AN EQUIVARIANT VAN KAMPEN SPECTRAL SEQUENCE MICHELE INTERMONT July 28, 1995. 1.Introduction Let G be a finite group, W a finite dimensional representation of G and (W + * *n) the representation (W Rn) with G acting trivially on Rn: For a based G-space X and a subgroup H of G; ssHW+n(X) is defined as the based set [S(W+n); X]H of H-homotopy classes of based H-maps from SW+n into X: This is a group if n 1; and an abelian group if n 2: This paper considers the problem of computing {ssHW+n(X _ Y )}n;H; n 0; H G; where X _ Y is the one point union of the based G-CW complexes X and Y: The approach is to construct a van Kampen spectral sequence indexed over H. The E2 term of this spectral sequence (which is the collection of E2 terms of spectral sequences associated to each H) has a nice algebraic description. Give* *n a based G-space X, the collection of based sets {ssHW+n(X)}n;H with all the prima* *ry homotopy operations between the sets is called a (W )-algebra and denoted by W (X): The`category of such objects is denoted (W )-al: There`is a coproduct functor : ((W )-al)2 ! (W )-al which has derived functors p: A G-space X is said to be W *-connected if for each subgroup H of G; the fixed point set XH* * is n-connected in the usual sense, where n is the dimension of W H: We can now sta* *te the main result: Theorem 1.1. Let X; Y be based W *-connected G-CW complexes. For each sub- group H of G there exists a first quadrant spectral sequence {Erp;q(H)} converg- ing to {ssHW+p+q(X _ Y )}p+q; p + q 0: For fixed p; the`collection of columns E2p;*:= {E2p;*(H)}H is a (W )-algebra isomorphic to W (X) pW (Y ): This result holds for arbitrary homotopy pushouts as well as for the wedge of based spaces. Even more generally, we will show that this result holds for poin* *ted homotopy colimits. That is, we will show that there is a spectral sequence con- verging to the equivariant homotopy groups of the pointed homotopy colimit of an I-diagram of based G-CW complexes, where I is any small category, and where the E2 term can be described nicely using derived functors. This will be more fully explained in section 7. We should note that although the category of (W )-algebras is not abelian, derived functors of the coproduct functor can be defined using the techniques of Quillen [22] and [23], as in [20, x5]. The second main result provides a range in which the coproduct functor is ad- ditive in this equivariant setting. Corollary 1.3 is an immediate application * *of Theorem 1.2. 1 2 MICHELE INTERMONT Theorem 1.2. If X, Y are based G-CW complexes such that X is (W + r)*- connected, Y is (W + s)*-connected and dim(W G)+min(r; s) 1; then for each H G; ssHW+i(X _ Y ) ~=* for 0 i min(r; s): When i dim(W H) + r + s; the natural maps X ,! X _ Y - Y induce an isomorphism of abelian groups ssHW+i(X _ Y ) ~=ssHW+i(X) ssHW+i(Y ): Corollary 1.3.If X and Y are based G-CW complexes satisfying the conditions of Theorem 1.2, then for each H G; there is an exact sequence of abelian group* *s: ssHW+q+2(X _ Y ) ! E22;q(H) ! E20;q+1(H) ! ssHW+q+1(X _ Y ) ! E21;q(H) ! 0 where q = r + s + dim(W H): Recovery of the van Kampen Theorems: For a fixed n and G-space X; let ss_W+n(X) denote the collection {ssHW+n(X)}H along with the primary operations relating these sets to one another. These objects form a category (with natural transformations as morphisms) with a coproduct. Lemma 4.1 will show that the groups E2p;0(H) (set, if p = 0) in Theorem 1.1 are all trivial, which means tha* *t the groups E20;1(H) in the lower left corners of the E2(H) terms are not affected by differentials. In light of this, E10;1(H) = E20;1(H) and, by Lemma 6.2, ` ss_W+1(X _ Y ) = ss_W+1(X) ss_W+1(Y ): This is the equivariant van Kampen theorem of Lewis [13]. In fact, Lewis' result applies not just to the wedge, but also to more general homotopy pushout diagra* *ms. When G is the trivial group, the spectral sequence constructed here reduces to that of [20], modulo a shift by the dimension of W . Once again, one can recove* *r the usual formula for the lowest dimensional non-trivial homotopy group of the wedge of two spaces by considering the lower left corner. On the Proof of 1.1: As in [20], the idea is to construct, for each based, W *- connected G-CW complex X; a nice simplicial space Xo which is, in a sense, a resolution of X by G-spheres. Given two such simplicial G-spaces, the simplicia* *l G- space Xo_Yo, defined as Xn_Yn in dimension n, has nice properties. In particula* *r, the work of Bousfield and Friedlander [2] yields a spectral sequence converging* * to based H-maps from SW into (Xo _ Yo): On the Organization of the Paper: To begin, some pertinent facts of equivari- ant homotopy theory are recalled in Section 2. Section 3 contains the construct* *ion of a nice simplicial resolution of X. In addition, Section 3 relates weak W -equiv* *alence to G-homotopy equivalence. Section 4 discusses the connectivity requirements for the G-CW complex X and its resolution, and presents a spectral sequence for each H converging to ssHW+*(X): In Section 5 it is shown that this spectral sequence* * also exists for the wedge of two properly connected G-CW complexes. The category of (W )-algebras is introduced in Section 6 and used to complete the proof of The- orem 1.1. Section 7 extends these results to pointed homotopy colimits, and giv* *es the proof of Theorem 1.2. Acknowledgements: I would like to thank William G. Dwyer, under whose direction this work was completed, and both the University of Notre Dame and the AN EQUIVARIANT VAN KAMPEN SPECTRAL SEQUENCE 3 Luce Foundation for financial support. I would also like to thank the referee f* *or his comments, and for the short proof of Theorem 7.1 presented here. 2.Preliminaries Throughout this paper G is assumed to be a finite group, and G-spaces are assumed to be left G-spaces. Recall that the basepoint of a G-space must be fix* *ed by the action of G: Let X+ denote a G-space X with a disjoint (fixed) basepoint added. If X and Y are pointed G-spaces, the set of based maps from X to Y , denoted Map*(X; Y ), is also a G-space, with G acting by conjugation. The G- maps from X to Y , MapG*(X; Y ); are the fixed points of this action. The set of based G-homotopy classes of based G-maps between X and Y is written as [X; Y ]G : When X; Y; and Z are compactly generated weak Hausdorff G-spaces, [Z ^ X; Y ]G ~=[Z; Map*(X; Y )]G . Most of the spaces involved in this paper will be based G-CW complexes. These are built from pointed G-cells of the form G=H+ ^ Dn using based attaching maps. A G-cell can only be attached to a G-cell with an isotropy subgroup of equal or greater order. In view of this, the subgroups of G can be ordered (with repetit* *ion): G H1; H2; . .H.m e |Hi| |Hj| ifi j where Hj is the isotropy subgroup of the attached cell. A useful fact concerning G-CW complexes is the following: Proposition 2.1.([7, x2.7]) Let f : Y ! Z be a G-map between G-CW complexes such that fH : Y H ! ZH is a homotopy equivalence for all H: Then f is a G- homotopy equivalence. Fixing a representation W of G; the one-point compactification of W is denoted by SW : This G-space is a sphere of dimension equal to the real dimension of W;* * with the compactification point as fixed basepoint. DW+1 is the reduced cone over S* *W : If V is another representation of G; V + W is the direct sum of the representat* *ions. The trivial representation of dimension n will be denoted simply by the integer* * n: The group G acts diagonally on the object G=H+ ^ SW+n , which we call a generalized G(W )-sphere. It can be thought of as a wedge of SW+n , where G acts not only on each wedge summand, but also by permuting the summands. For compactness of notation, we write SW+nH for G=H+ ^ SW+n : Of course, SW+nH is the boundary of a generalized disk G=H+ ^ DW+n+1 , abbreviated DW+n+1H. The set [SWH; X]G is actually a group whenever dim(W G) 1, and an abelian group whenever dim(W G) 2. The collection {[SWH; X]G }H , H a subgroup of G; is called the collection of W thequivariant homotopy groups (sets) of X. [SWH; X]G will be written as ssHW(X). Recall that [SWH; X]G ~=[SW ; X]H (see [7* *]), so the notation ssHW(X) is unambiguous. Note too, that ssHn(X) ~=ssn(XH ): Definition 2.2.([13, x1.1]) A G-space X is said to be W *-connected if, for each subgroup H of G; XH is n-connected in the ordinary sense where n is the dimensi* *on of W H: Remark If X is W *-connected, then ssHW(X) ' * for each subgroup H of G; as Lemma 4.1 will show. 4 MICHELE INTERMONT The orbit category of G is denoted by OG : The objects are G=H; where H is a subgroup of G; and the morphisms are G-maps. For any G-space X, we can define a functor ss_W(X) : OopG! Sets* which maps G=H to ssHW(X) and gives, for each G-map G=H ! G=K; a map ssKW(X) ! ssHW(X) [13]. 3.Techniques The Resolution Given a based G-space X, this section constructs a simplicial resolution, Xo,* * of X by wedges of generalized spheres. This resolution has the property that when X is a based, W *-connected G-CW complex, the simplicial (W )-algebra produced from Xo (see x6) is a free simplicial resolution of the (W )-algebra produced f* *or X. This simplicial space is constructed functorially by an inductive process. L* *et VX be the space given by the following pushout: W W W W+n - W W W W+n+1 H n1 h(SH )h_____ H n1 h(DH )h | | | | | | | | W W W|? W+n -|? H n1 f(SH )f_____________VX_ (Diagram 1 ) where the indexing G-map h runs through all the G-maps from DW+n+1H to X and f runs through all the G-maps from SW+nH to X: The upper horizontal arrow is induced by the inclusion of the boundary of a disk into the disk itself: @DW+n+* *1H = SW+nH ,! DW+n+1H: The left vertical map takes each wedge summand (SW+nH)h by the identity to the wedge summand (SW+nH)f where f : SW+nH ! X is the restriction of h to the boundary of DW+n+1H: G acts on VX in the natural way. This construction comes with natural maps (i)ffl : VX ! X which sends (SW+nH)f into X by the indexing map f and (DW+n+1H)h into X by the indexing map h. (ii)fi : VX ! V2X which sends (SW+nH)f G-homeomorphically to the copy of SW+nH in V2X indexed by the inclusion of the sphere into VX and which sends (DW+n+1H)h G-homeomorphically to the copy of DW+n+1H in V2X indexed by the inclusion. It can be checked that (V; ffl; fi) is a cotriple (or comonad, in the languag* *e of [15]). Following [11], there is an associated cellular simplicial G-space, Xo; augment* *ed by ffl : X0 ! X; with Xi:= Vi+1X; i 0: The face and degeneracy maps are given by dj = VjfflVp-j : Xp ! Xp-1 sj = VjfiVp-j : Xp ! Xp+1 for all 0 j p: A simplicial G-space is said to be cellular if each degeneracy map is an inclusion of G-CW complexes. Definition 3.1.The construction Xo described above is called the generalized sphere resolution of X. AN EQUIVARIANT VAN KAMPEN SPECTRAL SEQUENCE 5 The propositions which follow are equivariant versions of those found in [20,* * x2]. It is these properties of the generalized sphere resolution which will be appli* *ed in x6. Proposition 3.2.Let X be a based G-space, Xo its generalized sphere resolution. There exist G-contractible subcomplexes Cp in Xp such that (i)For all p 0 the quotient Xp=Cp is a wedge of generalized W -spheres SW+nH; n 1; H G: (ii)sjCp Cp+1 for all degeneracy maps sj : Xp ! Xp+1: (iii)For j = 0; . .;.p the map s0j: Xp=Cp ! Xp+1=Cp+1 induced by sj is an inclusion of wedges. Recall that when X is a simplicial space, ssqX is a simplicial group, and so * *has homotopy groups sspssqX itself. These groups can be calculated as the homology of a particular chain complex following [17, x17]. They will form the E2 term o* *f a spectral sequence in x4. Proposition 3.3.Let X be a W *-connected G-CW complex, with generalized sphere resolution Xo. If Yo denotes Map*(SW ; Xo) and Y = Map*(SW ; X) then (i)sspssHq(Yo) = 0 for all p 1 and q 1: (ii)For all q 1 the augmentation map induces an isomorphism ~= H ss0ssHq(Yo) ! ssq (Y ): This last proposition is established by showing that every cycle in the pthgr* *oup of the chain complex actually bounds an element (see [20, x2]). The G-CW Decomposition of SW For any subgroup K of G, the cellular structure of SW allows us to approach MapK*(SW ; -) inductively, using fibration sequences. We will use this to prove that a weak W -equivalence between (W - 1)*-connected based G-CW complexes is a G-homotopy equivalence. We will also use this in the proof of Lemma 4.1. The sphere SW is not technically a based G-CW complex unless the fixed point set (SW )G is connected. However, since W is a representation and (SW )H is its* *elf a sphere, it is not too difficult to see that we can choose a G-CW decomposition * *of SW which attaches pointed cells at each stage through basepoint-preserving attachi* *ng maps, even when dim(SW )G = 0. Let Z1 = (SW )G . We write Z1 Z2 . . . Zm = SW , where Zj, 2 j m, is the cofibre of f : SiHi! Zj-1, and f is a basepoint-preserving attaching map. Let Y be a G-space. Applying MapK*(-; Y ) to the cofibre sequence determined by Zj; j 2; yields a fibre sequence MapK*(Zj; Y ) ! MapK*(Zj-1; Y ) ! MapK*(SiHi; Y ): Q The term MapK*(SiHi; Y ) is isomorphic to lMap*(Si; Y Kl), where Kl = K \ g-1lHglfor representatives glof the orbits of the left action of K on G=H: Definition 3.4.[13, x3.6] Let f : X ! Y be a G-map. f is a weak W -equivalence if f induces a homotopy equivalence between mapping spaces MapH*(SW ; X) !' MapH*(SW ; Y ) for each H G: 6 MICHELE INTERMONT Theorem 3.5. Let X; Y be pointed, (W -1)*-connected G-CW complexes. Then f : X ! Y is a weak W -equivalence if and only if it is a G-homotopy equivalenc* *e. Proof.()) Proceed by induction on the order of G: If the order of G is 1; the r* *esult is clear, since then SW is just an ordinary sphere of dimension l = dim(W ): I* *n this case, the assumptions are that X and Y are (l-1)-connected and that f induces an isomorphism on all homotopy goups in and above dimension l. Thus, Whitehead's theorem applies. Now assume fH : XH ! Y H is an equivalence for all proper subgroups H of G: If we show that fG : XG ! Y G is an equivalence, then Proposition 2.1 will finish the proof in this direction. Notice that if SW = (SW )G , then, under * *the connectivity assumptions of the theorem, a weak W -equivalence is an ordinary weak equivalence, and again Whitehead's Theorem applies. If SW 6= (SW )G , then SW = Zm-1 [f DiHfor some proper subgroup H and some i. Consider the ladder: MapG*(SW ; X)----! MapG*(Zm-1 ; X)----! MapG*(Si-1H; X) ?? ? ? y fl ?yff ?yfi MapG*(SW ; Y )----! MapG*(Zm-1 ; Y )----! MapG*(Si-1H; Y ) Since f induces a weak W -equivalence by assumption, fl is a homotopy equiv- alence. Since MapG*(Si-1Hi; -) ~= i-1(-H ), fi is a homotopy equivalence by in- duction, and by the 5-lemma, ff is a homotopy equivalence above dimension 0. In dimension 0, notice that ss0MapG*(Si-1H; -) is trivial, since the dimension of * *the cell DiHmust be dimW H, and so ff is indeed a homotopy equivalence. Continuing ~= backwards through the decomposition of SW , we conclude that MapG*(Z1; X) ! MapG*(Z1; Y ): But Z1 is just (SW )G , so we have Map*((SW )G ; XG ) '!Map*((SW )G ; Y G): Since X and Y are (W - 1)*-connected, X '!Y: The implication in the other direction is obvious. __|_| 4.Connectivity and the Spectral Sequence Let Yo be a based, suitably connected simplicial G-space. This section applies the techniques of Bousfield-Friedlander in [2] to MapH*(SW ; Yo) to obtain the * *ho- motopy spectral sequence of Theorem 4.4. Lemma 4.1 establishes a connectivity condition for the spaces Yn 2 Yo which is sufficient for the convergence of the* * spec- tral sequence. Lemma 4.2 shows that the generalized sphere resolution of a G-CW complex satisfies this condition. Lemma 4.1. If Y is a based, W *-connected G-space, then MapG*(SW ; Y ) is 0- connected. Proof.Let Z1 = (SW )G . .Z.j+1= Zj[f Di+1H . . .Zm = SW be the G-CW decomposition of SW as in x 3. Since Y G is assumed to be dim W G-connected, ss0MapG*((SW )G ; Y ) = [(SW )G ; Y ] = *. Now continue by induction. Assume ss0MapG*(Zj; Y ) = *. Consider the homotopy fibration sequence ss1MapG*(SiH; Y ) ! ss0MapG*(Zj+1; Y ) ! ss0MapG*(Zj; Y ): AN EQUIVARIANT VAN KAMPEN SPECTRAL SEQUENCE 7 MapG*(SiH; Y ) ' i(Y H), and since i must be less than or equal to the dimension of W H, this is 1-connected. Thus, ss0MapG*(Zj+1; Y ) is trivial. __|_| Remark In particular, Lemma 4.1 says that ssHW of a based, W *-connected G-CW complex is trivial for all subgroups H of G. Lemma 4.2. Let X be a G-CW complex and let Xo be the generalized sphere resolution of X: Then Xi is W *-connected for each Xi2 Xo; i 0: Proof.By construction, Xiis a pushout for each i 0 (see Diagram 1 in x3). Since the upper horizontal arrow in this pushout is an inclusion of G-CW complexes, applying the K fixed point functor gives a homotopy pushout diagram for (Xi)K : By assumption, each space involved in the resolution is a suspension of SW or a cone over a suspension of SW ; and hence is at least W *-connected. Therefore, * *all of the spaces in the pushout diagram for (Xi)K are n-connected where n = dimW K: Now a standard argument with the van Kampen and Hurewicz theorems shows that (Xi)K is also n-connected, n equal to dimW K: This is true for all subgrou* *ps K of G; so Xi is W *-connected. __|_| Let [n] be the standard n-simplex. Recall [19] that for a simplicial space Xo; the geometric realization |Xo| is formed from _ ( [n] x Xn)=([n] x *) n0 by making identifications using the face and degeneracy maps. The problem with this realization, however, is that given two simplicial spaces and a simplicial* * map between them which is a homotopy equivalence on each level, it is not true that the map induced between the geometric realizations is a homotopy equivalence. For the realization to preserve homotopy equivalences, Segal showed in [18] that it is sufficient to require that the degeneracy maps of the simplicial spaces a* *re cofibrations. In the context of this paper all simplicial spaces are assumed to* * satisfy this condition. All simplicial spaces are also assumed to be based. It is clear by naturality that, if Xo is a simplicial G-space, then |Xo| is a* * G-space. For A a based G-space, H a subgroup of G, and Yo any simplicial G-space, let MapH*(A; Yo) denote the simplicial space defined by MapH*(A; Yo)n := MapH*(A; Yn); with the face and degeneracy maps given by composition with those from Yo: Lemma 4.3. If Yo is a simplicial G-space such that Yi is W *-connected for each i; then |MapH*(SW ; Yo)| ' MapH*(SW ; |Yo|) for all H G. Proof.This result is established for unbased maps in [6, x 5.4]. Now consider t* *he diagram: |MapH*(SW ; Yo)|----!|MapH (SW ; Yo)|----! |YoH| ?? ? ? yfl ?yff ?yfi MapH*(SW ; |Yo|)----!MapH (SW ; |Yo|)----! |Yo|H 8 MICHELE INTERMONT of fibrations._Since ff and fi are homotopy equivalences, so is fl. |_| In other words, realization commutes with equivariant loop spaces, subject to some connectivity conditions. The first quadrant homotopy spectral sequence for bisimplicial sets, establis* *hed in [2, B.5] by Bousfield-Friedlander, has a simplicial space analog [5, Appendi* *x]. For a connected simplicial space, the spectral sequence converges to the homoto* *py of the realization. In conjunction with Lemma 4.3, this immediately gives: Theorem 4.4. Let Yo be a simplicial G-space such that Yi is W *-connected for each i. Then for each H G there exists a first quadrant spectral sequence which converges strongly to the homotopy groups of MapH*(SW ; |Yo|) with E2p;q= sspssqMapH*(SW ; Yo): The generalized sphere resolution of a G-space Y is an example of a W *-conne* *cted simplicial G-space. The spectral sequence associated to this simplicial G-space* * col- lapses by Proposition 3.3 and Lemma 4.1. The collapse of the spectral sequence has interesting consequences, namely: Corollary 4.5.Let X be a W *-connected, based G-CW complex, and let Xo be its generalized sphere resolution. Then the augmentation map ffl : X0 ! X induces a G-homotopy equivalence |Xo| ! X: Proof.By Proposition 3.3, E2 = E1 in the spectral sequences of Theorem 4.4 which gives: ss0ssqMapH*(SW ; Xo) ~=ss0+qMapH*(SW ; |Xo|): Again by Proposition 3.3, ss0ssqMapH*(SW ; Xo) ~=ssqMapH*(SW ; X): Hence, ssqMapH*(SW ; |Xo|) ~=ssqMapH*(SW ; X): Now by Theorem 3.5, this weak W -equivalence is actually a G-homotopy equiva- lence. __|_| 5.The Spectral Sequence for a Wedge This section establishes a spectral sequence which converges to the equivaria* *nt homotopy groups of the wedge of two based, W *-connected G-CW complexes. This is done in Theorem 5.2. The next section will reformulate the E2 term of Theorem 5.2 with the use of derived functors to yield Theorem 1.1. Thus far it has been shown that for X a based, W *-connected G-CW complex and Xo its generalized sphere resolution, |Xo| is G-homotopy equivalent to X: T* *he proof of Theorem 5.2 depends on knowing that this holds for the wedge of two su* *ch complexes X and Y : that the realization of Xo _ Yo is G-homotopy equivalent to the wedge X _ Y: Lemma 5.1. If X and Y are based, W *-connected G-CW complexes with general- ized sphere resolutions Xo; Yo, then |Xo _ Yo| is G-homotopy equivalent to X _ * *Y: AN EQUIVARIANT VAN KAMPEN SPECTRAL SEQUENCE 9 Proof.By Corollary 4.5, there are G-homotopy equivalences f : |Xo| ! X and g : |Yo| ! Y: In particular, |Xo|K '! XK and |Yo|K '! Y K for all subgroups K of G: Now |Xo|K _ |Yo|K '! |Xo _ Yo|K andXK _ Y K !' (X _ Y )K , so |Xo _ Yo|K !' (X _ Y )K for all subgroups K of G: By Proposition 2.1, they are G-homotopy equivalent. __|_| Theorem 5.2. Let X; Y be based, W *-connected G-CW complexes. Let Xo; Yo be their respective generalized sphere resolutions. For each subgroup K of G th* *ere exists a spectral sequence converging to ssKW+p+q (X _ Y ) with E2p;q(K) = sspssKW+q(Xo _ Yo): Proof.The simplicial G-space Xo_ Yo is W *-connected in each dimension because both Xo and Yo are W *-connected. Now for each K G; Theorem 4.4 gives a spectral sequence for Xo _ Yo: Lemma 5.1 shows that |Xo _ Yo| is G-homotopy equivalent to X _Y so the spectral sequence can be said to converge to ssKW+p+q* *(X _ Y ): __|_| Remark Let Xo be any simplicial G-space. A G-map G=H ! G=K induces a map of simplicial G-spaces OE : MapK*(SW ; Xo) ! MapH*(SW ; Xo): When each Xi is W *-connected, OE induces a map of spectral sequences {Erp;q(K)} ! {Erp;q(H* *)} where Erp;q(-) is the spectral sequence of Theorem 5.2. Thus, Erp;q(-) can be considered a functor from OopGinto the category of spectral sequences. 6.(W )-Algebras In this section we discuss the structure of (W )-algebras, the motivating exa* *mple of which is the collection of equivariant homotopy groups of a G-space and all * *the operations between them. Notice that Theorem 5.2 provides, for each subgroup H of G, a spectral sequence. These are the spectral sequences of Theorem 1.1. The columns of the E2 terms of these spectral sequences, when considered together, have the structure of (W )-algebras, and it is this structure that will allow u* *s to understand the E2 terms as described in the main theorem. W Let (W ) denote the category whose objects are finite wedges ki=1SW+niHiof generalized spheres where Hi G; ni 1; and whose morphisms are G-homotopy classes of G-maps. Definition 6.1.A (W )-algebra is a contravariant functor A : (W ) ! Sets* with the property that it takes finite wedges to products in the sense that if * *U0; U1 are objects of (W ); then the natural inclusions ij : Uj ! U0 _ U1; j = 0; 1 induce a bijection A(U0 _ U1) ! A(U0) x A(U1): Example For a G-space X, define the (W )-algebra W X as the functor which takes the generalized sphere SW+nH to the group ssHW+n(X): Remarks 10 MICHELE INTERMONT (i)It will sometimes be helpful to think of a (W )-algebra A as a collection of sets, An;HW:= {A(SW+nH)n;H}, together with operations. For a G-map SW+mJ ! ki=1SW+niHi; these operations are of the form: An1;H1x . .x.Ank;Hk-! Am;J: (ii)(W )-algebras form a category, (W )-al; with natural transformations as morphisms. (iii)The category (W )-al has limits and colimits [15, xV,IX]. In particular then, it has coproducts. (iv)The -algebras of [20] are (W )-algebras with G = {e} and W = 0: (v) If A is a (W )-algebra and U0 is an object of (W ); then A(U0) is actual* *ly a group. However, if f is a morphism of (W ); A(f) is not necessarily a group homomorphism. The proposition below records the fact that (W )-algebras are generalizatio* *ns of Lewis' W -Mackey functors [13]. This will justify the claim in the introduct* *ion that the van Kampen theorem of [13] appears as the edge of the spectral sequence in Theorem 1.1. Proposition 6.2.Let A : (W )op ! Sets* be a (W )-algebra. Then A deter- mines a functor A_: OopG! Sets* which maps wedges to products in the sense of Definition 6.1. Proof.Since there are no non-trivial maps SW+mH ! SW+nK when m n, there are no non-trivial maps A(SW+nK) ! A(SW+mH ): Thus, the only maps into A(SWH) are from A(SWK). Hence, define A_(G=H) := A(SWH). If f : G=H ! G=K, define A_(f) := A(f ^ id). __|_| Definition 6.3.Let C be the category of pointed sets graded by pairs {n; H} whe* *re n is an integer greater than or equal to 1 and H is a subgroup of G: Then there is a natural forgetful functor U : (W )-al ! C which takes a (W )-algebra A to the collection {A(SW+nH)}n;H 2 C: This functor U has a left adjoint F defined on {Tn;H} 2 C by _ _ _ F(T ) := W ( (SW+nH) ); 2 Tn;H - {*} H n1 (proof analogous to that in [20, x4]). F(T ) is called the free (W )-algebra on* * T . Proposition 6.4.Let U0; U1 be elements of (W ): Then ` W (U0 _ U1) = W (U0) W (U1): Proof.Let T 2 C be the graded pointed set with F(T ) = W (U0) and let S 2 C be the graded pointed set with F(S) = W (U1): Then T _ S is the graded pointed set with F(T _ S)`= W (U0 _ U1): Since F is left adjoint to the forgetful funct* *or, F(T _ S) = F(T ) F(S); and the result follows. __|_| AN EQUIVARIANT VAN KAMPEN SPECTRAL SEQUENCE 11 We now recall the necessary definitions regarding simplicial objects over the category of (W )-algebras, and derived functors. Definition 6.5.A simplicial (W )-algebra is a contravariant functor Ao : ! (W )-al: Equivalently, this is a contravariant functor Ao : (W ) ! sSets* with the special property that it takes finite wedges isomorphically to products in the * *sense of Definition 6.1. Ao is said to be augmented by A if there is a map ffl : A0 !* * A such that ffl O d1 = ffl O d0. Proposition 6.6.Let Ao be a simplicial (W )-algebra. Then ssp Ao : (W ) ! sSets* ! Sets* is a (W )-algebra for p 0: Proof.Since the image of a (W )-algebra at any object of (W ) is actually a group, the image of Ao at any object is a simplicial group. Hence, sspAo can be computed by the methods of ([17, x3.6]). To see that sspAo is a (W )-algebra, i* *t is enough to note that Ao takes finite wedges to products and the functor ssp pres* *erves finite products. __|_| Definition 6.7.A free simplicial resolution of a (W )-algebra A is a simpli- cial (W )-algebra Ao augmented by A such that (i)For all i;Ai is a free(W )-algebra on a collectionTi2 C; (ii)sjTi Ti+1 for all degeneracy mapssj; 0 j i; (iii)sspAo = 0 forp 1; and ~= (iv)The augmentation map induces an isomorphismss0Ao ! A: Proposition 6.8.Let Xo be the generalized sphere resolution of X. Then W (Xo) is a free simplicial resolution of the (W )-algebra W (X): Proof.This follows immediately from Propositions 3.2 and 3.3. __|_| Definition 6.9.(Derived Functors) Let Ao; A0obe`free simplicial resolutions of the (W )-algebras A; A0respectively. Then Ao A0odenotes the simplicial (W )- algebra formed by taking the coproduct in`each dimension of the (W )-algebras Ap; A0p: Define the pth derived functor, p; of the coproduct functor as ` 0 ` 0 A pA := ssp(Ao Ao): The standard theory`of derived functors in`[22],[23], [1], and the arguments * *of [20] show that pis well-defined. That is, pdoes not depend on the choice of free simplicial resolutions for the (W )-algebras. Lemma 6.10. The zeroth derived functor of the coproduct`functor`is isomorphic to the coproduct functor itself. In other words, A 0A0~=A A0for (W )-algebras A and A0: 12 MICHELE INTERMONT Proof.This follows from the fact that for a simplicial (W )-algebra Ao, ss0Ao is isomorphic to colim Ao: __|_| Now all of the pieces are in place to prove the main theorem. Proof.(of Theorem 1.1) For each H; Theorem 5.2 established a spectral sequence converging to ssHW+*(X _ Y ) with E2p;*(H) = sspssHW+*(Xo _ Yo): For each p; the columns {E2p;*(H)}*;H form a (W )-algebra. Since W (Xp _ Yp) is a free (W )- algebra, Proposition 6.4 shows ` W (Xp _ Yp) = (W Xp) (W Yp) for all p 0: But (Xo`_ Yo) was defined by taking`the wedge on each level p, so W (Xo _ Yo) = (W Xo) (W Yo); and E2p;*= ssp(W Xo W Yo): By Proposi- tion 6.8, W Xo and W Yo are`free simplicial resolutions of W X; W Y respec- tively. So, E2p;*= (W X) p(W Y ): __|_| 7.Extensions As stated in the introduction, Theorem 1.2 provides a range in which the copr* *od- uct functor is additive in the setting of equivariant homotopy. Using the vanis* *hing of the derived functors of the coproduct functor corresponding to this additive range, one can recover the exact sequence of Corollary 1.3. Proof.(of Theorem 1.2) Non-equivariantly, the map from X * Y into the homo- topy fibre, F , of X _ Y ! X x Y is known to be a homotopy equivalence [9]. It is not difficult to show that if X and Y are G-spaces, then this map is a G-map. To show that it is a G-homotopy equivalence, by Proposition 2.1, it is enough to show that (X * Y )H ! F H is a homotopy equivalence for each H G. But (X *Y )H is homeomorphic to (XH )*(Y H), and F H is homeomorphic to the homotopy fibre of XH _ Y H ! XH x Y H. Ganea's non-equivariant result cited above now shows that (X * Y )H ~=(XH ) * (Y H) ! F H is a homotopy equivalence. Thus, for each H G, we have a long exact sequence . .s.sHW+*(X * Y ) ! ssHW+*(X _ Y ) ! ssHW+*(X x Y ) . . . of equivariant homotopy groups. Under the connectivity assumptions for X and Y; (XH ) * (Y H) is (2 dim(W H) + r + s)-connected for any H G; so X * Y is (2 dim(W *) + r + s) -connected. This means ssHW+i(X * Y ) vanishes for i dim(W H) + r + s which implies ssHW+i(X _ Y ) ~=ssHW+i(X x Y ) ~=ssHW+iX ssHW+iY for 0 i dim(W H) + r + s: Notice that ssHW+i(X _ Y ) ~=* for i min(r; s) since ssHW+iX ~=* when i r and ssHW+iY ~=* when i s: __|_| AN EQUIVARIANT VAN KAMPEN SPECTRAL SEQUENCE 13 Proof.(of Corollary 1.3) For X a (W + r)*-connected based G-CW complex, let X(r)odenote the free simplicial resolution of X formed as is the generalized sp* *here resolution, but using only generalized spheres SW+nH, n r + 1: By Theorem 1.2, ssHW+i(X(r)o_ Yo(s)) = (ssHW+iX(r)o) (ssHW+iYo(s)) for i r + s + dim(W H): Proposition 3.3 shows that applying ssp to this gives the trivial group for all p 1; and, of course, for p = 0 the assumed connectiv* *ity gives the trivial group. The terms sspssHW+i(X(r)o_ Yo(s)) are derived functors* * of the coproduct functor which are independent of the choice of free simplicial resolu* *tion. Thus, E2p;i(H) = sspssHW+i(Xo _ Yo) = * for p > 0 and i < q = r + s + dim(W H): This means that there are no non-zero differentials hitting or emanating from E21;q(H); and only one differential aff* *ecting E22;q(H): Thus, there is a short exact sequence 0 ! E12;q(H) ! E22;q(H) ! E20;q+1(H) ! E10;q+1(H) ! 0: In view of the filtration for a first quadrant spectral sequence, this gives th* *e exact sequence: ssHW+q+2(X _ Y ) ! E22;q(H) ! E20;q+1(H) ! ssHW+q+1(X _ Y ) ! E11;q(H) ! 0: __|_| The wedge of two based spaces is a particular case of a pointed homotopy colimit. It is natural to think of extending Theorem 1.1 to pointed homotopy colimits of arbitrary diagrams [3, XII 2.1]. Theorem 7.1. Let I be a small category, X_an I-diagram of based, W *-connected G-CW complexes. Then for each subgroup H of G, there is a first quadrant spec- tral sequence converging to ssHW+*(hocolim X_): For each p, the columns E2p;*:= {E2p;*(H)}H can be described as the pth derived functor of the colimit functor ((W )-al)I ! (W )-al: As in the case of the wedge, we need to construct free simplicial resolutions* * of I-diagrams of (W )-algebras. Once this has been done, Theorem 5.2 will again provide the spectral sequences, and it will remain only to reinterpret the E2 t* *erms. Definition 7.2.([20, x6.1]) Let I be a small category, and I0 the category obta* *ined from I by forgetting the non-identity morphisms. Let S be a category with copro* *d- ucts, and let D : SI0! SI be the functor which takes a collection {S(i)}i2Ito * *the free I-diagram on {S(i)}. The I-diagram D{S(i)} is defined at an object i0 2 I as a S(i): i2I;ff:i! i0 For a morphism fi : i0 ! i1 in I, D{S(i)}(fi) is defined to be the map which se* *nds the copy of S(i) indexed by ff to the copy indexed by fi O ff : i ! i1: 14 MICHELE INTERMONT The free functor D : SI0 ! SI has a right adjoint in the functor O which forgets all non-identity morphisms. The natural transformations j : DO ! id and O : id ! DO arising from this adjunction will be used in the definition of the cotriple given below. In particular, when S is the category of (W )-algebras, D can be precomposed with the free (W )-algebra functor F (considered as a functor of I0-diagrams) of Definition 6.3. The result is that given a collection T (i) := {T (i)n;H} i* *n the category CI0 of pointed sets graded over integers n 1 and subgroups H of G; then DF : CI0 ! (W )-alI is a free I-diagram of (W )-algebras generated by {T (i)}i. Combining the adjunctions of Definitions 6.3 and 7.2 we have: Proposition 7.3.Let A_be an I-diagram of (W )-algebras, U the forgetful functor of Definition 6.3. There is a natural isomorphism ~= Hom(W)-alI(D{F(T (i))}; A_) -! HomCI0({T (i)}; UOA_): Beginning with an I-diagram X_ of G-spaces, we use the resolution Xo con- structed in x3 for a G-space X to construct a similar resolution of X_. Namely, the cotriple (V; ffl; fi) induces a cotriple (V_; ffl_; fi_) which gives rise t* *o an associated simplicial I-diagram X_oof G-spaces augmented by X_. (Equivalently, Xo_can be thought of as an I-diagram of simplicial G-spaces.) The diagram V_(X_) is the f* *ree diagram DV O X_and the maps are ffl_:= j O DfflO and fi_:= DV OVO O DfiO: When X_is an I-diagram of G-CW complexes, X_osatisfies analogs of Proposi- tions 3.2, 3.3 (see [20, x6]). Once again, these propositions insure that the s* *pectral sequence of Theorem 4.4 collapses for each i, so that |X_o(i)| 'G X_(i): They a* *lso insure that the simplicial diagram, W (X_o), of (W )-algebras is a free simplic* *ial resolution of the diagram, W (X_), of (W )-algebras. By a free simplicial resol* *u- tion of an diagram, A_, we mean a simplicial diagram A_owhich is a free diagram in each simplicial dimension such that sspA_ois trivial for all p > 0 and the m* *ap ss0A_o! A_, induced by ffl_, is an isomorphism (see [20, x6]). The G-homotopy e* *quiv- alence of |X_o(i)| and X_(i) for each i 2 I and the following proposition estab* *lish the G-homotopy equivalence of their respective homotopy colimits. Proposition 7.4.Let A_ and B_ be I-diagrams of based, W *-connected G-CW complexes and f : A_! B_a map of I-diagrams which is a G-homotopy equivalence for each i 2 I. Then the map hocolim A_! hocolim B_, induced by f, is a G- homotopy equivalence. Proof.For H G, let A_H; B_Hbe the diagrams of fixed points. Then the map fH : hocolim A_H ! hocolim B_H is a homotopy equivalence [3, XII 4.2]. But hocolim (A_H) is isomorphic to (hocolim A_)H , and likewise for B, so Propositi* *on 2.1 applies. __|_| Proof.(of Theorem 7.1) X_ois a simplicial diagram of W *-connected G-CW com- plexes, since each X_(i) is a wedge of SVH+njj; j 1: Therefore, hocolim X_o is a simplicial W *-connected G-CW complex. For each H G, Theorem 4.4 and Lemma 4.3 provide a spectral sequence with E2p;q(H) = sspssHW+q(hocolim X_o) and which converges to ssHW+p+q|hocolim X_o| ~=ssHW+p+qhocolim |X_o|. Since X_ is a AN EQUIVARIANT VAN KAMPEN SPECTRAL SEQUENCE 15 diagram of W *-connected G-CW complexes, ssHW+p+qhocolim |X_o| is isomorphic to ssHW+p+qhocolim X_be Proposition 7.4 It remains to show that the E2 term has the stated description. First, we est* *ab- lish that hocolim X_oand colim X_oare G-homotopy equivalent in each simplicial dimension. 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