it has a final object (the decomposition with* * one component), but no initial object. Indeed, an initial object would be a decompo* *si- tion of Fn into a direct sum of lines, but such a decomposition is not unique. * *Notice also that Ln is a topological category, in the sense that the simplicial nerve * *of Ln has a natural topology on it (in fact, the sets of simplices are disjoint union* *s of homogeneous manifolds). So whenever we speak about homotopy limits or colimits over Ln (or any other topological category), it is understood that these are de* *fined in such a way as to take the topology of the category into account. In particul* *ar, when we speak about the geometric realization of Ln, we take the topology of Ln into account. We introduce a certain based simplicial space Oo;n, which is essentially a susp* *ension of the nerve of Ln. Let ^1be the maximal object of Ln. Let Tk;nbe the space of non-decreasing chains (0; : :;:k) of direct-sum decompositions of Fn such that k = ^1. Define Oo;nby taking Ok;n= (Tk;n)+ and letting the face and degeneracy maps be defined as follows: di((0; : :;:k)) = (0; : :;:^i; : :;:k) fori = 0; : :;:k ( (0; : :;:k-1) ifk-1 = ^1 dk((0; : :;:k)) = basepoint otherwise si((0; : :;:k)) = (0; : :;:i; i; : :;:k) fori = 0; : :;:k It is not hard to check that Oo;n is indeed a simplicial space, and its geometr* *ic realization is equivalent to the unreduced suspension of the geometric realizat* *ion of the category Ln with the final object deleted. Let eLnbe the geometric realizat* *ion WEISS DERIVATIVES 9 of the poset of proper, non-trivial direct sum decompositions, and let Ln be the geometric realization of Oo;n. Thus Ln is the unreduced suspension of eLn. Noti* *ce that Ln has a natural action of Aut (n). We are now ready to state the main theorem of this section Theorem 1.1. Let n 1. The n-th derivative of the functor V 7! BAut (V ) is the spectrum i Fj Map * LFn; 1 SAdn F Here SAdn is the one-point compactification of the adjoint representation of Au* *t(n). Notice that the spectrum has a natural action of Aut (n). Notice that it follows from the theorem that the n-th layer in the Weiss tower * *of BAut (V ) is the infinite loop space 1 Ad nV 1 Map * Ln; S nS h Aut(n) The proof of the theorem is given in section 5. For concreteness, we will only * *prove it for F = R. The proof of the complex case is virtually identical. 2. A partially equivariant description of the homotopy type of the derivatives In this section we compare the Weiss tower of the functor BAut (V ) with the Go* *od- willie tower of the identity, and use it to obtain information about the homoto* *py type of the spectra MRnand MCn. This idea is due to Goodwillie. Let Kn be the unreduced suspension of the category of non-trivial partitions of* * the set {1; : :;:n}. Recall from [AM99 ] that the Spanier-Whitehead dual of S1 ^ K* *n is the n-th Goodwillie derivative of the identity functor. Theorem 2.1. There exists an O (n - 1)-equivariant equivalence (1) Map *(Ln; 1 SAdn) ' Map *(S1 ^ Kn; 1 S0) ^n O(n - 1)+ Similarly, in the complex case there exists a U (n - 1)-equivariant weak equiva* *lence C 1 1 n Map *(LCn; 1 SAdn) ' Map *(S ^ Kn; S ) ^n U(n - 1)+ Proof.We will only deal with the real case in detail, the complex case being ve* *ry similar. Consider the fibration sequence of functors of V SV ! BO (V ) ! BO (V R) Let Pn be the n-th Weiss derivative of the functor V 7! SV . Thus Pn is a spect* *rum with an action of O (n). The above fibration sequence induces a fibration seque* *nce of derivatives (which is also a cofibration sequence) Pn ! MRn! MRn^ Sn 10 GREG ARONE It follows easily that Pn ' MRn^O(n-1)O(n)+ . On the other hand, it is easily s* *een that Pn is the n-th Goodwillie derivative of the identity functor induced from * *the symmetric group n to the group O (n). In other words, Pn ' Map *(S1 ^ Kn; 1 S0) ^n O(n)+ Notice that the right hand side can be rewritten as Map *(S1 ^ Kn; 1 S0) ^n O (n - 1)+ ^O(n-1)O (n)+ where n is considered a subgroup of O (n - 1) via the reduced regular representation. To summarize, there exists an O (n)-equivariant weak equivalence 1 1 0 (2)MRn^O(n-1)O(n)+ ' Map *(S ^ Kn; S ) ^n O(n - 1)+^O(n-1)O(n)+ This in fact implies that there exists an O (n-1)-equivariant equivalence as cl* *aimed in (1). One way to see the implication is to observe that (2) implies that ther* *e is a weak equivalence of functors i j V 7! MRn^ S(n-1)V h O(n-1) and i j V 7! Map *(S1 ^ Kn; 1 S0) ^n O(n - 1)+ ^ S(n-1)Vh O(n-1) it follows that the n - 1-th derivatives of these functors are equivalent, and * *these, by [W95 , example 5.7] are precisely the spectra in (1). For the complex case, consider the fibration sequence of functors (where V is * *un- derstood to be a complex vector space) SV ! BU (V ) ! BU (V C) It follows, by the same reasoning as above, that there is an equivalence of U (* *n - 1)- equivariant spectra MCn' Map *(S1 ^ Kn; 1 Sn) ^n U(n - 1)+ __| | __ 3.Cohomology of the derivatives and their homotopy orbits Theorem 3.1. (a) The spectrum MCn is contractible rationally for n > 1. The spectrum MRnis contractible rationally for n > 2. (b) The spectrum MCnis contractible unless n is a power of a prime. If n = pk >* * 1 then the homology of MCpkis all p-torsion. The cohomology of MCpkis free over Ak-1 and its cohomology with mod p coefficients is isomorphic (up to a shift), * *as an Ak-1-module to Ak-1 E , where E is a free module on one generator over the exterior algebra < __ci| 1 i pk - 1; i 6= pk - pj > generated by the looped Chern classes that are not Dickson classesi(so thejdegr* *ee of __c C pkV iis 2i - 1). The cohomology of the homotopy orbit spectrum Mpk^ S h U(pk) WEISS DERIVATIVES 11 __ __ is free over Ak-1 and is isomorphic to Ak-1 P, where P is a free module on one generator over the polynomial algebra Fp[d-1 ; d0; : :;:dk-1], where |dj| = 2(p* *k - pj). (c) The spectrum MRn is contractible unless n is a power of a prime or twice a power of an odd prime. The detailed statement is different for p = 2 and p odd. First, let p = 2, n = 2k. The mod 2 cohomology of the spectrum MR2kis free over Ak-2. It is isomorphic, as an Ak-2-module, to Ak-1 E , where E is as in parti(b) ofjthe theorem. The mod 2 cohomology of the homotopy orbit spectrum kV MR2k^ S2 h O(2k)is free over Ak-2 and is isomorphic, as an Ak-2-module to __ __ Ak-1 P where P is, again, as in part (b). i k j Now let p be odd, n = pk. The spectrum MRpk^ Sp V h O(pk)is contractible if V is odd-dimensional and is mod p-equivalent to i k j Map * Kpk; 1 Sp (V -1)h pk if V is even-dimensional. Its cohomology is Ak-1-free and is isomorphic to Ak-1 Fp[d0; : :;:dk-1]. Finally, let p be odd, n = 2pk. If V is odd-dimensional, then the spectra i k j i k j Map * Kpk; 1 Sp (V )h and Map * Kpk; 1 Sp (V +1) pk hpk are mod p homotopy equivalent. The cohomology of these spectra is Ak-1-free and is isomorphic to Ak-1 Fp[d2-1; d0; : :;:dk-1]. Proof.As far as cohomology calculations go, the complex case is simpler than the real case, so we deal with it first. In fact, the spectrum Map *(S1 ^ Kn; 1 Sn)* * ^n U (n - 1)+ , alias MCn, was analyzed in [A98 ] (see especially section 2 and th* *eorem 2.2). It is shown in [op. cit.] that this spectrum is contractible unless n is * *a power of a prime, and it is contractible rationally for all n. Let n = pk. It is also* * shown in [op. cit.] that the mod p cohomology of MCpkis free over Ak-1, and in fact t* *here is an isomorphism (up to a dimension shift) of Ak-1-modules i j H* MCpk ~=Ak-1 E where E is as in the statement of the theorem. Moreover, one can easily deduce * *that for any complex vector space V the cohomology of the homotopy orbit spectrum i k j k k MCpk^ Sp V h U(pk-1)' Map *(S1 ^ Kpk; 1 Sp ^ Sp V)hpk is free over Ak-1, and in fact there is an isomorphism (up to a dimension shift depending on V ) of Ak-1-modules i k k j H * Map *(S1 ^ Kpk; 1 Sp ^ Sp V)hpk ~=Ak-1 P 12 GREG ARONE where P is a free module on one generator over the Dickson algebra, which is the polynomial algebra Fp[d0; : :;:dk-1] where |dj| = 2(pk - pj). Now consider the cofibration sequence of spectra i k j i k j i k j MCpk^ Sp V h U(pk-1)! MCpk^ Sp V h U(pk)! MCpk^ Sp (V C)h U(pk) By theorem 2.1, the first of these three spectra is equivalent to i k k j Map * S1 ^ Kpk; 1 Sp ^ Sp V It is not very hard to show, using homology calculations in the spirit of [AM99* * ], that the second map is injective on cohomology (we plan to write a detailed expositi* *on of this in a separate paper). It follows that the spectral sequence associated * *with the filtration i j i kj H * MCpk^ S0 h U(pk)- H* MCpk^ S2p h U(pk)- . . . collapses, and that i j M i k k j j H * MCpk^ S0 h U(pk)~= H* Map *(S1 ^ Kpk; 1 Sp ^ Sp C )hpk j Clearly, the right hand side is isomorphic to Ak-1 P [d-1 ], where d-1 is a g* *en- erator of dimension pk - p-1 = pk. This completes the proof of the complex cas* *e. Now we move on to the real case. Here we have to distinguish between the cases * *of p = 2 and p odd, and between the cases of V being even or odd-dimensional. Let us start with the case p = 2. In this case the main result of [AD99 ] is valid * *for all V (both even and odd dimensional). By this we mean that the spectrum 1 1 V n Map * S ^ Kn; S hn is contractible (at the prime 2) unless n is a power of 2, for all V . If n = 2* *k then the natural map T k! Kpk where T kis the unreduced suspension of the geometric realization of the category of strict, non-zero, vector subspaces of Fkp(the "T* *its building" for GL k(Fp)) induces a mod 2 equivalence for all V i k j i k j Map * S1 ^ K2k; 1 S2 V h ! Map * S1 ^ Tk; 1 S2 V k 2k h GLk(F2)nF2 The target of this map is equivalent (up to a suspension) to the image of the kV suspension spectrum 1 S2hZ=2Zkunder the action of the Steinberg idempotent of GL k(F2). It is well known that the cohomology of this spectrum is free over Ak* *-1 if V is odd-dimensional and is free over Ak-2 if V is even-dimensional. In particu* *lar, it is always Ak-2-free. In fact, this cohomology is isomorphic (up to a dimensi* *on shift) to Ak-1 F2[d0; : :;:dk-1] as a graded vector space. The isomorphism is * *an isomorphism of Ak-1-modules if V is odd-dimensional and is an isomorphism of Ak-2-modules if V is even-dimensional. WEISS DERIVATIVES 13 It follows, using the same reasoning as in [A98 ], that the spectrum 1 1 V n MRn= Map * S ^ Kn; S ^n O(n - 1)+ is contractible mod 2 unless n is a power of 2 and that there exists a mod 2 equivalence i k j i k j Map * S1 ^ K2k; 1 S2 V ^2k O (2k-1)+ ! Map * S1 ^ Tk; 1 S2 V ^AffkO(2k-1)+ where Affk = GL k(F2) n Fk2. The target of this map is, again, equivalent to t* *he kV k image of the suspension spectrum of S2 ^Z=2ZkO(2 - 1)+ under the action of the Steinberg idempotent. In fact, it is a version of the spectrum constructed* * in [Mt85 ]. Its cohomology is free over Ak-2 (in fact, it is free over Ak-1 if V * * is odd-dimensional). Usingithe same argumentjas in the complex case, one concludes kV that the cohomology of MR2k^ S2 h O(2k)is free over Ak-2, and is isomorphic, as an Ak-2-module, to Ak-1 F2[d-1 ; d0; : :;:dk-1] (note: the freeness stateme* *nt is vacuous if k - 2 < 0). It remains to deal with the odd-primary case. Let p be an odd prime, and let all spaces be completed at p and all cohomology be taken with mod p coefficients. Consider again the cofibration sequence 1 1 nV R nV R nV n Map * S ^ Kn; S hn ! Mn ^ S h O(n)! Mn ^ S ^ S h O(n) From [AM99 ] we know that the spectrum Map *(Kn; 1 SnV ) hn is contractible unless n is a power of p or else n is twice a power of p and V is even-dimensio* *nal. Let us deal with the case n = pk first. From [AD99 ] we know that if V is odd- dimensional then i k j i k j Map * S1 ^ Kpk; 1 Sp V h ' Map * S1 ^ Tk; 1 Sp V pk h Affk On the other hand, we will see below (proposition 4.9) that if V is odd-dimensi* *onal then i kVj MRpk^ Sp h O(pk) is in fact contractible. It follows that if V is even-dimensional, then there* * are homotopy equivalences i k j i k j MRpk^ Sp V h O(pk)' Map * Kpk; 1 Sp (V -1)h ' pk i k j ' Map * T k; 1 Sp (V -1)h Aff k i k j It follows that if V is even-dimensional then the cohomology of MRpk^ Sp V h O* *(pk) is free over Ak-1 and in fact is isomorphic to Ak-1Fp[d0; : :;:dk-1]. The state* *ment about MRpkitself is derived similarly. 14 GREG ARONE Now let us consider the case n = 2pk. We know from [AM99 ] that the spectrum kV Map *(S1 ^ K2pk; 1 S2p )h2pk is contractible if V is odd and is equivalent to k(2V -1) Map *(S1 ^ Kpk; 1 Sp )hpk if V is even. It follows that the map i k j i k kj MR2pk^ S2p Vh O(2pk)! MR2pk^ S2p V^ S2p h O(2pk) Is an equivalence if V is odd-dimensional. On the other hand, it is not very di* *fficult to show that the map i k j i k kj MR2pk^ S2p Vh O(2pk)! MR2pk^ S2p V^ S4p h O(2pk) is injective on cohomology (we will prove this in detail in a separate paper). * * We k(2V -1) know that the fiber of this map is equivalent to Map *(S1 ^ Kpk; 1 Sp )hpk* * . The cohomology of this spectrum is free over Ak-1, andiis isomorphicjto Ak-1 kV Fp[d0; : :;:dk-1]. It follows that the cohomology of MR2pk^ S2p h O(2pk)is * *iso- morphic to Ak-1 Fp[(d-1 )2; d0; : :;:dk-1]. __|_| As was pointed out in [AM99 ], these results have an interpretation in terms o* *f the chromatic filtration in homotopy theory. The fact that a certain layers' cohomo* *logy is free over Ak means that the vi-periodic homotopy of this layer is trivial for i k. The fact that the Weiss tower converges exponentially rather than linearly implies that the tower actually converges in vi-periodic homotopy (this is expl* *ained in [AM99 ] for the Goodwillie tower of the identity evaluated at spheres). We * *obtain the following theorem as a corollary Theorem 3.2. Let Tn BAut(V ) be the n-th Taylor approximation of BAut (V ). In the complex case, the map BU (V ) ! TpkBU (V ) is an equivalence in vi-perio* *dic homotopy for i k. In the real case, the map BO (V ) ! T2pkBO (V ) is an equiva- lence in vi-periodic homotopy for i k. 4. Homology approximations of the layers We know now that the space LCnis contractible unless n is a power of a prime. We also know that if n = pk > 1 then LCnis p-complete. In this section we construct, for every prime p and positive integer k, an equivariant mod p homol* *ogy approximation of the space LCpk, and identify it as the symplectic Tits buildin* *g. We will also prove a vanishing result for the layers of BO (-) at odd primes. This section is closely related to [AD99 ]. For most of the section, the discussion * *will be confined to the complex case, so we use the notation Ln for LCn. We need to recall from [O94 ] the definition of certain subgroups Upkof U (pk). Recall that there exists a (unique up to conjugacy) transitive elementary Abeli* *an subgroup k pk. Thus k ~=(Z=pZ)k. The subgroup Upk U(pk) is analogous WEISS DERIVATIVES 15 in many ways to the subgroup k pk. Let oe0; : :;:oek-1 be the following basis for k: ae r r r+1 oer(i) = ii+-p(p - 1)piri1; :(:;:(pp--1)p1)pr +(1;mod:p:;:)pr+1( mod * *pr+1) Let = e2ssi=p. Define matrices A0; : :;:Ak-1; B0; : :;:Bk-1 by letting Br be * *the permutation matrix corresponding to oer and by defining ae[(i-1)=pr] (Ar)ij= 0 ii=6j= j where [-] denotes greatest integer. These matrices satisfy the commutator relat* *ions [Br; Ar] = . Iand all the other commutators equal the identity. Now we are rea* *dy to define the groups Upk U(pk) Definition 4.1. Upk=< u . I; Ar; Br | u 2 S1; 0 r k - 1 > U(pk) Thus there is a central extension 0 ! S1 ! Upk! (Z=pZ)2k! 0 Notice that [Upk; Upk] = Z=pZ S1 = Z(Upk) In particular, the commutator of any two elements is in the center. It follows that the commutator defines a non-degenerate, skew-symmetric, bilinear form on ss0(Upk) = (Z=pZ)2k. In other words, the commutator endows ss0(Upk) with the structure of a symplectic vector space over Fp. It is easy to see that conjugat* *ion by elements of U (pk) preserves the symplectic structure, and in fact it is sho* *wn in [O94 ] that the Weyl group of Upkin U (pk) is the symplectic group Sp 2k(Fp). Following standard terminology from symplectic geometry, we say that a subgroup V (Z=pZ)2k is isotropic if V V ? (with respect to the symplectic inner produc* *t), coisotropic if V ? V , Lagrangian if V = V ? and symplectic if V \ V ? = {0}. In particular, V has rank at most k if it is isotropic, has rank at least k if* * it is coisotropic, has rank exactly k if it is Lagrangian, and has an even rank if it* * is symplectic. We say that a subgroup H of U (pk) is admissible if H is a pullback* * of a diagram of the form Upki (Z=pZ)2k- V where V is a coisotropic subspace of (Z=pZ)2k. Let Ek be the poset of proper ad* *mis- sible subgroups of Upk. Clearly, this is isomorphic to the poset of proper cois* *otropic subspaces of the symplectic vector space (Z=pZ)2k. Let TSp (2k) = |Ek| be the unreduced suspension of the geometric realization of Ek. Thus TSp (2k) is the symplectic Tits building. Let Ak = NU(pk)(Upk) be the normalizer of Upkin U (pk* *). Thus there is a (non-split) extension 1 ! Upk! Ak ! Sp2k(Fp) ! 1 16 GREG ARONE Clearly, there is a natural action of Ak on TSp (2k) coming from conjugation, and in fact the restriction of the action to the subgroup Upkof Ak is trivial. * *We want to show that there exists an equivariant (with respect to the group inclus* *ion Ak ,! U (pk)) map TSp (2k) ! Lpk that is in some sense a mod p homology equivalence. To do this, we use a couple of different models for Ln. The models* * are analogous to the models for the poset of partitions of a finite set considered * *in [AD99 , section 4]. They are given in terms of posets of subgroups of U (n). Given a di* *rect- sum decomposition of Cn , let H be the subgroup of U (n) consisting of elemen* *ts g that preserve in the strong sense that each component of is invariant under the action of g . H is conjugate to a group of the form U (n1) x U(n2) x . .x.* *U(nk) where n1+ . .+.nk = n. Let Qn be the (topological) poset of all such subgroups * *of U (n) and let Q0n Qn be the subposet consisting of proper subgroups of U (n) of the specified form. It is clear that the association 7! H induces an isomorph* *ism of posets Ln ~= Qn. It follows that there are homeomorphisms Len ~=|Q0n| and Ln ~=|Q0n|. There is also a bigger poset of subgroups that gives a model for Ln. To describe it, we need a new definition. Let H U(n) be a subgroup. Consider Cn as the restriction of the standard representation of U (n) to H . Write Cn = V1m1 . . .Vkmk where V1; : :;:Vk are irreducible, pairwise non-isomorphic, representations of H Definition 4.2.With notation as above, we say that H is irreducible if k = 1 and m1 = 1. We say that H is bad if k = 1 and m1 > 1 and we say that H is good if it is neither irreducible nor bad and contains the center of U (n) (the circle * *group). Remark 4.3. The group Upkis irreducible. It is a well-known fact from projective representation theory of Abelian groups that every irreducible subgroup of U (p* *k) that is an extension of the center of U (pk) by an elementary abelian group is conjugate to Upk. It also easy to see that a proper subgroup H Upk(containing the circle) is bad if and only if it is symplectic. Let Fn be the poset of all good subgroups of U (n). Clearly, Q0n Fn. Proposition 4.4. The inclusion i : Q0n,! Fn is a homotopy equivalence. Proof.Let G 2 Fn. Let V1m1. .V.mkkbe the restriction of the standard represen- tation of U (n) to G. By definition of Fn, k > 1. Let Ui= Vimi for i = 1; : :;* *:k. It is easy to see that the subspaces Ui of Cn are uniquely determined by H (unl* *ike the subspaces Vi). Let G be the direct-sum decomposition given by U1; : :;:Uk and let HG be the corresponding element of Q0n. The assignment G 7! HG in- duces a map of posets Fn ! Q0n, and it is easy to see that it is a homotopy inv* *erse to the inclusion. __|_| It follows that there is a homotopy equivalence Ln ' |Fn| via U (n)-equivariant maps. Now we specialize again to the case n = pk (p a prime). By remark 4.3, the poset Ek of proper admissible subgroups of Upkis a subposet of Fpk. The WEISS DERIVATIVES 17 poset inclusion induces a map TSp (2k) ! Lpk that is equivariant with respect to the group inclusion Ak ! U(pk). Equivalently, we have a U (pk)-equivariant map U (pk)+ ^Ak TSp(2k) ! Lpk. We now need to recall another basic construction with collections of subgroups.* * Let G be a compact Lie group. Let C be a collection of subgroups of G (a set of clo* *sed subgroups of G closed under conjugation). Let E (C) be the universal space for C [E83 , LMS86 , AD99 ]. By [AD99 , section 2.10], there is a natural, G-equivari* *ant, map E (C) ! |C|, which is a homotopy equivalence. In fact, this statement is on* *ly proved in [op. cit.], (or rather in the reference [D97 ] cited there) for the * *case G finite, but it is obvious that the proof works just as well for compact Lie gro* *ups. Thus instead of looking at the map U (pk)+ ^Ak TSp (2k) ! Lpk we may look at the map U (pk)+ ^Ak E(Ek) ! E(Fpk). Our goal is to show that this map is a mod p homology equivalence. We will achieve this by analyzing the fixed point subspaces with respect to certain subgroups of U (pk). Recall that for a group * *G, a collection C and a subgroup H , the fixed point set E (C)H is homotopy equival* *ent to the realization of the poset of elements of C that contain H (again, this is* * proved in [D97 , 4.24] for a finite G, but the proof works for G compact Lie). In part* *icular, if H 2 C then E(C)H is contractible. Definition 4.5.A subgroup H of U (pk) is projective elementary Abelian if H contains the center of U (pk) and the quotient of H by the center of U (pk) is elementary Abelian. H is non-trivial (as a projective elementary abelian group)* * if it contains the center of U (pk) as a proper subgroup. In particular, Upkand its admissible subgroups are projective elementary Abelia* *n. Theorem 4.6. Let P be a projective elementary Abelian p-subgroup of U (pk). The map of fixed points k P P U (p )+ ^Ak E(Ek) ! ( E(Fpk)) is a mod p homology equivalence. Proof.The statement is obvious if k = 0 (both the domain and the target are S0 and the map is the identity). We use induction on k. Suppose first that P is not conjugate to a subgroup of Upk. Then obviously the source is contractible. On t* *he other hand, in this case P 2 Fpk by remark 4.3, so the target is contractible, * *and thus the map is an equivalence. Now assume the P is conjugate to a subgroup of Upk, so we may as well assume P Upkand P acts on E(Ek). Assume also that P contains the center of U (pk) as a non-trivial subgroup. Suppose, furthermore, that P is not symplectic. We claim that in this case both the domain and the target are again contractible. * *To see that the domain is contractible, observe that it is homeomorphic to k P P U(p )=Ak + ^ E(Ek) 18 GREG ARONE and so it is enough to show that E (Ek)P is contractible. This is equivalent to* * the poset of proper coisotropic subspaces_of Z=pZ2k containing P0 := ss0(P ). To se* *e that this poset is contractible,_let P0 = (P0\ P0?)? = P0?+ P0. By the_assumption_th* *at P is not symplectic, P0 is a proper subspace of Z=pZ2k. Clearly, P0 is coisotro* *pic and contains P0. Now let U be another_proper coisotropic subspace_of Z=pZ2k_ containing_P0. Observe that U? P0? P0. It follows that (U \ P0)? U \ P0, so U \ P0 is again a proper_coisotropic subspace of Z=pZ2k containing P0. Thus taking intersection with P0 defines a retraction of the poset of proper coisotr* *opic_ subspaces of Z=pZ2k containing P0 to its subposet_of_subspaces contained in P0 . This later poset is contractible because it has P0 as a maximal element. As for the target of the map, P not symplectic, so P is good, therefore the target is contractible. Now assume that P is symplectic, (but still contains the center of U(pk) as a p* *roper subgroup). It follows that P ~= Upk1for some 0 < k1 < k. By inspection, in this case our map of fixed points spaces k P P U (p )+ ^Ak E(Ek) ! ( E(Fpk)) is equivalent to the map U (pk-k1)+ ^Ak-k1 E(Ek-k1) ! E(Fpk-k1) which is a mod p homology equivalence by our inductive assumption on k. We have now shown that our map on fixed points is a mod p homology equivalence for all projective elementary Abelian P that contain the center of U(pk) as a p* *roper subgroup. It is well-known that the collection of non-trivial elementary Abeli* *an subgroups is ample [D97 ] for every compact Lie group containing an element of order p. It follows that the collection of non-trivial projective elementary Ab* *elian subgroups of U (pk) is ample for every closed subgroup of U (pk) that contains * *the center of U (pk) and also contains an element of order p after dividing by the * *center of U (pk). It follows easily that it is ample for every isotropy group of each * *of the spaces U(pk)+ ^Ak E(Ek) and E(Fk). It follows, by [AD99 , proposition 3.2] that this collection is ample for these two spaces (with, say, trivial coefficient s* *ystem). This means, by [op. cit., definition 3.1] (with the usual qualification that [o* *p. cit] only deals with finite groups and also that it deals with the unbased construct* *ion while we work with base spaces, but it is obvious that the results still hold) * *that the map on homotopy orbits k U(p )+ ^Ak E(Ek) h U(pk)! E(Fk)h U(pk) is a mod p homology equivalence. Since the group U (pk) is connected one can in fact show a little more, using an easy Borel-Serre spectral sequence argumen* *t. Namely, that the map U(pk)+ ^Ak E(Ek) ! E(Fk) WEISS DERIVATIVES 19 is itself a mod p homology equivalence. Since the action of the center of U (pk* *) is trivial on both spaces, the same is true for fixed points of the center. __|_| Corollary 4.7. There is a map TSp (2k) ! LCpkthat is equivariant with respect to the group inclusion Sp2k(Fp) n Upk! U (pk), which induces a mod p equivalence on the following homotopy orbit spectra i k j Map * LUpk; 1 SAdpk^ Sp Vh U(pk) i # j kV Map * TSp(2k); 1 SAdpk^ Sp h Sp 2k(Fp)nUpk Presumably, a similar result could be proved in the real case, but it would be a little more complicated, basically because there is more than one conjugacy cla* *ss of irreducible projective elementary abelian subgroups of O (2k). We will cont* *ent ourself with proving a vanishing result for the layers at odd primes. We know a* *lready that LRnis contractible unless n is a power of a prime or twice a power of a pr* *ime. We will now show that if p is an odd prime, n = pk then the n-th layer of the T* *aylor tower is sometimes contractible, even if the n-th derivative is not. Let p be a* *n odd prime, k a positive integer. Let E(FkR) be the universal space of the collectio* *n of all good subgroups of O (pk) (where the definition of a good subgroup is modified f* *rom U (n) to O (n) in an obvious way). Irreducible faithful projective representati* *ons of elementary Abelian groups (for odd p) have even dimension over the reals. It fo* *llows immediately that all the projective elementary Abelian p subgroups of O (pk) are good. Therefore, we have the following proposition Proposition 4.8. Let p be an odd prime, k a positive integer. The fixed point space (E (FkR))P is contractible for every projective elementary Abelian p subg* *roup P O(pk). The collection of projective elementary Abelian p subgroups is not ample for all isotropy groups of (E (FkR)) (it is only ample for groups containing an element* * of order p), but it is easy to see that it is ample (and also "reverse ample" in t* *he sense of [AD99 , Definition 3.6]) for all isotropy groups if one twists by the * *local coefficient system that comes from the sign representation of Z=2Z = ss0(O (pk)) (cf. [op. cit., Lemma 4.6]). It follows that if S is a sphere with an action of* * O (pk) such that the action of ss0(O (pk))ion thejtop homology of S is multiplication * *by -1 then the homotopy orbit space S ^ LRpkh O(pk)is contractible mod p. There is a similar Spanier-Whitehead dual result, as a consequence of reverse amplene* *ss. In particular, we have the following proposition Proposition 4.9. Let p be an odd prime, k a positive integer, V a real vector space of odd dimension. The homotopy orbit spectrum i k j Map * LRpk; 1 SAdpkSp Vh O(pk) 20 GREG ARONE is contractible mod p. Proof.It is easy to see that the action of ss0(O (n)) on the top homology of SA* *dn is trivial if n is odd. It follows that the action of ss0(O (pk)) on the top homol* *ogy of kV __ SAdpkSp is multiplication by -1. The proposition follows. |_| 5. Proof of Theorem 1.1 In this section we will describe our own model for the Weiss tower of the funct* *or V 7! Qk(Mor (W; W V )) The formulas are a little more elegant for the functor Qk(Mor (W; W V )+) , so* * we will work with this functor all along, and get rid of the extra basepoint at th* *e very end. For concreteness, we will only discuss the real case in detail. The constr* *uction goes through for the complex case with trivial modifications. Before plunging into the details of the construction, let us try to motivate it by discussing informally the case k = 1, i.e., the Taylor tower of the functor Q (Mor(W; W V )+) . By the Miller splitting, the n-th layer of the Taylor tower of this functor is the functor Ad n nV V 7! Q (S n ^ Mor(R ; W )+ ^ S )h O(n) Now observe that Mor (Rn; W ) hom (Rn; W ) SnW . Let nSW = SnW r Mor (Rn; W ). It follows, by Spanier-Whitehead duality, that there is a homoto* *py equivalence Q (SAdn ^ Mor(Rn; W )+ ^ SnV )h O(n)' i j ' 1 Map * SnW =nSW ; 1 SAdn ^ Sn(V W) h O(n) Moreover, by a suitable version of the Adams isomorphism [L98 ], one can replace homotopy orbits with fixed points. More precisely, there is a homotopy equivale* *nce i j 1 Map * SnW =nSW ; 1 SAdn ^ Sn(V W) h O(n) ' i j O(n) ' Map * SnW =nSW ; QSn(V W) We are looking for a natural model for TnQ (Mor(W; W V )+) (recall that Tn stands for "n-th Taylor polynomial of"). Thus we need to assemble these layers into a tower of fibrations. It turns out that there is a description in terms o* *f spaces of natural transformations between two functors of a certain small category, ve* *ry much in the spirit of [A99 ]. The small category is the (topological) category* * of subspaces of Rn and isometric inclusions. It turns out that i j TnQ (Mor(W; W V )+) ' NatERn SEW ; QSE(WV ) Here the functors on the right hand side are functors of E (and so the right ha* *nd side does not depend on E ). WEISS DERIVATIVES 21 Now we start with describing the construction in full detail and generality. We need to introduce a certain small category. It is a suitable version of the cat* *egory of chains of surjections of finite sets that was used in [AK98 ]. Definition 5.1.Let M = J op be the opposite category of J . For an object W of M , let MW be the full subcategory of M whose objects are vector subspaces * *of W . Let Mn = MRn. Definition 5.2.For k 0 let kM be the category defined as follows: an object of kM is a triple = ((lk; : :;:l1; l0); (ffk; : :;:ff1); {Ej}j2lk) where (1) (lk; : :;:l1; l0) is a k-tuple of finite sets, where l0 = {1} (l0 is a dumm* *y variable that does not carry any information, but it will be convenient to have) (2) ffi: lii li-1 is a surjection of sets (3) Ej are pairwise orthogonal objects of M , one for each j 2 lk A morphism i * * j ((lk; : :;:l0); (ffk; : :;:ff1);!{Ej}j2lk)(l0k; : :;:l00); (ff0k; : :;:ff01* *); {E0j}j2l0k is given by the following data: (1) a surjection of sets hi: li i l0ifor i = 0; 1; : :;:k. The surjections hi * *must satisfy the following admissibility condition (cf. [AK98 , definition 1.3]): * *For all j 2 li-1, hi maps the inverse image of j in li surjectively onto the inverse im* *age of hi-1(j) in l0i. (2) A morphism in aej;j0:Ej ! E0j0in M for each j02 l0kand each j 2 h-1k(j0). Let kMW be the full subcategory of kM whose objects are the ones for which the direct sum of the vector spaces associated with the elements of lk is a subspac* *e of W . Let kMn = kMRn . Next we define a contravariant functor from kMn to spaces. In fact, the functor depends on a choice of a tuple (W ; Xk; : :;:X0), where W is an object of J and Xk; : :;:X0 are based topological spaces. So we really will describe a functor * *from (kMn)op x J x Spacesk+1*to Spaces*. Let be an object of kMn, as in definition 5.2, let (W ; Xk; : :;:X0) be as above. Our functor is defined on objects as fo* *llows ^ ^ FW;Xk;:::;X0() = SWEj ^ X^lii j2lk ki0 It is easy to see how to make F (contravariantly) functorial in . Given a morph* *ism ! 0, the XisVare mapped diagonally, as prescribed by the surjections hi: lii l* *0i and the term j2lkSWEj is mapped diagonally as prescribed by the surjection * *hk and by the morphisms on the Ejs. We will be particularly interested in the case 22 GREG ARONE when the Xis are spheres. Suppose Xi= Smi for all i. Then we use the following notation S(W;mk;:::;m0).= FW;Xk;:::;X0() Recall that we want to find a model for the n-th Taylor polynomial of the funct* *or V 7! Qk(Mor (W; W V )+). We will start with attempting to do something more general (we will not quite succeed), namely to find a model for the n-th polyno* *mial approximation of m0Sm0 . .m.kSmk (Mor(W; W V )+) . Our model for this poly- nomial approximation is the following space of natural transformations of funct* *ors of i j NatkMn S(W;mk;:::;m0).; S(WV ;mk;:::;m0). We will use the notation Tn(W; V ; mk; : :;:m0) for this space of natural transformations. There exists a natural transformation m0Sm0 . .m.kSmk (Mor(W; W V )+) (3) # Tn(W; V ; mk; : :;:m0) This natural transformation is easy to understand, but cumbersome to describe. * *We will describe it for k = 0; 1, and the reader should be able to see without dif* *ficulty how to extend it by induction to all values of k. Let k = 0. In this case, kMn = Mn. So we want to construct a natural transfor- mation i j m0Sm0 Mor(W; W V )+ ! NatE2Mn SWE ^ Sm0; Sm0 ^ S(WV )E The target of this map is a subspace of the product space Y i j Map * WE m0Sm0S(WV )E E2Mn So we will construct a natural map into this product, and the image of this map will be inside the space of natural transformations. Consider the following cha* *in of maps m0Sm0 Mor(W; W V )+ Q # E2Mnm0Sm0 Mor(W E; (W V ) E)+ Q # E2Mn m0Sm0WE S(WV )E Q # E2Mn Map * WE m0Sm0S(WV )E here the first map is given by tensoring with the identity on E , the second map is given by taking the one-point compactifications of the linear isometries and* * the WEISS DERIVATIVES 23 third map is the obvious tautological map. It is easy to check that the image o* *f the composed map is in NatE2Mn SWE ^ Sm0; Sm0 ^ S(WV )E . This completes the discussion of the case k = 0. Now let k = 1. An object of 1Mn is a finite set l1 and a vector space Ej for ea* *ch j 2 l1 such that their direct sum is a subspace of Rn. We will denote a generic* * object of 1Mn by (E1; : :;:Ek), the underlying assumption being that l1 = {1; : :;:k}.* * So, we want to construct a natural transformation m0Sm0m1Sm1 Mor(W; W V )+ (4) # iV V j Nat1Mn j2l1SWEj ^ Sm1k ^ Sm0; Sm0 ^ Sm1k ^ j2l1S(WV )Ej Again, we think of the target as a subspace of the product of mapping spaces in* *dexed by objects of 1Mn Y W(Ej) m1km0Sm0Sm1kS(WV )(Ej) 1Mn To describe the map into this product, we fix an object (E1; : :;:Ek) of 1Mn and describe the map into the corresponding component. Thus we want to define a map m0Sm0m1Sm1 Mor(W; W V )+ ! W(Ej) m1km0Sm0Sm1kS(WV )(Ej) The map is defined as the following composite m0Sm0m1Sm1 Mor(W; W V )+ # m0Sm0m1kSm1k (Mor(W; W V )+) ^k V # m0Sm0m1kSm1k kj=1(Mor(W Ej; (W V ) Ej)+) i # Vk WE (WV )E j m0Sm0m1kSm1k j=1 j S j # m0Sm0m1kSm1kW(Ej) S(WV )(Ej) # W(Ej) m1km0Sm0Sm1kS(WV )(Ej) where the first map is taking the k-fold smash power of an element of m1Sm1(Mor (W; W V )+) the second map is given by tensoring with the identity on the Ejs, the third ma* *p is given by taking one-point compactification, the fourth map is smash product and the last map is the tautological map. It is tedious but straightforward to veri* *fy that this map gives rise to the map that we need (as indicated in (4)). The natural transformation of (3) is probably not useful in general, but it bec* *omes useful when one lets m0; : :;:mk go to infinity and passes to the colimit. In * *this 24 GREG ARONE case, the target becomes a model for the n-th polynomial approximation of the source, and the map becomes a model for the approximation map. To see this, we have to analyze the target of this map. In particular, we will provide an induc* *tive description of the target (inductive on n). Definition 5.3.Let kPn be the full subcategory of kMn whose objects are the objects of kMn that are not objects of kMW for W a strict subspace of Rn. The category kPn plays the role of the category nOk-1Mn of [AK98 ]. Proposition 5.4. The category kPn is a groupoid. Proof.Straightforward, and similar to the proof of [AK98 , proposition 2.3] __* *|_| For an object of kM let Aut () be the group of automorphisms of in kM . It follows from proposition 5.4 that there is a homotopy equivalence i j NatkPn S(W;mk;:::;m0).; S(WV ;mk;:::;m0).' Y i jAut() ' Map * SW(Ej)+mklk+...+m0l0; S(WV )(Ej)+mklk+...+m0l0 where the indexing set for the right hand side contains one object from each co* *n- nected component of the groupoid kPn. For an object = ((lk; : :;:l1; l0); (ffk; : :;:ff1); {Ej}j2lk) of kPn, the group Aut () acts on SW(Ej)+mklk+...+m0l0= S(W;mk;:::;m0). Let SW(Ej)+mklk+...+m0l0 SW(Ej)+mklk+...+m0l0 be the singular set for this action, i.e., the subspace of SW(Ej)+mklk+...+m0l0* * on which Aut () does not act freely. Theorem 5.5. Let Tn = Tn(W; V ; mk; : :;:m0). There exists a pullback square of the following form Q W(E )+m l +...+m l (WV )(E )+m l +...+m lAut() Tn ! Map * S j k k 0 0; S j k k 0 0 # # Q W(E )+m l +...+m l (WV )(E )+m l +...+m l Aut() Tn-1 ! Map * S j k k 0 0; S j k k 0 0 Here the products on the right side are indexed on a set of representing object* *s of the connected components of kPn. The vertical map on the left is the restriction map induced by inclusion of subcategories. The vertical map on the right is the product of the obvious restriction map, thus it is a fibration and therefore th* *e square is a homotopy pullback as well as a strict pullback. WEISS DERIVATIVES 25 Proof.The proof is very similar to the proof of [AK98 , theorem 2.5]. First of * *all, recall that one can write the space of natural transformations of functors of k* *Mn as an inverse limit over the twisted arrow category akMn. The objects of akMn are morphisms ! in kMn, and a morphism ( ! ) ! (0 ! 0) is a commutative square 0 # # ! 0 Given two contravariant functors F , G from kMn, define the (covariant) functor hom a(F; G) from akMn to spaces by ( ! ) 7! Map *(F (); G()) It is then easy to see that there is a homeomorphism NatkMn(F; G) ~=limhom a(F; G) For the purposes of this proof, let F () = S(W;mk;:::;m0). and let G() = S(WV ;mk;:::;m0). Our next step is to write akMn as a union of two subcategories. Let akM1nbe the full subcategory of akMn whose objects are the arrows ! such that is an object of dimension exactly n. Let akM2nbe the full subcategory of akMn whose objects are the arrows ! such that is of dimension strictly less than n. It * *is easily seen that the nerve of akMn is the union of the nerves of akM1nand akM2n. It follows that there exists a pullback square 1 limakMnhom a(F; G) ! limakMnhom a(F; G) # # 2 a M1\a M2 limakMnhom a(F; G) ! lim k n k nhom a(F; G) We claim that this pullback square is equivalent to the square in the statement* * of the theorem. To see this, we have to analyze the four spaces in the square. T* *he upper left space is obvious. The analysis of the upper right and the lower left* * corners is rather straightforward and closely imitates the analysis of the analogous sp* *aces in the proof of theorem 2.5 in [AK98 ] (see especially page 13 of [op. cit.]). * *We leave the details to the reader. We will analyze the lower right corner. Thus we want* * to analyze 1\a M2 limakMn k nhoma(F; G) Recall that the objects of akM1n\ akM2nare the arrows ! such that is of dimension n and is of dimension less than n. It is not hard to see that two arrows ! and 0! 0 are in the same connected component of akM1n\ akM2n if and only if and 0 are in the same component of kPn. Thus there is a bijecti* *on 26 GREG ARONE between the connected components of akM1n\ akM2nand of kPn and a limit over akM1n\ akM2nsplits as a product indexed by the connected components of kPn. Now fix a connected component D of akM1n\ akM2nand let be a fixed object in the corresponding connected component of kPn. Let # be the subcategory of akM1n\ akM2nwhose objects are arrows of the form ! (where is now our fixed object) and whose morphisms are commutative squares = # # ! 0 where the top arrow is the identity. It is clear that Aut () acts on the catego* *ry # and easy manipulations of limits show that there is a homeomorphism 1\a M2 Aut() limakMn k nhom a(F; G) ~=Map *(colim# F (); G()) To finally relate this space to the space in the lower right corner in the squa* *re diagram of our theorem, one has to analyze the space colim# F (). We state the result in a proposition. Again, the proof is straightforward, and is closely gi* *ven by the union of lemma 2.7 and lemma 2.8 in [AK98 ]. Proposition 5.6. Consider the obvious map colim# F () ! F () This map is a cofibration, and its image is precisely SW(Ej)+mklk+...+m0l0 It follows that the lower right corner of our square diagram is as claimed. It * *is easy to check that the vertical maps in the diagram are as claimed in the theorem. * *__|_| It follows that the homotopy fiber of the restriction map Tn(W; V ; mk; : :;:m0) ! Tn-1(W; V ; mk; : :;:m0) is homotopy equivalent to the product Y i Aujt() Map * S(W;mk;:::;m0).= S(W;mk;:::;m0).; S(WV ;mk;:::;m0). This, in turn, implies the following corollary Corollary 5.7. The (homotopy) direct limit of the homotopy fiber of the restric* *tion map Tn(W; V ; mk; : :;:m0) ! Tn-1(W; V ; mk; : :;:m0) as W; m0; : :;:mk ! 1 is homotopy equivalent to the product (indexed, as usual, by representatives of connected components of kPn) Y i j 1 1 SAdn ^ SnV h Aut() WEISS DERIVATIVES 27 Proof.By definition, the space S(W;mk;:::;m0)., is precisely the singular set * *of the action of Aut on the space S(W;mk;:::;m0).. It follows that there are homo* *topy equivalences i Aujt() Map * S(W;mk;:::;m0).= S(W;mk;:::;m0).; S(WV ;mk;:::;m0). ' i Autj() Map * S(W;mk;:::;m0).= S(W;mk;:::;m0).; S(WV ;mk;:::;m0).^ EAut ()+ ' i Aujt() ' Map * S(W;mk;:::;m0).; S(WV ;mk;:::;m0).^ EAut ()+ Taking the limit and using the Adams isomorphism for incomplete universes [L98 * *], one obtains the desired result. __|_| From now on, we let Tn(V ) = hocolimW;m0;:::;mk!1Tn(W; V ; m0; : :;:mk) Notice, however, that we have been somewhat vague about what we mean by "letting W; m0; : :;:mk go to infinity". For the W variable, we mean just the homotopy colimit over the diagram R0 ,! R1 ,! . ... For the mi variables, normally one can just think of them as natural numbers, which form an ordered set and thus a category. This is good enough for most purposes, but on at least one occasion we will have to think of the mis as finite sets rather than just natural numbers (* *and accordingly one thinks of mi7! Smi as a functor on finite sets rather than natu* *ral numbers). It is a well-known lemma of B"okstedt that both homotopy colimits have the same homotopy type. The reason that we will want to use the finite sets mod* *el is its good multiplicativity properties. This idea is, again, due to B"okstedt* *, who used it to give a rigorous construction of topological Hochshild homology. To analyze further the map (3) we compare it with the standard filtrations of of m Sm X and Mor (W; W V ) due to May-Milgram and Miller respectively. Let Cm (X) be the standard little-cube model for m m (X) [Ma72 ]. Cm (X) has a nat- ural filtration by subspaces FnCm (X) and the subquotient FnCm (X)=Fn-1Cm (X) is homeomorphic to C(Rm ; n)+ ^n X^n where C(Rm ; n) is May's space of lit- tle cubes in Rm . Similarly, recall from [Mi85 , C86 ] that there is a filtrat* *ion of Mor (W; W V ) by subspaces Fn Mor(W; W V )+ such that there is a homeomor- phism Fn Mor(W; W V )+=Fn-1Mor (W; W V )+ ~=SAdn ^ SnV ^O(n)Mor (Rn; W )+ Combining all these filtrations, one obtains a multi-filtration of the space m1Sm1 . .m.kSmk Mor (W; W V )+ by spaces Fn1Cm1(Fn2Cm2(. .F.n0Mor(W; W V ))). We now consider the total filtration of this multi-filtration. Thus let [ Fn(m1; : :;:mk; W; V ) = Fn1Cm1(Fn2Cm2(. .F.n0Mor(W; W V )+)) n1+...nk+n0n 28 GREG ARONE be the space of total filtration n. All our constructions are functorial in m1;* * : :;:mk and W . We will abbreviate the limit space hocolimm1;:::;mk;W!1Fn(m1; : :;:mk; W; V ) as FnQk(O = O(V )). The following lemma describes the quotient FnQk(fiber(O = O(V ))=Fn-1Qk(O = O(V )) Its proof is straightforward and is left to the reader. Lemma 5.8. There is a homotopy equivalence FnQk(O = O(V ))=Fn-1Qk(O = O(V )) ' _ ' SAdn ^ SnV h Aut() where the wedge sum is indexed, as usual, on a set of representatives of the co* *nnected components of kPn This filtration of Qk(O = O(V )+) stably splits, by Miller's and Snaith's split* *ting results. Notice that the subquotient space in lemma 5.8 is homotopy equivalent * *to the homotopy fiber described in corollary 5.7. This explains why Tn(V ) should * *be a good model for Qk+1(O = O(V )+) (however, the reader is warned once again that the space Tn(W; V ; mk; : :;:m0) is probably not a good model for anything befo* *re passing to a limit). We are now ready to state the main theorem of this section. Theorem 5.9. The composed map QFnQk(O = O(V )+) # Qk+1(O = O(V )+) # Tn(V ) is a homotopy equivalence of functors of degree n Notice that the theorem implies that Tn(V ) is the n-th Taylor polynomial of the functor Qk+1fiber(O = O(V )+). Proof.We follow the strategy of the proof of theorem 2.21 in [AK98 ]. The proof* * is by induction on n. The verification for the case n = 0 is trivial (both the dom* *ain WEISS DERIVATIVES 29 and the target are Q(S0)). Assume the claim is true for n - 1. Consider the diagram QFn-1Qk(O = O(V )) ! Tn-1(V ) # " QFnQkfiber(O = O(V )) ! Tn(V ) # " iW j Q SAdn ^ SnV h Aut() ! fiber(Tn(V ) ! Tn-1(V )) The vertical columns are fibration sequences. We want to show that this diagram is homotopy equivalent to one of the form A ! A # " A x B ! A x B # " B ! B Thus, we want to show that the two columns give the same split fibration sequen* *ce with the arrows reversed, and that the horizontal arrows are homotopy equivalen* *ces. It is not hard to check that the upper square commutes. By contrast, the lower square does not commute, not even up to homotopy. However, it is easy to see th* *at it commutes on elements of highest filtration, and this implies that it commute* *s after passing to the n-th layers. The upper horizontal arrow is an equivalence by our induction assumption. It follows that the central horizontal arrow is an equiva* *lence on the (n - 1)-th Taylor polynomials. The bottom map is an equivalence by lemma 5.8 and corollary 5.7. It is easy to see now that the bottom two vertical maps,* * as well as the bottom horizontal map, induce equivalences on the n-th layers of We* *iss' towers. Thus the middle arrow is a transformation of functors of degree n that * *is an equivalence on the (n - 1)-th polynomials and on the n-th layers, therefore * *it is an equivalence. __|_| The next step is to use our models for the Taylor polynomials of the functor Qk+1(fiber(O = O(V ))+) to construct models for the Taylor polynomials of the c* *osim- plicial space [k] 7! Qk+1(fiber(O = O(V ))+). We will use this cosimplicial sp* *ace to describe the n-th layer of the functor fiber(O = O(V )) and thus of the func* *tor BO (V ) (cf. [AK98 , sections 3 and 4]). Consider the functor m0Sm0 . .m.kSmk (Mor (W; W V )+). Let 0 i k and let m0Sm0 . .^.mi^Smi.m.k.Smk (Mor (W; W V )+) 30 GREG ARONE be the same functor with miSmi omitted. Consider the natural map m0Sm0 . .^.mi^Smi.m.k.Smk (Mor (W; W V )+) # m0Sm0 . .m.kSmk (Mor (W; W V )+) Lemma 5.10. There exist natural maps for i = 0; : :;:k ffii: Tn(W; V ; m0; : :;:^mi; : :;:mk) ! Tn(W; V ; m0; : :;:mk) such that the diagrams m0Sm0 . .^.mi^Smi.m.k.Smk (Mor (W; W V )+) ! Tn(W; V ; : :;:^mi; : :;:) # # ffii m0Sm0 . .m.kSmk (Mor (W; W V )+) ! Tn(W; V ; m0; : :;:mk) commute. Proof.If i > 0 then the map ffii is given by smashing with the identity map on Slimi. If i = 0 then the map is more complicated and we will describe it in det* *ail. We need to describe a map Natk-1Mn S(W;mk;:::;m1).; S(WV ;mk;:::;m1). # NatkMn S(W;mk;:::;m0).; S(WV ;mk;:::;m0). For the purposes of this proof, we write an object of k-1Mn as a triple ((lk; : :;:l1); (ffk; : :;:ff2); {Ej}) thus we shift the indexing. Now suppose we are given an element of i j Natk-1Mn S(W;mk;:::;m1).; S(WV ;mk;:::;m1). thus for each 2 k-1Mn we are given a map S(W;mk;:::;m1).! S(WV ;mk;:::;m1).. We want to assign to this data a map S(W;mk;:::;m0).! S(WV ;mk;:::;m0).for each 2 kMn. Of course, these maps should satisfy some compatibility conditions, to assemble into a natural transformation. Let = ((lk; : :;:l0); (ffk; : :;:ff1); {Ej}) be an object of kMn. For each element e of l1 let ff-1(e) be the object of k-1Mn given by all the inverse images of this element in l2; : :;:lk plus the relevan* *t Ejs. We are given a certain map associated with ff-1(e) for each e. Take the smash product of all these maps and take the smash product of the resulting map with the identity map on Sm0 . This is the map we associate with . To check that this is well defined and satisfies the assertion of the lemma amounts to a tedious b* *ut straightforward unraveling of the definitions. __|_| WEISS DERIVATIVES 31 Now consider the semi-cosimplicial space [k] 7! Qk+1(fiber(O = O(V ))+). By sem* *i- cosimplicial we mean that we take into account only the coface maps, but not the codegeneracy maps. Semi-cosimplicial spaces are good enough for our purposes because they still have "realizations", i.e., total spaces. Let Tnk+1(V ) = hocolimW;m0;:::;mk!1Tn(W; V ; m0; : :;:mk) (i.e., Tnk+1(V ) is what we called Tn(V ) until now, but now we want to keep tr* *ack of the variable k). Using the previous lemma, we can construct a semi-cosimplic* *ial space [k] 7! Tnk+1(V ) and a morphism of semi-cosimplicial spaces Qk+1(fiber(O = O(V ))+) ! Tnk+1(V ) By theorem 5.9, Tnk+1(V ) is the n-th Taylor polynomial of Qk+1(fiber(O = O(V )* *)+) and in fact the map of semi-cosimplicial spaces induces on the level of total s* *paces the map from fiber(O = O(V )) to its n-th Taylor polynomial. Now we can use the semi-cosimplicial space [k] 7! Tnk+1(V ) to write a formula * *for the n-th layer of the Weiss tower of the functor V 7! BO (V ). Indeed, the n-th layer of this functor is given by the semi-cosimplicial space [k] 7! fiber(Tnk+* *1(V ) ! Tnk+1-1(V )). It follows from corollary 5.7 and its proof that this semi-cosim* *plicial space is equivalent to the semi-cosimplicial space whose k-cosimplices space is* * the homotopy colimit, as W; m0; : :;:mk go to 1, of the spaces Y i jO(n) Map * O (n)+ ^Aut() SW(Ej)+mili ; S(WV )(Ej)+mili ^ EO (n)+ It is easy to see that in this cosimplicial space the coface map di is given by* * smashing with the identity map on Smilifor all i including i = 0. Now, taking mis to be finite sets, and taking the homotopy colimit over finite sets, one can use B"ok* *stedt's addition functor to construct a homotopy equivalence from this semi-cosimplicial space to the one given by Y O(n) [k] 7! Map * O(n)= Aut()+; QO(n) SnV ^ EO (n)+ where QO(n) stands for equivariant stable homotopy with respect to the O (n)- universe [kRkn (see page 27 of [AK98 ] for more details on a very similar situa* *tion). It is easy to see that this cosimplicial space can be rewritten as nV O (n) [k] 7! Map * Ok;n; QO(n) S ^ EO (n)+ with the coface maps being induced by the face maps in the simplicial space Oo;* *n. It follows that the total space of this cosimplicial space is equivalent to nV O(n) Map * Ln; QO(n) S ^ EO (n)+ This, in turn, implies, by the Adams isomorphism for incomplete universes [L98 * *], that the n-th layer of the Weiss tower of BO (V ) is equivalent to i j 1 Map * Ln; 1 SAdnSnV h O(n) 32 GREG ARONE This completes the proof of theorem 1.1. 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