AN EQUIVARIANT VAN KAMPEN SPECTRAL SEQUENCE. July 28, 1995.

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1 AN EQUIVARIANT VAN KAMPEN SPECTRAL SEQUENCE MICELE INTERMONT July 28, Introduction Let G be a finite group, W a finite dimensional representation of G and (W + n) the representation (W R n ) with G acting trivially on R n. ForabasedG-space X and a subgroup of G, πw +n (X) is defined as the based set [S(W +n),x] of -homotopy classes of based -maps from S W +n into X. This is a group if n 1, and an abelian group if n 2. This paper considers the problem of computing {πw +n (X Y )} n,, n 0, 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. The E 2 term of this spectral sequence (which is the collection of E 2 terms of spectral sequences associated to each ) has a nice algebraic description. Given a based G-space X, the collection of based sets {πw +n (X)} n, with all the primary homotopy operations between the sets is called a Π(W )-algebra and denoted by Π W (X). The category of such objects is denoted Π(W )-al. There is a coproduct functor :(Π(W)-al) 2 Π(W )-al which has derived functors p. A G-space X is said to be W -connected if for each subgroup of G, the fixed point set X is n-connected in the usual sense, where n is the dimension of W. We can now state the main result: Theorem 1.1. Let X, Y be based W -connected G-CW complexes. For each subgroup of G there exists a first quadrant spectral sequence {Ep,q()} r converging to {πw +p+q (X Y )} p+q, p + q 0. For fixed p, the collection of columns Ep, 2 := {Ep, ()} 2 is a Π(W )-algebra isomorphic to Π W (X) p Π W (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 pointed homotopy colimits. That is, we will show that there is a spectral sequence converging 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 E 2 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 Π(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, 5]. The second main result provides a range in which the coproduct functor is additive in this equivariant setting. Corollary 1.3 is an immediate application of Theorem

2 2 MICELE 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 G, πw +i (X Y ) = for 0 i min(r, s). When i dim(w )+r + s, the natural maps X X Y Y induce an isomorphism of abelian groups πw +i (X Y ) = πw +i (X) π W +i (Y ). Corollary 1.3. If X and Y are based G-CW complexes satisfying the conditions of Theorem 1.2, thenforeach G, there is an exact sequence of abelian groups: π W +q+2 (X Y ) E2 2,q () E2 0,q+1 () π W +q+1 (X Y ) E2 1,q () 0 where q = r + s +dim(w ). Recovery of the van Kampen Theorems: For a fixed n and G-space X, let π W +n (X) denote the collection {πw +n (X)} 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 Ep,0 2 () (set,ifp = 0) in Theorem 1.1 are all trivial, which means that the groups E0,1 2 () in the lower left corners of the E2 () terms are not affected by differentials. In light of this, E0,1() =E0,1() 2 and, by Lemma 6.2, π W +1 (X Y )=π W +1 (X) π 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 diagrams. 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 recover 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 X which is, in a sense, a resolution of X by G-spheres. Given two such simplicial G-spaces, the simplicial G- space X Y, defined as X n Y n in dimension n, has nice properties. In particular, the work of Bousfield and Friedlander [2] yields a spectral sequence converging to based -maps from S W into (X Y ). On the Organization of the Paper: To begin, some pertinent facts of equivariant homotopy theory are recalled in Section 2. Section 3 contains the construction of a nice simplicial resolution of X. In addition, Section 3 relates weak W -equivalence 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 converging to πw + (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 Π(W )-algebras is introduced in Section 6 and used to complete the proof of Theorem 1.1. Section 7 extends these results to pointed homotopy colimits, and gives 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

3 AN EQUIVARIANT VAN KAMPEN SPECTRAL SEQUENCE 3 Luce Foundation for financial support. I would also like to thank the referee for 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 fixed 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, Map G (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 ausdorff 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/ + D n 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 repetition): G 1, 2,... m e i j if i j where j is the isotropy subgroup of the attached cell. A useful fact concerning G-CW complexes is the following: Proposition 2.1. ([7, 2.7]) Let f : Y Z be a G-map between G-CW complexes such that f : Y Z is a homotopy equivalence for all. Then f is a G- homotopy equivalence. Fixing a representation W of G, the one-point compactification of W is denoted by S W. This G-space is a sphere of dimension equal to the real dimension of W, with the compactification point as fixed basepoint. D W +1 is the reduced cone over S W. If V is another representation of G, V + W is the direct sum of the representations. The trivial representation of dimension n will be denoted simply by the integer n. The group G acts diagonally on the object G/ + S W +n, which we call a generalized G(W )-sphere. It can be thought of as a wedge of S W +n,whereg acts not only on each wedge summand, but also by permuting the summands. For compactness of notation, we write S W +n for G/ + S W +n. Of course, S W +n is the boundary of a generalized disk G/ + D W +n+1, abbreviated D W +n+1. The set [S W,X] G is actually a group whenever dim(w G ) 1, and an abelian group whenever dim(w G ) 2. The collection {[S W,X] G}, a subgroup of G, is called the collection of W th equivariant homotopy groups (sets) of X. [S W,X] G will be written as πw (X). Recall that [SW,X] G = [S W,X] (see [7]), so the notation πw (X) is unambiguous. Note too, that π n (X) = π n (X ). Definition 2.2. ([13, 1.1]) A G-space X is said to be W -connected if, for each subgroup of G, X is n-connected in the ordinary sense where n is the dimension of W. Remark If X is W -connected, then π W (X) for each subgroup of G, as Lemma 4.1 will show.

4 4 MICELE INTERMONT The orbit category of G is denoted by O G. The objects are G/, where is a subgroup of G, and the morphisms are G-maps. For any G-space X, we can define a functor π W (X) :O op G Sets which maps G/ to πw (X) and gives, for each G-map G/ G/K, amapπw K (X) π W (X) [13]. 3. Techniques The Resolution Given a based G-space X, this section constructs a simplicial resolution, X,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 Π(W )-algebra produced from X (see 6) is a free simplicial resolution of the Π(W )-algebra produced for X. This simplicial space is constructed functorially by an inductive process. Let VX be the space given by the following pushout: n 1 +n h (SW - ) h n 1 h (DW +n+1 ) h n 1? -? f +n (SW ) f VX (Diagram 1) where the indexing G-map h runs through all the G-maps from D W +n+1 to X and f runs through all the G-maps from S W +n to X. The upper horizontal arrow is induced by the inclusion of the boundary of a disk into the disk itself: D W +n+1 = S W +n D W +n+1. The left vertical map takes each wedge summand (S W +n ) h by the identity to the wedge summand (S W +n ) f where f : S W +n X is the restriction of h to the boundary of D W +n+1.gacts on VX in the natural way. This construction comes with natural maps (i) ɛ : VX X which sends (S W +n ) f into X by the indexing map f and (D W +n+1 ) h into X by the indexing map h. (ii) β : VX V 2 X which sends (S W +n ) f G-homeomorphically to the copy of S W +n in V 2 X indexed by the inclusion of the sphere into VX and which sends (D W +n+1 ) h G-homeomorphically to the copy of D W +n+1 in V 2 X indexed by the inclusion. It can be checked that (V,ɛ,β) is a cotriple (or comonad, in the language of [15]). Following [11], there is an associated cellular simplicial G-space, X, augmented by ɛ : X 0 X, with X i := V i+1 X, i 0. The face and degeneracy maps are given by d j = V j ɛv p j : X p X p 1 s j = V j βv p j : X p X p+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 X described above is called the generalized sphere resolution of X.

5 AN EQUIVARIANT VAN KAMPEN SPECTRAL SEQUENCE 5 The propositions which follow are equivariant versions of those found in [20, 2]. It is these properties of the generalized sphere resolution which will be applied in 6. Proposition 3.2. Let X be a based G-space, X its generalized sphere resolution. There exist G-contractible subcomplexes C p in X p such that (i) For all p 0 the quotient X p /C p is a wedge of generalized W -spheres S W +n,n 1, G. (ii) s j C p C p+1 for all degeneracy maps s j : X p X p+1. (iii) For j =0,...,p the map s j : X p/c p X p+1 /C p+1 induced by s j is an inclusion of wedges. Recall that when X is a simplicial space, π q X is a simplicial group, and so has homotopy groups π p π q X itself. These groups can be calculated as the homology of a particular chain complex following [17, 17]. They will form the E 2 term of a spectral sequence in 4. Proposition 3.3. Let X be a W -connected G-CW complex, with generalized sphere resolution X.IfY denotes Map (S W,X ) and Y = Map (S W,X) then (i) π p π q (Y )=0for all p 1 and q 1. (ii) For all q 1 the augmentation map induces an isomorphism π 0 π q (Y ) = π q (Y ). This last proposition is established by showing that every cycle in the p th group of the chain complex actually bounds an element (see [20, 2]). The G-CW Decomposition of S W For any subgroup K of G, the cellular structure of S W allows us to approach Map K (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 S W is not technically a based G-CW complex unless the fixed point set (S W ) G is connected. owever, since W is a representation and (S W ) is itself a sphere, it is not too difficult to see that we can choose a G-CW decomposition of S W which attaches pointed cells at each stage through basepoint-preserving attaching maps, even when dim(s W ) G =0. LetZ 1 =(S W ) G. We write Z 1 Z 2 Z m = S W,whereZ j,2 j m, is the cofibre of f : S i i Z j 1,andf is a basepoint-preserving attaching map. Let Y be a G-space. Applying Map K (,Y) to the cofibre sequence determined by Z j,j 2, yields a fibre sequence Map K (Z j,y) Map K (Z j 1,Y) Map K (Si i,y). The term Map K (S i i,y) is isomorphic to l Map (S i,y K l ), where K l = K g l for representatives g l of the orbits of the left action of K on G/. g 1 l Definition 3.4. [13, 3.6] Let f : X Y be a G-map. f is a weak W -equivalence if f induces a homotopy equivalence between mapping spaces Map (SW,X) Map (SW,Y) for each G.

6 6 MICELE 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 equivalence. Proof. ( ) Proceed by induction on the order of G. If the order of G is 1, the result is clear, since then S W is just an ordinary sphere of dimension l =dim(w ). In 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 f : X Y is an equivalence for all proper subgroups of G. If we show that f G : X G Y G is an equivalence, then Proposition 2.1 will finish the proof in this direction. Notice that if S W =(S W ) 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 S W (S W ) G,then S W = Z m 1 f D i for some proper subgroup and some i. Consider the ladder: Map G (SW,X) Map G (Z m 1,X) Map G (Si 1,X) γ α β Map G (S W,Y) Map G (Z m 1,Y) Map G (S i 1,Y) Since f induces a weak W -equivalence by assumption, γ is a homotopy equivalence. Since Map G (S i 1 i, ) = Ω i 1 ( ), β is a homotopy equivalence by induction, and by the 5-lemma, α is a homotopy equivalence above dimension 0. In dimension 0, notice that π 0 Map G (Si 1, ) is trivial, since the dimension of the cell D i must be dim W,andsoα is indeed a homotopy equivalence. Continuing backwards through the decomposition of S W, we conclude that Map G (Z 1,X) = Map G (Z 1,Y). But Z 1 is just (S W ) G,sowehave Map ((S W ) G,X G ) Map ((S W ) 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 Y be a based, suitably connected simplicial G-space. This section applies the techniques of Bousfield-Friedlander in [2] to Map (S W,Y ) to obtain the homotopy spectral sequence of Theorem 4.4. Lemma 4.1 establishes a connectivity condition for the spaces Y n Y which is sufficient for the convergence of the spectral 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 Map G (S W,Y) is 0- connected. Proof. Let Z 1 =(S W ) G...Z j+1 = Z j f D i+1 Z m = S W be the G-CW decomposition of S W as in 3. Since Y G is assumed to be dim W G -connected, π 0 Map G ((SW ) G,Y) = [(S W ) G,Y] =. Now continue by induction. Assume π 0 Map G (Z j,y)=. Consider the homotopy fibration sequence π 1 Map G (S i,y) π 0 Map G (Z j+1,y) π 0 Map G (Z j,y).

7 AN EQUIVARIANT VAN KAMPEN SPECTRAL SEQUENCE 7 Map G (Si,Y) Ωi (Y ), and since i must be less than or equal to the dimension of W, this is 1-connected. Thus, π 0 Map G (Z j+1,y) is trivial. Remark In particular, Lemma 4.1 says that π W of a based, W -connected G-CW complex is trivial for all subgroups of G. Lemma 4.2. Let X be a G-CW complex and let X be the generalized sphere resolution of X. Then X i is W -connected for each X i X,i 0. Proof. By construction, X i is a pushout for each i 0 (see Diagram 1 in 3). 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 (X i ) K. By assumption, each space involved in the resolution is a suspension of S W or a cone over a suspension of S W, and hence is at least W -connected. Therefore, all of the spaces in the pushout diagram for (X i ) K are n-connected where n =dimw K. Now a standard argument with the van Kampen and urewicz theorems shows that (X i ) K is also n-connected, n equal to dim W K. This is true for all subgroups K of G, so X i is W -connected. Let [n] be the standard n-simplex. Recall [19] that for a simplicial space X, the geometric realization X is formed from ( n 0 [n] X n )/( [n] ) 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 are 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 X is a simplicial G-space, then X is a G-space. For A abasedg-space, a subgroup of G, andy any simplicial G-space, let Map (A, Y ) denote the simplicial space defined by Map (A, Y ) n := Map (A, Y n ), with the face and degeneracy maps given by composition with those from Y. Lemma 4.3. If Y is a simplicial G-space such that Y i is W -connected for each i, then Map (SW,Y ) Map (SW, Y ) for all G. Proof. This result is established for unbased maps in [6, 5.4]. Now consider the diagram: Map (S W,Y ) Map (S W,Y ) Y γ α β Map (SW, Y ) Map (S W, Y ) Y

8 8 MICELE INTERMONT of fibrations. Since α and β are homotopy equivalences, so is γ. In other words, realization commutes with equivariant loop spaces, subject to some connectivity conditions. The first quadrant homotopy spectral sequence for bisimplicial sets, established in [2, B.5] by Bousfield-Friedlander, has a simplicial space analog [5, Appendix]. For a connected simplicial space, the spectral sequence converges to the homotopy of the realization. In conjunction with Lemma 4.3, this immediately gives: Theorem 4.4. Let Y be a simplicial G-space such that Y i is W -connected for each i. Thenforeach G there exists a first quadrant spectral sequence which converges strongly to the homotopy groups of Map (SW, Y ) with E 2 p,q = π p π q Map (S W,Y ). The generalized sphere resolution of a G-space Y is an example of a W -connected simplicial G-space. The spectral sequence associated to this simplicial G-space collapses 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 X be its generalized sphere resolution. Then the augmentation map ɛ : X 0 X induces a G-homotopy equivalence X X. Proof. By Proposition 3.3, E 2 = E in the spectral sequences of Theorem 4.4 which gives: π 0 π q Map (S W,X ) = π 0+q Map (S W, X ). Again by Proposition 3.3, π 0 π q Map (SW,X ) = π q Map (SW,X). ence, π q Map (S W, X ) = π q Map (S W,X). NowbyTheorem3.5,thisweakW -equivalence is actually a G-homotopy equivalence. 5. The Spectral Sequence for a Wedge This section establishes a spectral sequence which converges to the equivariant 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 E 2 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 X its generalized sphere resolution, X is G-homotopy equivalent to X. The proof of Theorem 5.2 depends on knowing that this holds for the wedge of two such complexes X and Y : that the realization of X Y is G-homotopy equivalent to the wedge X Y. Lemma 5.1. If X and Y are based, W -connected G-CW complexes with generalized sphere resolutions X,Y,then X Y is G-homotopy equivalent to X Y.

9 AN EQUIVARIANT VAN KAMPEN SPECTRAL SEQUENCE 9 Proof. By Corollary 4.5, there are G-homotopy equivalences f : X X and g : Y Y. In particular, X K X K and Y K Y K for all subgroups K of G. Now X K Y K X Y K and X K Y K (X Y ) K, so X Y 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 X,Y be their respective generalized sphere resolutions. For each subgroup K of G there exists a spectral sequence converging to πw K +p+q (X Y ) with E 2 p,q (K) =π pπ K W +q (X Y ). Proof. The simplicial G-space X Y is W -connected in each dimension because both X and Y are W -connected. Now for each K G, Theorem 4.4 gives a spectral sequence for X Y. Lemma 5.1 shows that X Y is G-homotopy equivalent to X Y so the spectral sequence can be said to converge to π K W +p+q (X Y ). Remark Let X be any simplicial G-space. A G-map G/ G/K induces a map of simplicial G-spaces φ : Map K (SW,X ) Map (SW,X ). When each X i is W -connected, φ induces a map of spectral sequences {Ep,q r (K)} {Er p,q ()} where Ep,q( ) r is the spectral sequence of Theorem 5.2. Thus, Ep,q( ) r canbe considered a functor from O op G into the category of spectral sequences. 6. Π(W )-Algebras In this section we discuss the structure of Π(W )-algebras, the motivating example 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 of G, a spectral sequence. These are the spectral sequences of Theorem 1.1. The columns of the E 2 terms of these spectral sequences, when considered together, have the structure of Π(W )-algebras, and it is this structure that will allow us to understand the E 2 terms as described in the main theorem. Let Π(W ) denote the category whose objects are finite wedges k +ni i=1 SW i of generalized spheres where i G, n i 1, and whose morphisms are G-homotopy classes of G-maps. Definition 6.1. A Π(W )-algebra is a contravariant functor A : Π(W ) Sets with the property that it takes finite wedges to products in the sense that if U 0,U 1 are objects of Π(W ), then the natural inclusions i j : U j U 0 U 1, j =0, 1 induce a bijection A(U 0 U 1 ) A(U 0 ) A(U 1 ). Example For a G-space X, define the Π(W )-algebra Π W X as the functor which takes the generalized sphere S W +n to the group πw +n (X). Remarks

10 10 MICELE INTERMONT (i) It will sometimes be helpful to think of a Π(W )-algebra A as a collection of sets, A n, := {A(S W +n ) n, }, together with operations. For a G-map S W +m J k +ni i=1 SW i, these operations are of the form: A n1, 1 A nk, k A m,j. (ii) Π(W )-algebras form a category, Π(W )-al, with natural transformations as morphisms. (iii) The category Π(W )-al has limits and colimits [15, V,IX]. In particular then, it has coproducts. (iv) The Π-algebras of [20] are Π(W )-algebras with G = {e} and W =0. (v) If A is a Π(W )-algebra and U 0 is an object of Π(W ), then A(U 0 ) is actually a group. owever, if f is a morphism of Π(W ), A(f) is not necessarily a group homomorphism. The proposition below records the fact that Π(W )-algebras are generalizations of Lewis W -Mackey functors [13]. This will justify the claim in the introduction that the van Kampen theorem of [13] appears as the edge of the spectral sequence in Theorem 1.1. Proposition 6.2. Let A :Π(W) op Sets be a Π(W )-algebra. Then A determines a functor A : O op G Sets which maps wedges to products in the sense of Definition 6.1. Proof. Since there are no non-trivial maps S W +m S W +n K when m n, there are no non-trivial maps A(S W +n K ) A(S W +m ). Thus, the only maps into A(S W ) are from A(SK W ). ence, define A(G/) :=A(SW ). If f : G/ G/K, define A(f) :=A(f id). Definition 6.3. Let C be the category of pointed sets graded by pairs {n, } where n is an integer greater than or equal to 1 and is a subgroup of G. Then there is a natural forgetful functor U :Π(W)-al Cwhich takes a Π(W )-algebra A to the collection {A(S W +n )} n, C. This functor U has a left adjoint F defined on {T n, } C by F(T ):=Π W ( n 1 µ (S W +n ) µ ), µ T n, { } (proof analogous to that in [20, 4]). F(T ) is called the free Π(W )-algebra on T. Proposition 6.4. Let U 0,U 1 be elements of Π(W ). Then Π W (U 0 U 1 )=Π W (U 0 ) Π W (U 1 ). Proof. Let T Cbe the graded pointed set with F(T )=Π W (U 0 )andlets C be the graded pointed set with F(S) =Π W (U 1 ). Then T S is the graded pointed set with F(T S) =Π W (U 0 U 1 ). Since F is left adjoint to the forgetful functor, F(T S) =F(T ) F(S), and the result follows.

11 AN EQUIVARIANT VAN KAMPEN SPECTRAL SEQUENCE 11 We now recall the necessary definitions regarding simplicial objects over the category of Π(W )-algebras, and derived functors. Definition 6.5. A simplicial Π(W )-algebra is a contravariant functor A : Π(W )-al. Equivalently, this is a contravariant functor A : Π(W ) ssets with the special property that it takes finite wedges isomorphically to products in the sense of Definition 6.1. A is said to be augmented by A if there is a map ɛ : A 0 A such that ɛ d 1 = ɛ d 0. Proposition 6.6. Let A be a simplicial Π(W )-algebra. Then is a Π(W )-algebra for p 0. π p A : Π(W ) ssets Sets Proof. Since the image of a Π(W )-algebra at any object of Π(W ) is actually a group, the image of A at any object is a simplicial group. ence, π p A can be computed by the methods of ([17, 3.6]). To see that π p A is a Π(W )-algebra, it is enough to note that A takes finite wedges to products and the functor π p preserves finite products. Definition 6.7. A free simplicial resolution of a Π(W )-algebra A is a simplicial Π(W )-algebra A augmented by A such that (i) For all i, A i is a free Π(W )-algebra on a collection T i C, (ii) s j T i T i+1 for all degeneracy maps s j, 0 j i, (iii) π p A =0for p 1, and = (iv) The augmentation map induces an isomorphism π 0 A A. Proposition 6.8. Let X be the generalized sphere resolution of X. Then Π W (X ) is a free simplicial resolution of the Π(W )-algebra Π W (X). Proof. This follows immediately from Propositions 3.2 and 3.3. Definition 6.9. (Derived Functors) Let A,A be free simplicial resolutions of the Π(W )-algebras A, A respectively. Then A A denotes the simplicial Π(W )- algebra formed by taking the coproduct in each dimension of the Π(W )-algebras A p, A p. Define the p th derived functor, p, of the coproduct functor as A p A := π p (A A ). The standard theory of derived functors in [22],[23], [1], and the arguments of [20] show that p is well-defined. That is, p does not depend on the choice of free simplicial resolutions for the Π(W )-algebras. Lemma The zeroth derived functor of the coproduct functor is isomorphic to the coproduct functor itself. In other words, A 0 A = A A for Π(W )-algebras A and A.

12 12 MICELE INTERMONT Proof. This follows from the fact that for a simplicial Π(W )-algebra A, π 0 A is isomorphic to colim A. Now all of the pieces are in place to prove the main theorem. Proof. (of Theorem 1.1) For each, Theorem 5.2 established a spectral sequence converging to π W + (X Y ) with E2 p, () =π p π W + (X Y ). For each p, the columns {E 2 p, ()}, form a Π(W )-algebra. Since Π W (X p Y p ) is a free Π(W )- algebra, Proposition 6.4 shows Π W (X p Y p )=(Π W X p ) (Π W Y p ) for all p 0. But (X Y ) was defined by taking the wedge on each level p, so Π W (X Y )=(Π W X ) (Π W Y ), and E 2 p, = π p (Π W X ΠW Y ). By Proposition 6.8, Π W X and Π W Y are free simplicial resolutions of Π W X, Π W Y respectively. So, E 2 p, =(Π W X) p (Π W Y ). 7. Extensions As stated in the introduction, Theorem 1.2 provides a range in which the coproduct functor is additive in the setting of equivariant homotopy. Using the vanishing 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 ΩX ΩY into the homotopy fibre, F,ofX Y 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 (ΩX ΩY ) F is a homotopy equivalence for each G. But (ΩX ΩY ) is homeomorphic to Ω(X ) Ω(Y ), and F is homeomorphic to the homotopy fibre of X Y X Y. Ganea s non-equivariant result cited above now shows that (ΩX ΩY ) = Ω(X ) Ω(Y ) F is a homotopy equivalence. Thus, for each G, we have a long exact sequence...π W + (ΩX ΩY ) π W + (X Y ) π W + (X Y )... of equivariant homotopy groups. Under the connectivity assumptions for X and Y, Ω(X ) Ω(Y )is (2 dim(w )+r + s)-connected for any G, so ΩX ΩY is (2 dim(w )+r + s) -connected. This means πw +i (ΩX ΩY ) vanishes for i dim(w )+r + s which implies πw +i (X Y ) = πw +i (X Y ) = πw +i X π W +i Y for 0 i dim(w )+r + s. Notice that πw +i (X Y ) = for i min(r, s) since πw +i X = when i r and πw +i Y = when i s.

13 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) denote the free simplicial resolution of X formed as is the generalized sphere resolution, but using only generalized spheres S W +n, n r +1. By Theorem 1.2, π W +i(x (r) Y (s) )=(π W +ix (r) ) (π W +iy (s) ) for i r + s +dim(w ). Proposition 3.3 shows that applying π p to this gives the trivial group for all p 1, and, of course, for p = 0 the assumed connectivity gives the trivial group. The terms π p πw +i (X(r) Y (s) ) are derived functors of the coproduct functor which are independent of the choice of free simplicial resolution. Thus, Ep,i 2 () =π pπw +i (X Y )= for p>0andi<q= r + s +dim(w ). This means that there are no non-zero differentials hitting or emanating from E1,q 2 (), and only one differential affecting E2,q(). 2 Thus, there is a short exact sequence 0 E2,q () E2 2,q () E2 0,q+1 () E 0,q+1 () 0. In view of the filtration for a first quadrant spectral sequence, this gives the exact sequence: πw +q+2 (X Y ) E2 2,q () E2 0,q+1 () π W +q+1 (X Y ) E 1,q () 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 of G, there is a first quadrant spectral sequence converging to π W + (hocolim X). For each p, the columns E2 p, := {E 2 p, ()} can be described as the p th derived functor of the colimit functor (Π(W )-al) I Π(W )-al. As in the case of the wedge, we need to construct free simplicial resolutions of I-diagrams of Π(W )-algebras. Once this has been done, Theorem 5.2 will again provide the spectral sequences, and it will remain only to reinterpret the E 2 terms. Definition 7.2. ([20, 6.1]) Let I be a small category, and I 0 the category obtained from I by forgetting the non-identity morphisms. Let S be a category with coproducts, and let D : S I0 S I be the functor which takes a collection {S(i)} i I to the free I-diagram on {S(i)}. TheI-diagram D{S(i)} is defined at an object i 0 I as S(i). i I,α:i i 0 For a morphism β : i 0 i 1 in I, D{S(i)}(β) is defined to be the map which sends the copy of S(i) indexed by α to the copy indexed by β α : i i 1.

14 14 MICELE INTERMONT ThefreefunctorD : S I0 S I has a right adjoint in the functor O which forgets all non-identity morphisms. The natural transformations η : DO id and χ : id DOarising from this adjunction will be used in the definition of the cotriple given below. In particular, when S is the category of Π(W )-algebras, D can be precomposed with the free Π(W )-algebra functor F (considered as a functor of I 0 -diagrams) of Definition 6.3. The result is that given a collection T (i) :={T (i) n, } in the category C I0 of pointed sets graded over integers n 1 and subgroups of G, then DF : C I0 Π(W )-al I is a free I-diagram of Π(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 Π(W )-algebras, U the forgetful functor of Definition 6.3. There is a natural isomorphism = om Π(W )-ali (D{F(T (i))},a) om C I 0 ({T (i)}, UOA). Beginning with an I-diagram X of G-spaces, we use the resolution X constructed in 3 for a G-space X to construct a similar resolution of X. Namely, the cotriple (V,ɛ,β) induces a cotriple (V,ɛ,β) which gives rise to an associated simplicial I-diagram X of G-spaces augmented by X. (Equivalently, X can be thought of as an I-diagram of simplicial G-spaces.) The diagram V(X) isthefree diagram DVOX and the maps are ɛ := η DɛO and β := DVχVO DβO. When X is an I-diagram of G-CW complexes, X satisfies analogs of Propositions 3.2, 3.3 (see [20, 6]). Once again, these propositions insure that the spectral sequence of Theorem 4.4 collapses for each i, sothat X (i) G X(i). They also insure that the simplicial diagram, Π W (X ), of Π(W )-algebras is a free simplicial resolution of the diagram, Π W (X), of Π(W )-algebras. By a free simplicial resolution of an diagram, A, we mean a simplicial diagram A which is a free diagram in each simplicial dimension such that π p A is trivial for all p>0andthemap π 0 A A, induced by ɛ, is an isomorphism (see [20, 6]). The G-homotopy equivalence of X (i) and X(i) foreachi I and the following proposition establish 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 I. Then the map hocolim A hocolim B, induced by f, isaghomotopy equivalence. Proof. For G, leta,b be the diagrams of fixed points. Then the map f : hocolim A hocolim B is a homotopy equivalence [3, XII 4.2]. But hocolim (A ) is isomorphic to (hocolim A), and likewise for B, so Proposition 2.1 applies. Proof. (of Theorem 7.1) X is a simplicial diagram of W -connected G-CW complexes, since each X(i) isawedgeofs V +nj j,j 1. Therefore, hocolim X is a simplicial W -connected G-CWcomplex. Foreach G, Theorem 4.4 and Lemma 4.3 provide a spectral sequence with Ep,q 2 () =π pπw +q (hocolim X )and which converges to πw +p+q hocolim X = πw +p+q hocolim X. Since X is a

15 AN EQUIVARIANT VAN KAMPEN SPECTRAL SEQUENCE 15 diagram of W -connected G-CW complexes, πw +p+q hocolim X is isomorphic to πw +p+qhocolim X be Proposition 7.4 It remains to show that the E 2 term has the stated description. First, we establish that hocolim X and colim X are G-homotopy equivalent in each simplicial dimension. Recall that X is a free simplicial diagram, so the non-equivariant result of [20, 6.16] gives that the map (hocolim X n ) (colim X n ) is a G-homotopy equivalence for all and all simplicial dimensions n. We apply Proposition 2.1 to conclude that hocolim X n colim X n is a G-homotopy equivalence. Now, for p 0, the collection {Ep, 2 ()}, can be rewritten as π p Π W colim X. Again, since X is a free diagram, Π W colim X = colim ΠW X by Proposition 7.3. Of course, Π W X is a free simplicial resolution of Π W X,soπ p colim Π W X is, by definition analogous to Definition 6.9, the p th derived functor of the colimit functor. References 1. M. André, Méthode simpliciale en algèbre homologique et algèbre commutative, Lecture Notes in Mathematics, 32, Springer, Berlin (1967). 2. A.K. Bousfield and E.M. Friedlander, omotopy theory of Γ-spaces, spectra, and bisimplicial sets, Geometric Applications of omotopy Theory II, Lecture Notes in Mathematics, 658, Springer, Berlin (1978) A.K. Bousfield and D.M. Kan, omotopy Limits, Completions and Localizations, Lecture Notes in Mathematics, 304, Springer, New York (1972). 4. G.E. Bredon, Equivariant Cohomology Theories, Lecture Notes in Mathematics, 34, Springer- Verlag, Berlin (1967). 5. R. Brown and J.L. Loday, van Kampen theorems for diagrams of spaces, with an appendix by M. Zisman, Topology, 26 (1987) S.R. Costenoble and S. Waner, Fixed set systems of equivariant infinite loop spaces, Trans. Amer. Math. Soc., 326 (1991) T. tom Dieck, Transformation Groups, Walter de Gruyter, Berlin, W.G. Dwyer, D.M. Kan, and C.R. Stover, An E 2 model category structure for pointed simplicial spaces, J. Pure and Applied Algebra, 90 (1993) T. Ganea, A generalization of the homology and homotopy suspension, Comment. Math. elv., 39 (1965) J. ollender and R.M. Vogt, Modules of topological spaces, applications to homotopy limits and E structures, Arch. Math., 59 (1992) P.J. uber, omotopy theory in general categories, Math. Annalen, 144 (1961) S. Illman, Smooth equivariant triangulations of G-manifolds for G a finite group, Math. Ann., 233 (1978) L.G. Lewis, Jr, Equivariant Eilenberg-MacLane spaces and the equivariant Seifert-van Kampen and suspension theorems, Topology and its Applications, 48 (1992) L.G. Lewis, Jr, J.P. May, M. Steinberger, with contributions by J.E. McClure, Equivariant Stable omotopy Theory, Lecture Notes in Mathematics, 1213, Springer, Berlin (1986). 15. S. MacLane, Categories for the Working Mathematician, Graduate Texts in Mathematics, 5, Springer, New York, J.P. May, The Geometry of Iterated Loop Spaces, Lecture Notes in Mathematics, 271, Springer, Berlin, J.P. May, Simplicial Objects in Algebraic Topology, Univ of Chicago, Chicago, G. Segal, Categories and cohomology theories, Topology, 13 (1974) G. Segal, Classifying spaces and spectral sequences, Inst.. Et. Sci. Math., 34 (1968) C.R. Stover, A van Kampen spectral sequence for higher homotopy groups, Topology, 29 (1990) D.G. Quillen, Spectral sequences of a double semi-simplicial group, Topology, 5 (1966) D.G. Quillen, omotopical Algebra, Lecture Notes in Mathematics 43, Springer, Berlin (1967).

16 16 MICELE INTERMONT 23. D.G. Quillen, On the (co-)homology of commutative rings, Proc. Symp. Pure Math. XVII, Amer. Math. Soc., Providence, RI (1970) G.W. Whitehead, Elements of omotopy Theory, Graduate Texts in Mathematics, 61, Springer, New York, Mesa State College, Grand Junction, CO 81502