EXACTNESS OF HOMOTOPY FUNCTORS OF SPACES NATHAN WODARZ Abstract.We will provide an analysis of the generalized Atiyah-Hirzebruch spectral sequence (GAHSS), which was introduced by Hakim-Hashemi and Kahn. To do so, we introduce a new class of functors, called n-exact fun* *c- tors, which are analogous to Goodwillie's n-excisive functors. In the st* *udy of these functors, we introduce a new spectral sequence, the homological Ba* *rratt- Goerss spectral sequence (HBGSS), which has properties similar to those * *of the classical Barratt-Goerss Spectral Sequence on homotopy. We close by giving an identification of the E2 term of the GAHSS in the case of 2-ex* *act functors on Moore spaces. 1.Introduction Given a functor F from the category of pointed finite CW complexes to itself * *and a finite CW complex X, there is a generalized Atiyah-Hirzebruch spectral sequen* *ce (GAHSS) converging to H*(F X), where F X is filtered by applying F to the CW structure of X. In [13], Hakim-Hashemi and Kahn study the GAHSS, and identify the E2 term in the event that F is homotopy exact, which will be defined below. They also determine a criterion on F for determining when the spectral sequence collapses. The motivation for their work can be traced back to the famous theor* *em of Dold's which states that the homology of the symmetric product SP kX depends only on the homology of the space X [6]. The GAHSS can be thought of as a rough comparison of the homology of X with the homology of F X. This article continues the study of the GAHSS a step further. We will first classify homotopy exact functors, and then generalize the concept of exactness * *to define n-exact functors. This definition is similar to Goodwillie's definition * *of n- excisiveness [10]. If X is a Moore space and n = 2, identification of the E2 te* *rm is possible by using Baues' quadratic tensor product [2, 3]. Unfortunately, the difficulties seen in extending the work of [13] lead to the conclusion that the filtration of the spectral sequence is too coarse to be use* *ful, and another path must be taken to obtain further progress. The author has derived a more tractable spectral sequence containing similar information which will be studied in a future article. Most of the results in this work originally appeared in the author's Ph.D. th* *esis, conducted under the supervision of Donald W. Kahn at the University of Min- nesota. The author would also like to thank Paul Goerss and Stewart Priddy of Northwestern University for helpful suggestions (mathematical and otherwise) du* *r- ing the writing of this article. We begin by setting some basic notation and terminology. We will work in the category CW * of finite pointed CW complexes and basepoint preserving maps. ____________ 2000 Mathematics Subject Classification. Primary 55P65, Secondary 55T25. 1 2 NATHAN WODARZ Let * denote the zero object in CW *. We denote the space of all maps in CW * from X to Y by map *(X, Y ). The space map *(X, Y ) is topologized by giving it the compact-open topology. We will use X ' Y to indicate that X and Y are homotopy equivalent. Unless otherwise stated, we will use the word "functor" to refer to an endofu* *nctor of CW *. We assume that all functors are continuous, meaning that the natural function map *(X, Y ) ! map*(F X, F Y ) is continuous when the mapping spaces are given the compact-open topology. Fur- thermore, functors will be assumed to be reduced, that is F (*) ' *. As we are working with reduced functors in CW *, we assume that all homology theories are reduced. We now present the GAHSS. If X is a CW complex, let Xp denote the p-skeleton of X. We may filter F X as follows: * = F (X-1) F (X0) F (X1) . . .F (X). This filtration gives rise to a filtered chain complex, which in turn yields a * *spectral sequence converging to H*(F X) (actually, we can get one converging to E*(F X) where E is an arbitrary generalized homology theory). If F is the identity func* *tor, it is easy to see that we obtain the classical Atiyah-Hirzebruch spectral seque* *nce. In fact, the filtration gives rise to the cellular chain complex for X. There a* *re other cases when H*O F is itself a generalized homology theory. In these cases, we ag* *ain obtain the classical AHSS. This case was studied by Hakim-Hashemi and Kahn, and requires the concept of a homotopy exact functor. We review their work now. Let C(f, g) denote the double mapping cylinder of f and g. We recall that giv* *en a commutative square f X _____//Y, g || || fflffl|fflffl| Z _____//W we have a natural map : hocolim(Y f X !g Z) ' C(f, g) ! W . If is a homotopy equivalence, the square is said to be homotopy co-Cartesian (or simply co-Cartesian). Definition 1.1. A functor F is said to be homotopy exact if whenever X is a co-Cartesian square, the diagram F X is co-Cartesian as well. Example 1.2. The functor F (X) = X ^ K for a fixed space K is homotopy exact. In fact, as we will see, up to natural equivalence, this is the only possibilit* *y. Remark 1.3. We note that the concept of homotopy exactness doesn't require the functor F to be reduced. In this event, we get a number of additional examples. In particular, if we fix a space K, then F (X) = X _ K and F (X) = X x K are both homotopy exact, as is the Borel construction F (X) = X xG EG for any group G. Details of these examples are presented in [12]. Although these examples are interesting in their own right, we will not consider them in this example. The major result of [13] involving homotopy exact functors was EXACTNESS OF HOMOTOPY FUNCTORS OF SPACES 3 Theorem 1.4. If F is a homotopy exact functor, then given any generalized ho- mology theory E*, E* O F is itself a generalized homology theory. Furthermore, * *in this case, the GAHSS for X coincides with the classical AHSS for E* O F . This theorem will guide the present work in two ways. In the next section, we will identify homotopy exact functors. In subsequent sections, we will general* *ize the concept of homotopy exactness and analyze the GAHSS in this situation. 2.Homotopy Exact Functors The major result of this section is the classification of homotopy exact func* *tors. In particular, if F is homotopy exact, then F is naturally equivalent to the fu* *nctor X ^ F (S0). We recall that we require all functors to be continuous. With this assumption, we may define a map jF :X ^ F (Y ) ! F (X ^ Y ) for any spaces X and Y . When it is clear from the context, we will use j to denote this map. Note that we ar* *e not yet assuming that F is homotopy exact. The map is given by j(x ^ y) = F (ix)(y) where ix: Y ! X ^ Y is the map sending y to x ^ y. We first note that this map is in fact continuous. Lemma 2.1. j is continuous for any choice of F , X or Y Proof.Continuity of the functor F means that the map map *(Y, X ^ Y ) ! map*(F Y, F (X ^ Y )) is continuous, and hence, so is X ! map *(Y, X ^ Y ) ! map *(F Y, F (X ^ Y )), where X ! map *(Y, X ^ Y ) is the adjoint of the identity map X ^ Y ! X ^ Y . This map obviously sends x to ix. We notice that the composite then sends x to F (ix), and its adjoint is j. Since j is in the image of the adjunction, it mus* *t be continuous. The virtue of j is that it provides a natural transformation. In particular, * *we see Lemma 2.2. The map j is natural in X, Y and F . Proof.This can be proved by a straight-forward diagram chase. Details are left * *to the reader. This brings us to the main result of this section. Theorem 2.3. Let F be a reduced homotopy exact functor. Then the natural transformation j :X ^ F Y ! F (X ^ Y ) is a homotopy equivalence. Proof.Suppose that the diagram f X _____//Y g || || fflffl| fflffl| Z _____//W is co-Cartesian. If o :F ! G is a natural transformation of homotopy exact func- tors such that o is a homotopy equivalence on X, Y and Z, then o must be an equivalence on W as well, as we have equivalences F (W ) ~ C(F (f), F (g)) ~!C(G(f), G(g)) ~!G(W ) 4 NATHAN WODARZ where o :F (W ) ! G(W ) is the composite. We know that any space X in CW * can be built from * and S0 by finitely many iterations of mapping cones and wedges, so by the preceding argument, it suffic* *es to prove the theorem for the cases X = * and X = S0. However, both of these cases are obvious from the definition of j. By setting Y = S0, we then immediately get the desired corollary. Corollary 2.4. A functor F is homotopy exact if and only if F is naturally equi* *v- alence to the functor X ^ F (S0). Remark 2.5. We recall that a commutative square X _____//Y | | | | fflffl|fflffl| Z _____//W is Cartesian if the natural map X ! holim(B ! D C) is a homotopy equiva- lence. Goodwillie defines a functor L to be linear if whenever X is a co-Cartesian s* *quare, the square L(X) is Cartesian [9]. (Recall that we are assuming L reduced). We note as well that if F is homotopy exact, the composite functor L O F is obviou* *sly linear. In particular, this holds for Q O F , where Q = 1 1 . Any linear func* *tor L is known to be naturally equivalent to the functor 1 (X ^ L(S)), where S is the sphere spectrum. Additionally, ss* O L is a homology theory. Thus, the preceding theorem may be regarded as an unstable version of the result on linear functors. The practical consequence of Corollary 2.4 is that for homotopy exact functor* *s F , the identification given in Theorem 1.4 is less useful than was initially antic* *ipated. In fact, the GAHSS reduces to the K"unneth Theorem in this case. 3. n-exact Functors To make further progress in our analysis of the GAHSS, we will develop the connection between homotopy exact and linear functors seen at the end of the preceding section. In order to do so, we must initially recall some notions and results from [10]. Let S be a set. We may consider its power set P(S) as a poset, and hence a sm* *all category, in the usual way. A functor X: P(S) ! CW * is said to be an S-cube. If S = {1, 2, 3, . .,.n}, we refer to X simply as an n-cube. A 0-cube is simply a * *space, i.e., the functor corresponding to the set S = ;. Given an S-cube X, we are interested in two related functors. Let P0(S) be the full subcategory of P (S) with objects all non-empty subsets of S, and let P 1(* *S) the full subcategory consisting of all proper subsets. Let Xi denote the compos* *ite functor Pi(S) ! P(S) ! CW *. We denote the homotopy limit of X0 by h0(X) and the homotopy colimit of X1 by h1(X). For any S-cube X, there are natural maps a: X(;) ! h0(X) and b: h1(X) ! X(S). We say that X is (homotopy) Cartesian (resp. (homotopy) co-Cartesian) if the map a is a homotopy equivalence (resp. b* * is a homotopy equivalence). For later use, we also recall that X is k-Cartesian (r* *esp. k-co-Cartesian) if the map a is k-connected (resp. b is k-connected). EXACTNESS OF HOMOTOPY FUNCTORS OF SPACES 5 Finally, an S-cube X is strongly co-Cartesian if every 2-face of X is co-Cart* *esian. It is easy to see that an n-cube X is strongly co-Cartesian if and only if ever* *y m-face of X with 2 m n is co-Cartesian. In particular, X itself is co-Cartesian. We recall that Goodwillie defines a functor F to be n-excisive if given any strongly co-Cartesian (n + 1)-cube X, the diagram F (X) is Cartesian. It is imm* *e- diate from the definition that 1-excisiveness is the same as linearity. Followi* *ng this lead, we define a functor F to be n-exact if for any strongly co-Cartesian (n +* * 1)- cube X, the diagram F (X) is co-Cartesian. Homotopy exactness then corresponds to 1-exactness. We will occasionally have reason to refer to functors which are* *n't reduced. The definition can be used without change in that case, although some properties won't apply. We note the following results, which may be proved in a similar manner to the analogous results in [10]. Theorem 3.1. Let F be (n - 1)-exact. Then F is n-exact. Theorem 3.2. Let F be a functor from CW m*! CW *such that F is ni-exact when considered as a functor in the i-th variable. If : CW * ! CW m*is the diagonal functor, then the composite F O is n-exact, where n = n1 + . .n.m. We note that this gives many examples of n-exact functors, as seen in this example. Example 3.3. The following are all examples of n-exact functors. (1) The n-fold smash product X^n = X ^ . .^.X. (2) The n-fold Cartesian product Xn. (3) If G is a subgroup of the symmetric group n, then the functor Xn=G is n-exact. In particular, the symmetric product functor SP nis n-exact. Some aspects of the Goodwillie calculus are also found in n-exact functors, although no corresponding tower has been found. Given a functor F , we will define cross-effect functors ^criF for i 1. Th* *ese functors differ from the usual cross-effect, such as the one used by Johnson [1* *5], but they have similar properties. We will occasionally assume in the following * *that a functor F takes values in simply-connected spaces. If this fails to be the ca* *se, we will replace F by either O F or 2 O F , whichever is required to meet the hypothesis. As our ultimate goal is the calculation of H*(F (X)), this presents* * no problem. Recall from Goodwillie [10] that given a sequence of spaces S = X, X1, . .X.n and cofibrations X ! Xifor i 1, we can construct a strongly co-Cartesian n-cu* *be XS , called a pushout cube. If T {1, . .,.n}, we can define XS (T ) to be the* * union of the Xi along X for each i 2 T . We agree to set XS (;) = X. Every strongly co-Cartesian cube admits an equivalence from a pushout cube. To define the cross-effect, we let n 2. Given a sequence of n spaces S = X1, . .,.Xn, we define a strongly co-Cartesian n-cube ^XS by setting it to be t* *he pushout cube XT for the sequence T = *, X1, . .,.Xn. Remark 3.4. If n = 1, we obtain the unique map * ! X1. If n = 2, this is simply the co-Cartesian square 6 NATHAN WODARZ * ________//X1 . | | | | fflffl| fflffl| X2 ____//_X1 _ X2 If X ! Y is a map of (n - 1)-cubes (i.e., a natural transformation of functors)* *, it may be considered as an n-cube. Thus, if Z is an n-cube, we can write Z = X ! Y (actually, we can do so in n different ways). If n 2, let T be the collection of n - 1 spaces X1, . .,.Xn-1. We can then easily see that ^XS= ^XT! (^XT _ Xn). We may now define the cross-effect. Definition 3.5. If F is a functor, we define the nth cross effect of F as the f* *unctor c^rnF (X1, X2, . .,.Xn) = hocofib(h1(F (^XS)) ! F (hocolim^XS)), where the given map is the natural one, and S is the sequence X1, X2, . .,.Xn. Note that we have ^cr1F (X) ' F (X). We note the following immediate facts about ^crnF , which are analogous to pr* *op- erties of the usual cross-effect. Recall that the total cofiber "cX of an n-cube X is defined iteratively. For * *a 0- cube X = X, we define "cX to be the space X. If X is an n-cube with n 1, we m* *ay view X as a map of (n - 1)-cubes X0! X00. The total cofiber "cX is then merely * *the homotopy cofiber of the induced map "cX0! "cX00. It is known that the homotopy type of the total cofiber doesn't depend on the directions chosen to decompose * *X. See, for example, Goodwillie [10]. Proposition 3.6. The cross-effect ^crnF satisfies all of the following properti* *es. (1) Let S = X1, . .,.Xn. Then ^crnF (X1, . .,.Xn) ' "c(F (^XS)). (2) The cross-effect ^crnF (X1, . .,.Xn) is homotopy equivalent to the homot* *opy cofiber of the map ^crn-1F (X1, . .,.^Xi, . .,.Xn) _ ^crn-1F (X1, . .,.Xi^+ 1, . .,.Xn) ! c^rn-1F (X1, . .,.Xi_ Xi+1, . .,.Xn) for any 1 i < n. (3) c^rnF is symmetric. That is, for any permutation oe in the symmetric gro* *up n, we then have a homotopy equivalence c^rnF (X1, . .,.Xn) ! ^crnF (Xoe(1), . .,.Xoe(n)). (4) c^rnF is reduced in every variable. (5) If F is n-exact then ^crn+1F vanishes. (6) There is a natural isomorphism H*(F (X _ Y )) ~=H*(F X) H*(F Y ) H*(c^r2F (X, Y )). (7) Let F be n-exact with n > 1. Then ^cr2F is (n - 1)-exact in each variabl* *e. More generally, ^criF is (n-i+1)-exact in each variable. We then see that we must have ^crnF (X1, . .X.n) ' X1 ^ . .^.Xn ^ ^crnF (S0, . .,.S0). Proof.(1) and (2) are immediate from the description of the total cofiber given* * by Goodwillie. EXACTNESS OF HOMOTOPY FUNCTORS OF SPACES 7 (3) is obvious from the fact that the total cofiber is independent of the dir* *ections chosen to take the homotopy cofibers. We prove (4) by induction on n. If n = 0, this is by assumption. If n > 0, we may assume by (3) that X1 = *. By the description of the cross effect in (2), ^crnF (*, X2, . .,.Xn) is given by the homotopy cofiber of the map ^crn-1F (*, X3, . .,.Xn) _ ^crn-1F (X2, . .,.Xn) ! ^crn-1F (X2, . .,.Xn). By the inductive hypothesis, the summand ^crn-1F (*, X3, . .,.Xn) is contractib* *le, hence the map is a homotopy equivalence, and the homotopy cofiber is contractib* *le, as desired. To prove (5), take any sequence of spaces S = X1, . .,.Xn+1. If F is n-exact, then the (n + 1)-cube F (^XS) is co-Cartesian. Thus, it has trivial total cofib* *er. To show (6), we consider the following sequence of spaces, which is a homotopy cofibration sequence by the definition of cross-effect. F X _ F Y ! F (X _ Y ) ! ^cr2F (X, Y ) The long exact homology sequence of this cofibration sequence can be seen to sp* *lit. Note that, as F X and F Y are both retracts of F (X _ Y ), we have a natural map H*(F (X _ Y )) ! H*(F X) H*(F Y ) which gives the desired split exact sequenc* *e. The proof of (7) can be found later in this section. Toward the end of proving the last part of the preceding Proposition, we give* * a series of technical lemmas. Lemma 3.7. Let Z be an n-cube, with n 2. Choose a decomposition of Z as above, and write Z = X ! Y, where X and Y are (n - 1)-cubes. Z is co-Cartesian if and only if the diagram h1(X)______//h1(Y) | | | | fflffl| |fflffl colimX _____//colimY is co-Cartesian. Proof.Assume that Z is co-Cartesian. Let S = {1, 2, 3, . .,.n}, and let U rep- resent the restriction of Z to the full subcategory of P 1(S) which omits the s* *et {1, 2, 3, . .,.n - 1}. We can then refer to results presented by Goodwillie [1* *0] to obtain a pushout square. h1(X)_______//hocolim(U) | | | | fflffl| fflffl| hocolim(X)_______//h1(Z) By naturality, we may extend this diagram to h1(X)_______//hocolim(U) fflffl| fflffl| hocolim(X)_______//h1(Z) | offlffl| |o| hocolim(Z) | fflffl| offlffl| colim(X)_______//colim(Z) 8 NATHAN WODARZ As the lower vertical maps are equivalences, the square made up of the outer ve* *rtices is co-Cartesian. We then obtain a commutative diagram h1(X) ____________//_QQhocolim(U)~55 | QQ((Q lllll | | h (Y) | | 1 | fflffl| | fflffl| colim(X)______|_____//_colim(Z) QQ | ~ l55 QQ(( fflffl|lll colim(Y) The fact that the two arrows pointing upward are equivalences then tells us that the desired diagram is co-Cartesian. Conversely, we can reverse the process to find that the square hocolim(X)______//h1(Z) |o| || fflffl| fflffl| colim(X)______//colim(Z) is co-Cartesian. As the left vertical map is an equivalence, we can conclude th* *at so is the right vertical map. As colim(Z) = Z(S), this gives the desired result. Let Z = X ! Y be a map of n-cubes. We may define an auxiliary n-cube hocofib(Z) by setting its value on a set S to be the homotopy cofiber of the map X(S) ! Y(S). The cube hocofib(Z) obviously depends on the choice of decompo- sition of Z as a map X ! Y, but this will always be clear from the context. We note then the following lemma Lemma 3.8. Let Z = X ! Y be a map of n-cubes. If Z is co-Cartesian, then so is hocofib(Z). Conversely, if hocofib(Z) is co-Cartesian and every space in * *Z is simply connected, then Z is co-Cartesian. Proof.Assume that Z is co-Cartesian. Then the square hocolim(X1)_____//hocolim(Y1) | | | | fflffl| fflffl| colimX _________//colimY is co-Cartesian by the above lemma. Take the homotopy cofibers of the rows. The* *se are homotopy equivalent by the fact that the square is co-Cartesian. The homoto* *py cofiber of the top row is cofib(h1(X) ! h1(Y)). As homotopy colimits commute, this is precisely the same as h1(hocofib(Z)), The homotopy cofiber of the bottom row is colim(hocofib(Z)), by definition. We thus have an equivalence h1(hocofib(Z)) ~!colim(hocofib(Z)), hence hocofib(Z) must be co-Cartesian. Conversely, if hocofib(Z) is co-Cartesian, the homotopy cofibers of the rows * *in the above diagram are homotopy equivalent. As each of the spaces involved is simply connected, this implies that the diagram is co-Cartesian. By the previous lemma, we have that Z is co-Cartesian. Given a functor F , define a functor FY by setting FY (X) = hocofib(F (X) ! F (X _ Y )). EXACTNESS OF HOMOTOPY FUNCTORS OF SPACES 9 We note that we obviously have ^cr2F (X, Y ) ' hocofib(F (Y ) ! FY (X)). Proof of Proposition 3.6 (7).By symmetry, we only need prove that the functor ^cr2F (X, Y ) is (n - 1)-exact as a functor of the first variable. The other st* *atements then follow immediately. Let X be a strongly co-Cartesian n-cube. Then the n-cube X _ Y obtained by wedging Y to each space in X is also strongly co- Cartesian. Thus, we also have that the (n + 1)-cube Y = X ! X _ Y is strongly co-Cartesian. Since F is n-exact, F (Y) is co-Cartesian as well. We can also vi* *ew this cube as F (X) ! F (X_Y ). Since this is co-Cartesian, the n-cube hocofib(F* * (Y)) given along the above map is also co-Cartesian. It is obvious from the definiti* *ons that this n-cube is identical to FY (X). Hence, FY is an (n - 1)-exact functor. Now, let * represent the n-cube composed entirely of one-point spaces. This is obviously strongly co-Cartesian, so we have a map FY (*) ! FY (X) of co-Cartesi* *an n-cubes. By the above comment, we see that hocofib(FY (*) ! FY (X)) is actually ^cr2F (X, Y ). By the previous lemma, this is co-Cartesian, so we are done. Remark 3.9. Baues includes a similar proof for the cross-effect of a 2-excisive functor in [3]. We now construct a spectral sequence which is similar to the one Barratt intr* *o- duced in [1]. The best reference for this can be found in Goerss [8]. Let Sk be the sequence X, X1, . .,.Xk. We assume that we have a collection of cofibrations fi: X ! Xi. Let T kbe the sequence CX, Cf1, Cf2, . .,.Cfk, where CX is the cone on X and Cfirepresents the homotopy cofiber of fi. Notice that if we set Sk+1 = X, X1, . .,.Xk, CX, then the pushout cube XS k+1is the same as the map of pushout cubes XS k! XT k. Moreover, the cube XT kis equivalent to the cube ^XT 0kfor the sequence T0k= Cf1, Cf2, . .,.Cfk. Fix a sequence of spaces and cofibrations Sk = X, X1, . .,.Xk as above. Define Si= X, X1, . .,.Xk, CX, . .,.CX for any i > k, where there are i-k copies of CX. Let T ibe defined similarly to above for i k. As above, we then have for any i k a co-Cartesian cube XS i+1= XS i! XT i. Given a functor F , and i k, we define "Fi= "c(F (XS i)) and Fi= "c(F (XT i* *)). By definition of the total cofiber, this means that for every i k, we have a hom* *otopy cofibration sequence F"i! Fi! "Fi+1. We can use these cofibration sequences to define an exact couple and spectral sequence. Set Dp,q= HqF"-pfor p + k 0 and Dp,q= Hp+q-kF"-kfor p + k 0. Set Ep,q= HqF-p for p + k 0 and Ep,q= 0 otherwise. These definitions give an exact couple. We see that the E1 term of the corresponding spectral sequence wi* *ll be made up of cross-effects. Notice that for any q, we have a sequence of maps in homology . .!.Hq+i+1"Fk+i+1! Hq+i"Fk+i! . .!.HqF"k This implies that we can filter HqF"kby the subgroups im(Hq+i"Fk+i! HqF"k). In the event that F is n-exact, all but finitely many of these subgroups are trivi* *al, 10 NATHAN WODARZ and the spectral sequence will converge to this filtration. In particular, we h* *ave the following. Theorem 3.10 (Homological Barratt-Goerss spectral sequence). If F is an n- exact functor and Sk is a sequence as given above, then there is a spectral seq* *uence with the following properties: (1) The spectral sequence is a second quadrant sequence of homological type (2) The E1 term is completely contained in the band -n p -k. Specificall* *y, we have E1p,q= Hq(c^r-pF (Cf1, . .,.Cfk, X, . .,. X)) for -n p -k. (3) The spectral sequence converges to the above filtration of H*F"k. Here t* *he diagonal p + q = a actually provides the filtration for Ha+kF"k. We will concentrate on the spectral sequence following from the sequence S = X, Y , where f :X ! Y is a cofibration. In the event that F is n-exact, we noti* *ce that we have convergence to a filtration of the homology of the homotopy cofiber of the map F f :F X ! F Y . In particular, if we take Y = CX, we obtain a spectral sequence converging to the homology of F X. In this case, we also have the identification of the E1 term as E1p,q~=Hq(c^r-pF ( X, . .,. X)). The spectral sequence we obtain is then an obvious analogue of the Barratt-Goer* *ss desuspension spectral sequence. 4. Consequences of the HBGSS The spectral sequence of Theorem 3.10 has strong consequences for the behavior of an n-exact functor F . The first of these applications looks at constraints * *that the HBGSS puts on the homology of F X. We will say that a space X is m-acyclic if Hi(X) = 0 for all i m. Every spa* *ce is considered to be (-1)-acyclic. Given an n-exact functor F , we will let c(F * *) be the largest k such that ^crjF (S0, . .,.S0) is k-acyclic for all j. Lemma 4.1. Suppose that F is an n-exact functor. Let m > 0, and let f :X ! Y be a cofibration. We suppose that X and Cf are both m-acyclic. Additionally, suppose that F X is (m + c(F ))-acyclic and CFf is (m + c(F ) + 1)-acyclic. Then F (Cf) is itself (m + c(F ) + 1)-acyclic. Proof.We use induction on n. The result is obvious for all m if n = 1, as then F (Cf) ' CFf. We now assume the result is true for (n - 1)-exact functors. For 2 r n, we examine the column E1-r,*of the HBGSS corresponding to the sequence S = X, Y . This spectral sequence converges to H*(CFf). We know that E1-r,q= Hq(c^rrF (Cf, X, . .,. X)) and that c^rrF is (n - (r - 1))-exact, so we can assume the result for ^crrF . Since ^crrF is a functor in r variables* *, we can apply the inductive hypothesis on each variable successively, and we find t* *hat ^crrF (Cf, X, . .,. X) is at least (rm + r + c(F ))-acyclic. In particular, th* *e E1 term vanishes at and below the line q = -(m + 1)p + c(F ), except for possibly * *in the column p = -1. If we do have a q m + c(F ) + 1 with E1-1,q6= 0, this term will survive to E1 and we find that Hq(CFf) 6= 0, contradicting the hypothesis. EXACTNESS OF HOMOTOPY FUNCTORS OF SPACES 11 Thus, the smallest possible q with E1-1,q6= 0 is q = m + c(F ) + 2, which compl* *etes the proof, as E1-1,q= Hq(F (Cf)). Lemma 4.2. Let F be n-exact. Then F (Sm ) is (m + c(F ))-acyclic. Proof.The proof is by induction on m. We begin with the case m = 0, which is obvious from the definitions. Assume the result for Sm . Then, as we have a cofibration sequence Sm ! CSm ! Sm+1 , we satisfy the hypotheses of Lemma 4.1. Hence, we can conclude that F Sm+1 is (m + c(F ) + 1)-acyclic. W Lemma 4.3. If F is n-exact then F ( ff2ASm ) is (m + c(F ))-acyclic for m 0 and for any finite A. W Proof.The homology of F ( ff2ASm ) splits as a direct sum of the form M Hq(c^riF (Sm , . .,.Sm )). By the preceding lemma, we can assume that these are all trivial for q m + c(* *F ), hence the result follows. These lemmas put us in a position to give the first of our results on the hom* *ology of F X. We take here the usual convention that any space is (-1)-connected. Theorem 4.4. If F is n-exact and X is m-connected, then F X is (m+1+c(F ))- acyclic for any m -1. In particular, F X is always m-acyclic. Proof.The proof is by induction on the skeleton of X. We may assume that Xi is contractible for i m. We then let i = m + 1. Since Xm+1 is homotopy equivalent to a wedge of copies of Sm+1 , we can apply Lemma 4.3 to see that F Xm+1 must be (m+1+c(F ))-acyclic. If i > m+1, we may assume that F Xi-1 is (m+1+c(FW))- acyclic. Since Xi is equivalent to the homotopy cofiber of a map Si-1 ! Xi-1, we can apply Lemma 4.1 to show that F Xi is itself (m + 1 + c(F ))-acyclic. Obviously, if F X happens to be simply connected, we can make the stronger assertion that F X is m-connected. This will be the case whenever F takes simply connected spaces to simply connected spaces. It is immediate from the classific* *a- tion that any 1-exact functor satisfies this condition, and a simple van Kampen argument suffices to prove it for 2-exact functors. At this point, it is unkno* *wn whether this holds for n-exact functors with n 2. As the above proof indicates, we can certainly draw stronger conclusions about H*(F X). We observe that the vanishing line in Lemma 4.1 leads immediately to the following Freudenthal-type theorem. Theorem 4.5. Let F be n-exact, and X an m-connected space. Then there is an isomorphism Hi(F X) ~!Hi+1(F ( X)) valid for all i 2m + c(F ). Moreover, the map H2m+1+c(F)(F X) ! H2m+2+c(F)(F ( X)) is onto. These results mimic the classical results on homotopy groups. In fact, the si* *m- ilarity runs even deeper, as we can extend our results into the "metastable" ra* *nge to obtain an EHP-type sequence. The fact that this is true shouldn't be a surpr* *ise, 12 NATHAN WODARZ as our spectral sequence is merely a homological version of the classical homot* *opy spectral sequence. Theorem 4.6. Let F be n-exact, and X an m-acyclic space. Then there is a long exact sequence (1) H3m+c(F)+2(F ( X)) H!H3m+c(F)+2(c^r2F ( X, X)) ! . .!.Hi(F X) !E H H P0 i+1(F ( X)) ! Hi+1(c^r2F ( X, X)) ! Hi-1(F X) ! . . . Proof.In the region q 3m + c(F ), the HBGSS is concentrated in the columns p = -2 and p = -1. Standard spectral sequence arguments then suffice to derive the long exact sequence. In addition, we note that, by applying Theorem 4.5 to ^cr2F ( X, X) in each variable, we obtain an isomorphism oe :Hi-1(c^r2F (X, X)) ~!Hi+1(c^r2F ( X, X)) on this range. Thus, the long exact sequence of Theorem 4.6 can be rewritten as 0 (2) H3m+c(F)+2(F ( X)) H!H3m+c(F)(c^r2F (X, X)) ! . .!.Hi(F X) !E H H0 P i+1(F ( X)) ! Hi-1(c^r2F (X, X)) ! Hi-1(F X) ! . . . Corollary 4.7. If F is 2-exact, the above EHP-type sequence applies in all degr* *ees. Proof.The HBGSS is entirely concentrated in the columns p = -1 and p = -2 in this case. We now identify the maps appearing in this sequence. It's obvious from the construction that the map E :Hi-1(F X) ! Hi(F ( X)) is induced by the natural map F X ! F ( X). The maps H and P are less obvious, but are still exactly what they should be it the cases we're interested in. Theorem 4.8. Let F be a 2-exact functor which takes values in simply-connected spaces. Let ~: X ! X _ X denote the usual comultiplication map on X. The maps H, H0, P and P 0found in the long exact sequences (1) and (2) are given as follows. (1) H :Hn(F ( X)) ! Hn(c^r2F ( X, X)) is the composite Hn(F ( X)) Hn(F(~))!Hn(F ( X _ X)) i Hn(c^r2F ( X, X)), where the surjection is the projection induced by the natural splitting. (2) H0= oe O H. (3) P :Hn(c^r2F (X, X)) ! Hn(F X) is the composite Hn(c^r2F (X, X)) ,! Hn(F (X _ X)) Hn(Fr)!Hn(F X), where the inclusion is given by the natural splitting and r: X _ X ! X is the folding map. (4) P 0= P O oe-1. Proof.This is essentially Proposition 3.5 in Baues [3]. EXACTNESS OF HOMOTOPY FUNCTORS OF SPACES 13 The bounds on the above results are sharp. As an example, we take F X = X ^X and X = Sm . Since c(F ) = -1, Theorem 4.5 gives the trivial result Hi(S2m ) ~!Hi+1(S2m+2 ) for i < 2m. Also, we obtain the trivial surjection H2m(S2m ) i H2m+1(S2m ). In this case, we have ^cr2F (X, Y ) = (X ^ Y ) _ (Y ^ X), and Theorem 4.6 red* *uces to the short exact sequence 0 ! H2n+2(S2n) ! H2n+2(S2n+2 _ S2n+2) ! H2n(S2n) ! 0. By the identifications given above, we see that the first map must be the diago* *nal, and the second map subtraction. A more interesting example comes from taking F = SP 2, the symmetric square. We recall (see Hatcher, Example 4K.5, [14]) that the homology of the symmetric square of Sn is the following. For n 2, we get the formula H*(SP 2(Sn)) ~=H*(Sn) H*( n+1RP n-1). As in the case of X^X, we see that c(F ) = -1, so Theorem 4.5 gives an isomorph* *ism Hi(SP 2(Sn)) ~!Hi+1(SP 2(Sn+1)) valid for i < 2n, as can be seen from the formula. We also have the trivial sur* *jection 0 = H2n(SP 2(Sn)) ! H2n+1(SP 2(Sn+1)) = 0 if n is odd, and a surjection Z ~=H2n(SP 2(Sn)) ! H2n+1(SP 2(Sn+1)) ~=Z=2 if n is even. The potentially non-trivial portion of the EHP sequence is 0 0 ! H2n+2(SP 2(Sn+1)) H!H2n(c^r2SP 2(Sn, Sn) P!H 2 n 2 n+1 2n(SP (S )) ! H2n+1(SP (S )) ! 0. As we saw above, this sequence depends on whether n is odd or even. We note that ^cr2SP(2X, Y ) ' X ^ Y . If n is odd, we obtain the sequence 0 0 ! Z H!Z ! 0 ! 0 ! 0 If n is even, we obtain 0 ! 0 ! Z P!Z E!Z=2 ! 0. Hence, we can identify P as multiplication by 2. This information, together with the identification of the GAHSS for 2-exact functors on Moore spaces in Section* * 5 allows for the determination of SP 2(M) for any Moore space M (cf. [17, 16]). In addition to the above theorems, the HBGSS also gives a Whitehead-type uniqueness theorem. In particular, for n-exact functors F and G, the naturality* * of the spectral sequence together with induction on n obviously implies the follow* *ing theorem. 14 NATHAN WODARZ Theorem 4.9. Let o :F ! G be a natural transformation of n-exact functors taking values in simply-connected spaces. If for any i 1, the induced map o :c^riF (S0, . .,.S0) ! ^criG(S0, . .,.S0) is a homotopy equivalence, then o is a natural equivalence of functors. Proof.Our proof is by induction on n. For n = 1, the theorem is obvious. Assume the conclusion for (n - 1)-exact functors. By applying the inductive hypothesis* * to each cross-effect functor, we may assume that we have a natural equivalence o :c^riF ! ^criG for each i 2. It obviously suffices to show that whenever o is an equivalence for X and Y , then it is for X _ Y and Cf for any map f :X ! Y . By the natural splitting given in Part 6 of Proposition 3.6, the conclusion h* *olds for X _ Y . It only remains to show it for Cf. Let E denote the exact couple of the spectral sequence converging to CFf, and let E0denote the exact couple giving the spectral sequence converging to CGf. We will denote the individual groups of the latter exact couple by E0*,*and D0*,*.* * The transformation o induces a map of exact couples E! E0 which, by the inductive hypothesis, gives an isomorphism E*,*! E0*,* except possibly for p = -1. These isomorphisms, together with the fact that Dp,*= D0p,*= 0 for p < -n imply that Dn-1,*~=En,*and D0n-1,*~=E0n,*. Thus, o induces an isomorphism Dn-1,*~=D0n-1,*. A five lemma argument then implies that we have isomorphisms Dp,*~=D0p,* for p -2, and hence, for all p, since we know that we have a homotopy equival* *ence CFf ! CGf by assumption. Another five lemma argument then implies that o induces an isomorphism H*(F (Cf)) = E-1,*~! E0-1,*= H*(G(Cf)), which finished the proof. Remark 4.10. Analogously to the classical Whitehead theorem, it's not enough to show F and G with homotopy equivalent cross-effects on S0. The homotopy equivalence must be given by a natural transformation. We will now construct a pair of 2-exact functors with homotopy equivalent cross-effects on S0 which are not naturally equivalent. Our construction of the functors begins with the following observation. Lemma 4.11. Let F and G be reduced n-exact functors taking values in simply connected spaces, and let o :F ! G be a natural transformation. Then H = hocofib(o) is again an n-exact functor, and the cross-effect ^criH(X1, X2, . .,* *.Xi) is the homotopy cofiber of the induced map ^criF (X1, X2, . .,.Xi) ! ^criG(X1, X2, . .,.Xi). EXACTNESS OF HOMOTOPY FUNCTORS OF SPACES 15 Proof.We first show that H is n-exact. Let X be a strongly co-Cartesian (n + 1)- cube. Then the cubes F (X) and G(X) are co-Cartesian, as F and G are n-exact. Then, by Lemma 3.8, we see that H(X) = hocofib(F (X) ! G(X)) is co-Cartesian as well. It remains to show that the cross-effects are as indicated. By iteration, it * *suffices to prove this for the first cross-effect. To that order, we examine the follow* *ing diagram, which is commutative by the naturality of o. The non-trivial maps are the obvious ones. *OooG(X)__OG(Y )__//G(X _ Y ) | OO OO | | | * ooF_(X) _ F (Y_)//_F (X _ Y ) fflffl| fflffl| fflffl| * oo_____*____________//* If we take the homotopy colimits of the columns of this diagram, we obtain * H(X) _ H(Y ) ! H(X _ Y ) which has homotopy colimit ^cr2H(X, Y ). Similarly, if we take the homotopy col- imits of the rows, we obtain c^r2G(X, Y ) ^cr2F (X, Y ) ! * which has homotopy colimit hocofib(c^r2F (X, Y ) ! c^r2G(X, Y )). As homotopy colimits commute, we have the natural equivalence ^cr2H(X, Y ) ' hocofib(c^r2F (X, Y ) ! ^cr2G(X, Y )), which is what we claimed. We recall the map jF :X ^F (S0) ! F (X) defined at the beginning of Section 2. Note that if F is an n-exact functor, then, by the preceding lemma, the functor hocofib(j) is n-exact as well. Let F 0denote hocofib(jF ) for the functor F (X)* * = 2(X ^ X) and let G0 denote hocofib(jG ) for the functor G(X) = 2(SP 2(X) _ SP 2(X)). Since the original functors are obviously 2-exact, so are F 0and G0. Additionally, since the source of j is 1-exact, its cross-effects vanish, and t* *he cross- effects of F 0and G0are merely ^cr2F(0S0, S0) ' ^cr2F (S0, S0) and ^cr2G0(S0, S* *0) ' ^cr2G(S0, S0). We now show that F 0and G0satisfy have isomorphic homology on 0-spheres, but not in a natural manner. We initially note that F 0(S0) and G0(S0) are contract* *ible, since they are the homotopy cofibers of homotopy equivalences. We now must only show that c^r2F (S0, S0) ' c^r2G(S0, S0), since all higher cross-effects are tr* *ivial by 2-exactness. We have ^cr2F (X, Y ) ' 2((X ^ Y ) _ (Y ^ X)). Similarly, the cross-effect of 2 O SP 2can be seen to be 2(X ^ Y ), so the wedge of two copi* *es will have cross-effect 2(X ^ Y ) _ 2(X ^ Y ). Finally, we note that F 0and G0 are not equivalent functors. Take X = Sn-1 with n > 2. Note that, by the definition of F 0, the sequence 0 ! H2n(S2n) ! H2n(F 0(Sn-1)) ! 0 is exact. Similarly, the sequence 0 ! H2n-2(SP 2(Sn-1)) H2n-2(SP 2(Sn-1)) ! H2n(G0(Sn-1)) ! 0 is exact. Thus, for n 2, we see that F 0(Sn) and G0(Sn) differ, hence F 0and * *G0 are not naturally equivalent. 16 NATHAN WODARZ We now turn out attention to the analyticity of n-exact functors. For this, we recall more terminology and notation from [10]. We let ~ be an integer with ~ * * -1, and use Cto denote the class of all strongly co-Cartesian (n + 1)-cubes X such * *that for any s 2 {1, . .,.n+1}, we have that X(;) ! X({s}) is ks-connectedPwith ks * * ~. A functor F is then said to be En(c, ~) if F (X) is (-c + ks)-Cartesian for a* *ll X 2 C. Definition 4.12. A functor F is stably n-excisive if there are numbers c and ~ such that F is En(c, ~). F is ae-analytic if there is some number q such that F* * is En(nae - q, ae + 1) for all n 1. The condition that F be ae-analytic is known to be sufficient for the Goodwil* *lie tower of F to converge to F for ae-connected spaces [11]. Using analogous metho* *ds to the proof of Theorem 4.4, we can use the HBGSS to prove Theorem 4.13. Any n-exact functor which takes values in simply-connected spaces is 1-analytic. Let X be an S-cube with |S| = m 1. Given a subset T S, we may define the T -face of X, denoted @T X, to be the T -cube defined by @T X(U) = X(U) for U T . Suppose that we have a function k :P (S) ! Z having the properties that @T X is k(T )-co-Cartesian and k(U) k(T ) whenever U T . We allow the possibility that k(T ) = 1. Let T be the set of all partitions of S by non-empty sets. Given a partition T 2 T, define X l(T ) = 1 - m + k(Tff). Tff2T Lemma 4.14. Under the above hypotheses, X is l-Cartesian, where l = min l(T ). T2T Proof.This is Theorem 2.5 in [10]. Proof of Theorem 4.13.(Cf. Theorem 2.3 in [10]). Let X be a strongly co-Cartesi* *an m-cube with m 1, and set S = {1, 2, . .,.m}. We will use X(i) to refer to the space X({i}) for readability. Let F be an n-exact functor taking values in simp* *ly- connected spaces. Suppose that the map X(;) ! X(s) is s-connected for all s 2 S. If m = 1, we can apply Theorem 4.4 and the reason* *ing in the proof of Lemma 4.1 to see that the map F (X(;)) ! F (X(1)) is k1-connected, hence we may assume m > 1. If m > n+1, all T -faces with n+1 |T | < m are co-Cartesian, as F is n-exac* *t, hence k(T ) = 1. We may then restrict ourselves to looking at partitions of S i* *nto sets with no more than n elements. We then look at the connectivity of "c(F (@T* * X)) for all T S with j = |T | n. Since all strongly co-Cartesian cubes admit equivalences from pushout cubes we may assume that the cube @T X is a pushout cube for the sequence S = EXACTNESS OF HOMOTOPY FUNCTORS OF SPACES 17 X(;), X(s1), . .,.X(sj), where T = {s1, . .,.sj}. Define Fias we did in constru* *cting the HBGSS, and let conn(Fi) denote the connectivity of Fi. Set k0= min{conn(Fj+i)|i 1}, and set t = conn(X(;)). By applying the reasoning in Lemma 4.1, we can see that X conn(Fj+i) j + i - 1 + ks + it. s2T Thus, we know that X k0 j + ks s2T and the map "c(F (@T X)) !PFj must be at least k0-connected. As conn(Fj) (j - 1) + s2Tks, we can say the same for k(T ) = conn("c(F (@T X))). Let T = {Tff} be a partition of S with 1 |Tff| n for every Tff. Thus we have _ ! X X X k(Tff) |Tff| - |T | + ks, ff ff s2S which tells us that ( ) X X X min k(Tff) m - m + ks = ks. ff s2S s2S Hence by Lemma 4.14, F (X) is k-Cartesian, where X k = (1 - m) + ks, s2S and F is Em (m - 1, 1) for all m 1, hence 1-analytic, as claimed. 5. The GAHSS for 2-exact Functors In this section, we review Baues' work with quadratic Z-modules and the qua- dratic tensor product, and use this to examine the E2 term of the GAHSS in the case where F is 2-exact and X is a Moore space M(A, n) with n 2. References for these topics are [2, 3]. We assume that all Moore spaces are simply-connect* *ed. Let T be a functor from a category C with a zero object and coproducts into an Abelian category A . Suppose that T (0) = 0. We can then define the cross-effect of T by T (X|Y ) = ker{(T r1, T r2): T (X _ Y ) ! T (X) x T (Y )} where r1 and r2 are the obvious retractions. We obtain the usual splitting prop* *erty, so T (X _ Y ) ~=T (X) T (Y ) T (X|Y ). Such a functor T is linear if the cross-effect T (X|Y ) = 0 for all X and Y . T* * is said to be quadratic if T (X|Y ) is linear in each variable. Baues' study of qu* *adratic functors led to his introduction of quadratic modules and tensor products. 18 NATHAN WODARZ Definition 5.1. A quadratic Z-module (henceforth simply quadratic module), writ- ten M = Me H!Mee!P Me, consists of a pair of Z-modules Me and Mee, together with a pair of homomorphis* *ms H :Me ! Mee(the Hopf map) and P :Mee! Me (the Whitehead product map), such that HP H = 2H and P HP = 2P . Remark 5.2. It is possible to (and Baues does) define a quadratic A-module for * *an arbitrary Abelian group A. We are not interested in that development here. Definition 5.3. If M and N are quadratic modules, a morphism f :M ! N is a pair of homomorphisms fe: Me ! Ne and fee: Mee! Neemaking all the relevant diagrams commute. We may define direct sums, direct products, kernels and cokernels "coordinate- wise" in the obvious manner. It is then straight-forward to verify that the cat* *egory QM of quadratic modules is an Abelian category. The reason for the names of the maps in a quadratic module M becomes a little more apparent when examining situations where quadratic modules arise. Baues has shown how to derive quadratic modules in a natural way from quadratic functors C ! Ab , where C is an additive category. In particular,if we take C to be the category of simply-connected homotopy co-commutative cogroups, the construction applies. If X is a two-fold suspension, then X is obviously such a space, so all of the spaces Sn are in C for n 2. Given any element X of C, we have maps X ~!X _ X r!X where ~ is the comultiplication map and r is the folding map. Applying any func* *tor F then gives the obvious sequence F X F(~)!F (X _ X) F(r)!F X. On applying homology, we obtain H*(F (X _ X)) ~=H*(F X) H*(F X) H*(c^r2F (X, X)), by Part 6 of Proposition 3.6. Let r :H*(F (X _ X)) ! H*(c^r2F (X, X)) be the restriction map, and i: H*(c^r2F (X, X)) ! H*(F (X _ X)) the injection. We may then define maps H = r O H*(F (~)) and P = H*(F (r)) O i. Baues proves Theorem 5.4. The data H*(F X) H!H*(c^r2F (X, X)) P!H*(F X) defines a quadratic module, which we denote H*F {X}. Remark 5.5. The maps H and P in Theorem 5.4 are the same as the maps H and P in the EHP-type sequences of Theorem 4.6. Given an Abelian group A and a quadratic module M, we may define the qua- dratic tensor product A M as follows. Let G be the free Abelian group generat* *ed by all the symbols a m and [a, b] n, where a, b 2 A, m 2 Me and n 2 Mee. A M is the quotient of G given by imposing the following set of relations, wh* *ere a, a0, b, b02 A, m, m02 Me and n, n02 Mee. (1) a (m + m0) = a m + a m0 EXACTNESS OF HOMOTOPY FUNCTORS OF SPACES 19 (2) [a + a0, b] n = [a, b] n + [a0, b] n (3) [a, b + b0] n = [a, b] n + [a, b0] n (4) [a, b] (n + n0) = [a, b] n + [a, b] n0 (5) (a + a0) m = a m + a0 m + [a, a0] H(m) (6) [a, a] n = a P (n) Remark 5.6. The following facts are obvious from the above definition. (1) The quadratic tensor product in quadratic in the first variable and line* *ar in the second. The cross-effect (A|B) M is simply the usual tensor product A B Mee. (2) Any Abelian group G can be treated as a quadratic module M = G ! 0 ! G. Then for any Abelian group A, the quadratic tensor product A M is the same as the usual tensor product A G. For any quadratic module M, we can use the above description to see that we have a natural isomorphism Z M ~=Me. Since the quadratic tensor product is quadratic in the first variable, we can use the above identification of the cro* *ss-effect to show that we have a natural isomorphism Zn M ~=(Me)n (Mee)n(n-1)=2. Given an Abelian group A, let denote the diagonal map A ! A A, and let r: A A ! A denote the addition map. For any quadratic functor T :Ab ! Ab, we obtain a quadratic module T {Z} = T (Z) H!T (Z|Z) P!T (Z) where H is the composite T (Z) T(!)T (Z Z) ! T (Z|Z) and P is the composite T (Z|Z) ! T (Z Z) T(r)!T (Z). Here, the unlabeled maps denote the obvious projection and inclusion respective* *ly. For any such functor T :Ab ! Ab, there is a map ~: A T {Z} ! T (A) called the tensor approximation of T . The definition of this map can be found in [2].* * If A is finitely generated and free, then ~ is an isomorphism. It is not an isomorph* *ism in general, however. The following results on quadratic tensor products appear in the literature. Theorem 5.7. The quadratic tensor product is right exact in the first variable. Remark 5.8. We take here a non-additive definition of right exactness, given by Baues in [2]. An equivalent definition is given by Bouc [5]. Suppose that A1 ! A0 ! A ! 0 is an exact sequence of Abelian groups, and T :Ab ! Ab an arbitrary functor with T (0) = 0. We say that T is right exact if the sequence F (A1|A0) F (A1) ! F (A0) ! F (A) ! 0 is exact. This is obviously a generalization of the notion of right exactness * *of a linear functor. 20 NATHAN WODARZ Given an Abelian group A and a quadratic module M, we may take a free resolution of A A1 ! A0 ! A ! 0. Using the simplicial methods of Dold and Puppe for derived functors of a non-li* *near functor [7], we obtain a chain complex C* given by (A1|A1) M ! ((A1|A0) M) (A1 M) ! A0 M. By right-exactness, the 0th homology group of C* is simply A M. The 1st and 2nd homology groups of C* are the quadratic torsion products. We define A *0M = H1(C*) and A *00M = H2(C*). In the event that M = G ! 0 ! G is simply an Abelian group, A*00M is trivial and A*0M = Tor(A, G), the usual torsi* *on product of Abelian groups. As with the quadratic tensor product, both quadratic torsion products are quadratic in the first variable and linear in the second. * *The identification of A M for any finitely generated free Abelian A given above m* *akes calculation of the quadratic tensor and torsion products fairly easy. At this point, for a 2-exact functor F , we will use Qi to denote the compos- ite HiO F , so Qi(X) = Hi(F X) and Qi(X, Y ) = Hi(F X, F Y ). We will write Qi{X} for the quadratic module denoted HiF {X}. We will use Qi(X|Y ) to denote Hi(c^r2F (X, Y )). This is easily shown to be equivalent to the definition of c* *ross- effect for the functor Qigiven above. Analogously to the tensor approximation g* *iven above, we have a natural homomorphism ~: A Qp+q{Sp} ! Qp+q(M(A, p)) for any finitely generated Abelian group A with p 2. This map is defined as follo* *ws. Let a, b 2 A, m 2 Qp+q(Sp) and n 2 Qp+q(Sp|Sp). We may view a and b as elements of ssp(M(A, p)) = A, and so we can set ~(a m) = Qp+q(a)(m) ~([a, b] n) = P Qp+q(a|b)(n) If A is free Abelian, then ~ is an isomorphism. We denote the cokernel of ~ by ~Qp+q(M(A, p)). Proposition 3.12 of [3] shows that for any finitely generated Abelian group A, we have a natural exact sequence 0 ! A *0Qp+q{Sp} ! ~Qp+q+1(M(A, p)) ! A *00Qp+q-1{Sp} ! A Qp+q{Sp} ~!Qp+q(M(A, p)) ! ~Qp+q(M(A, p)) ! 0 We are now in position to compute the E2 term of the GAHSS in the event that F is 2-exact and X is a Moore space. Let M denote the Moore space M(A, p). If A1 ! A0 ! A is a free resolution of A, we set Y = M(A0, p) and X = M(A1, p). We may then take Y to be the p-skeleton of M, and the Moore space is the mapping cone of X ! Y . We note that a consequence is that the E1 term of the GAHSS is restricted to two column* *s. In particular, we have that the only non-trivial differential E1p+1,q= Qp+q+1(M, Y ) @!Qp+q(Y ) = E1p,q. Examine the following diagram, which appears in a somewhat different form in [3* *]. EXACTNESS OF HOMOTOPY FUNCTORS OF SPACES 21 A0 Qp+q+1{Sp} ____Qp+q+1(Y_) || i|| fflfflfflffl|~ fflffl| A Qp+q+1______/Qp+q+1(M)/_ j|| fflffl| Qp+q+1(M, Y ) @|| fflffl| A0 Qp+q{Sp} ______Qp+q(Y_) || i|| fflfflfflffl|~ fflffl| A Qp+q{Sp}______/Qp+q(M)/ j|| fflffl| Qp+q(M, Y ) The column is exact, as it is part of the obvious long exact sequence in homo* *logy. From this diagram, we can see that im ~ = imi = kerj ~=coker@ = E2p,q. Similarly, we obtain ~Qp+q+1(M(A, p)) = cokeri ~=imj = ker@ = E2p+1,q. Hence, we obtain the following identification of the the E2 term of the GAHSS. Theorem 5.9. Using the above notation, the E2 term of the GAHSS for a 2-exact functor on a Moore space is given by the exact sequence. 0 ! A *0Qp+q{Sp} ! E2p+1,q! A *00Qp+q-1{Sp} !d A Q p ~ 2 p+q{S } ! Ep,q! 0 In the event that A *00Qp+q-1{Sp} = 0 (which happens on a regular basis in practice), we get the obvious identifications E2p,q~=A Qp+q{Sp} E2p+1,q~=A *0Qp+q{Sp} Remark 5.10. Further progress on the GAHSS along the lines of this article seems unlikely. The differential in the E1 term of the GAHSS contains quite a bit of information, and appears to be virtually in any case more difficult than the pr* *esent. There is hope for a similar program, however. We note that we may define an ad hoc spectral sequence by setting E2p,q= A Qp+q{Sp} E2p+1,q= A *0Qp+q{Sp} E2p+2,q= A *00Qp+q{Sp} 22 NATHAN WODARZ and taking d : E2p+2,q-1= A *00Qp+q-1{Sp} !d A Qp+q{Sp} = E2p,qto be the only non-trivial differential, where d is the map in the statement of Theorem 5* *.9. 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