THE GRAY FILTRATION ON PHANTOM MAPS L^E MINH HA AND JEFFREY STROM Abstract This paper is a study of the Gray index of phantom maps. We give a new, tower theoretic, definition of the Gray index, which allows us to study the naturality properties of the Gray index in some detail. McGibbon and Roitberg have shown that if f* is surjective on rational cohomology, then the induced map on phantom sets is also surjective. We show that if f* is surjective just in dimension k, then f induces a surjection on a certain subquotient of the phantom set. If the condition holds for all k, we recover McGibbon and Roitberg's theorem. There is a dual result, and a theorem on phantom maps into spheres which holds one dimension at a time as well. Finally, we examine the set of phantom maps whose Gray in- dex is infinite. The main theorem is a partial verification of our conjecture that if X and Y are nilpotent and of finite type, then every phantom map f : X -! Y must have finite index. Math. Subject Classifications: 55S37, 55P30, 55P62, 20G10, 20J05 Keywords: Phantom Maps, Gray Index, Inverse Limit, lim1, 1. Introduction A map f : X -! Y is a phantom map if its restriction to each n- skeleton, f|Xn , is trivial; we denote the set of pointed homotopy classes of phantom maps from X to Y by Ph(X; Y ). This paper begins with the observation, due to Brayton Gray, that if Ph(X; Y ) 6= * (here X and Y are both nilpotent and of finite type), then there is a dimension n 1 such that both Hn (X; Q) 6= 0 and ssn+1(Y ) Q 6= 0. Usually there will be more than one dimension in which this condi- tion is satisfied. How can we tell which dimensions are relevant to a given phantom map f? We should expect two different answers: if we find that dimension n is relevant when talking about the domain, then dimension n + 1 should be relevant when talking about the target. The basic idea of this paper is that the least dimension which is relevant to f (in terms of the domain) is G(f), the Gray index of f. 1 2 L^E MINH HA AND JEFFREY STROM The Gray index gives rise to a filtration on the set of phantom maps from X to Y , which we call the Gray filtration. Our results concern this filtration and how it behaves under maps between domains or targets; they support our claim about the Gray index as follows. A theorem of McGibbon and Roitberg [8] shows that if OE : A -! B induces a surjection in rational cohomology then OE* : Ph(B; Y ) -! Ph(A; Y ) is also surjective (there is also a dual result involving the rational ho- motopy of the target). It is reasonable to expect that f should be in the image of OE* as long as OE is surjective in the dimensions that are relevant to f. One of our main theorems implies that if OE* is surjective in rational cohomology for all dimensions above G(f), then f is in the image of OE*, modulo the set of phantoms with infinite Gray index; if Y ' Z for some Z, then there is no indeterminacy (we also get a dual result). Even better, if OE is surjective in rational cohomology only in dimension G(f) then f is in the image of OE*, at least up to a certain indeterminacy which doesn't contain f. Dually, if OE induces a rational homotopy surjection in dimension G(f) + 1, then f is in the image of OE*, modulo the same indeterminacy. Before we begin, we would like to express our thanks to Chuck McGibbon for his help and encouragement as we have learned about phantom maps. 2. The Gray Index The Gray index was first defined by Gray in his thesis [3]; it has also been studied by McGibbon and Strom in [10 ]. It is important to note that this definition differs by one from Gray's original definition [3] (ours is bigger). Let f : X -! Y be a phantom map. Then f|Xk ' * for each k, and so there are factorizations of f f X F________________//Y<< FF yyy FFF yyy F""F yy f X=Xk Generally, there will be many possible choices for f. Definition The Gray index of f is the least integer k such that the map f cannot be chosen to be a phantom map. We write G(f) = k; if there is no such integer, then G(f) = 1. THE GRAY FILTRATION ON PHANTOM MAPS 3 We will denote the set of all phantom maps f : X -! Y with G(f) k by Phk(X; Y ). Thus, Phk+1(X; Y ) = Im(Ph(X=Xk; Y ) -! Ph(X; Y )). If f : X -! Y is a phantom map, then the image of f must be con- tained in the basepoint component of Y , or else the restriction of f to the 0-skeleton of X will be essential. Also, f must remain phantom when restricted to each component of X. If we give X a CW decompo- sition with one 0-cell in each component, we see that f factors through a phantom map X=X0 -! Y , and so Ph(X; Y ) = Ph1(X; Y ). There is a dual definition of phantom maps in terms of the connective covers of the target. A map f : X -! Y is phantom if f factors (up to homotopy) as in the diagram Y<< fezzzz | zz | zzf fflffl| X ______//_Y for each k 0. We can then define a dual Gray index, G0(f), to be the least integer k such that the map efcannot be chosen to be a phantom map. We have seen that we should expect that the least relevant dimension for the target should be one more than the least relevant dimension for the domain. This is in fact the case. Proposition 1. If f : X -! Y is a phantom map, then G0(f) = G(f) + 1. Proofs will be given in the last section. It follows that Phk(X; Y ) = Im(Ph(X; Y ) -! Ph(X; Y )). This result also shows that G(f) is independent of any choices that were made in giving X a CW decomposition. We will make essential use of the following alternative description of the Gray index in terms of inverse towers. According to Bousfield and Kan ([1], pages 254-255), Ph(X; Y ) is naturally isomorphic to a lim1 set: Ph(X; Y ) ~=lim1n[X; Y (n)]: Observe that in the notation Y (n)the order of operations is ambigu- ous. In this paper, as in most other papers about phantom maps, passing to the loop space is the last step. Thus, Y (n)really means (Y (n)). Let us write Gn = [X; Y (n)] and G(n)k= Im(Gn -! Gk): 4 L^E MINH HA AND JEFFREY STROM We have a commutative diagram of surjections of towers {Gn} G_____//{G(n)k} GGG | GGG | G##Gfflffl| {G(n)k-1} which induces a commutative diagram of surjections after taking lim1 pk 1 (n) lim 1nGn _____//_limnGk LLL LLL |j pk-1LLL%%Lffklffl|| lim1nG(n)k-1: The maps pk define equivalence relations on Ph(X; Y ): say f ~k g if and only if pk(f) = pk(g). The commutativity of the diagram shows that f ~k g implies f ~k-1 g, but the reverse need not be true. We will interpret lim1nG(n)kas the set of ~k-equivalence classes of Ph(X; Y ). Theorem 2. Let f : X -! Y be a phantom map. Then G0(f) is the least integer k such that f 6~k *, and G(f) is the greatest integer k such that f ~k *. In other words, we have Phk(X; Y ) = Ker(pk). These sets give us a natural filtration Ph(X; Y ) = Ph1(X; Y ) Ph2(X; Y ) . . .Phk(X; Y ) . . . which we call the Gray filtration. Typically, one studies a filtered group by examining the subquotients. Unfortunately, the pointed set Ph(X; Y ) does not generally have a group structure. Nevertheless, we can use the equivalence relations ~k to make sense of subquotients of this filtration. Write Phk(X; Y )=Phl(X; Y ) = pl(Phk(X; Y )) lim1nG(n)l: Clearly, this definition agrees with the usual notion whenever Ph(X; Y ) happens to be a group. Our final result in this section is that the Gray filtration can only decrease if the condition in Gray's principle is met. Proposition 3. Assume X and Y are nilpotent spaces of finite type. If either Hk(X; Q) = 0 or ssk+1(Y ) Q = 0, then Phk(X; Y ) = Phk+1(X; Y ): THE GRAY FILTRATION ON PHANTOM MAPS 5 This means that if either H*(X; Q) is bounded above by n or ss*(Y ) Q is bounded above by n + 1, then the Gray filtration is finite. More precisely, Phn(X; Y ) = Ph1 (X; Y ); we will see in Corollary 11 below that in fact Phn(X; Y ) = *. On the other hand, if either H*(X; Q) is bounded below by n or ss*(Y ) Q is bounded below by n + 1, then every phantom map f : X -! Y has G(f) n. Example Let us consider the Gray index of phantom maps X -! Sk+1. For any k, we have Ph(X; Sk+1) = Phk(X; Sk+1): If k is even, then Sk+1 has only one nontrivial rational homotopy group, and so Phk+1(X; Sk+1) = *: If k is odd, then there are two dimensions to consider, and we find that Phk+1(X; Sk+1) = Ph2k(X; Sk+1) and Ph2k+1(X; Sk+1) = *: 3. Maps Subject to Rational Conditions Now we turn our attention to the problem of determining how the Gray filtration behaves with respect to maps between targets or domains. Theorem 4. If OE : A -! B induces surjections OE* : Hm (B; Q) -! Hm (A; Q) in the range k m l then pl+1OE*(Phk(B; Y )) = Phk(A; Y )=Phl+1(A; Y ): All spaces are to be nilpotent and of finite type. Suppose OE induces a rational cohomology surjection only in dimen- sion k. Then Theorem 4 says that if G(f) = k (so f 6~k+1 *), then there is a phantom map f0 ~k+1 f such that f0 is in the image of OE*. In particular, the map OE* : [B; Y_]__//[A; Y ] is nontrivial. Here is a concrete example. Example There are phantom maps CP1 - ! S2 _ S2 of every even Gray index [10 ]. Therefore, if f : CP1 - ! Y is a map which is nonzero in rational cohomology, then our discussion applies, and so the map f* 1 2 2 [Y; S2 _ S2] -! [CP ; S _ S ] 6 L^E MINH HA AND JEFFREY STROM must be nontrivial. The following example shows how Theorem 4 can be used to con- struct exact sequences of phantom sets. Example Let f : X -! S2n be an essential phantom map. If f 6' *, then the composition f 2n oe 2n+1 X -! S -! S is nontrivial, and it follows that G(f) = 2n - 1. Since oe is a rational (2n - 2)-equivalence, it follows that if f ' *, then G(f) = 4n - 2. This means that there is a phantom map ef: X -! S2n<4n - 2> lifting f as in the diagram S2n<4n9-92> efs s | s | s s f fflffl| X _________//S2n In other words, there is an exact sequence of pointed sets Ph(X; S2n<4n - 2>) -! Ph(X; S2n) -! Ph(X; S2n+1) -! * : The surjectivity of Ph(X; S2n) -! Ph(X; S2n+1) follows from Theo- rem 4 and Corollary 11 below. It should be expected that we will get a dual version of Theorem 4, and we do. Theorem 5. If OE : Y - ! Z induces surjections ssm (Y )Q -! ssm (Z) Q in the range k m l, then plOE*(Phk-1(A; Y )) = Phk-1(A; Z)=Phl(A; Z): Of course, all these spaces must be nilpotent and of finite type. In particular, if f : A -! Z is a phantom map with G(f) = k (so f 6~k+1 *) and OE* is surjective in dimension k+1, then there is f0 ~k+1 f such that f0 2 OE*(Ph(A; Y )). Fibrations and cofibrations do not give rise to exact sequences of phantom maps. However, the composite of the maps induced by a cofibration or a fibration must be trivial. This allows us to pass from Theorems 4 and 5, which tell us how nearly surjective induced maps are, to Theorem 6, which tells us how nearly trivial induced maps are. Theorem 6. Let OE : A -! B be a map between finite type nilpotent spaces, and suppose that OE* : Hm (B; Q) -! Hm (A; Q) is zero for k m l. Then OE*(Phk(B; Y )) Phl+1(A; Y ): THE GRAY FILTRATION ON PHANTOM MAPS 7 Dually, if OE : Y - ! Z and OE* : ssm (Y ) Q -! ssm (Z) Q is zero for k m l then OE*(Phk-1(A; Y )) Phl(A; Z): We conclude this section by using the Gray index to put a topology on the set Ph (X; Y ). Example 7. The sets Uk(f) = {g | g ~k f} form the basis for a topol- ogy on Ph(X; Y ). Observe that, in this topology, Ph1 (X; Y ) is pre- cisely the closure of the singleton set consisting of the the trivial map. If Ph1 (X; Y ) = *, then the topology is Hausdorff. Since OEl: X -! X(l)is an (l - 1)-equivalence, Theorem 4 shows that OE*l(Ph(X(l); Y )) \ Ul(f) 6= ;: In other words the union of the OE*l(Ph(X(l); Y )) is dense in Ph(X; Y ). Suppose that X is nilpotent and of finite type and that Y is a fi- nite complex. A theorem of Zabrodsky [14 ] shows that [X(l); Y ] = Ph(X(l); Y ). The commutativity of the diagram [X(l+1); Y ] OO| RRRRROEl+1RR | RRR | RR((R [X(l); Y ]____OEl//_Ph(X; Y ) [X; Y ] gives us a map coliml[X(l); Y ] -! Ph(X; Y ) The above discussion shows that the image of this map is dense in Ph(X; Y ). Or, if we take the closure of the image, we find that ____________________ Ph(X; Y ) = Im(coliml; [X(l); Y:]) 4. Phantom Maps into Spheres In Theorem 1 of [8], McGibbon and Roitberg show that for finite type nilpotent X, the conditions a Ph(X; Sk+1) = * forgallWk; b there is a map X -! Snffsuch that i_ j g* : H* Snff; Q -! H*(X; Q) is an isomorphism; c Ph(X; Y ) = * for all finite type nilpotent Y ; are equivalent. This result is actually a reflection of a feature of phan- tom maps that holds in one dimension at a time. 8 L^E MINH HA AND JEFFREY STROM Theorem 8. Let X be nilpotent and of finite type. Each of the follow- ing statements implies the next: 1 Ph(X; Sk+1) = *; g Q 2 there is a map X -! Sk+1 such that iY j g* : Hk Sk+1; Q -! Hk(X; Q) is surjective; 3 Phk(X; Y ) = Phk+1(X; Y ) for all finite type nilpotent Y . Theorem 8 implies a modified version of Theorem 1 in [8]. If con- dition 1 is true for all k, then we find that Ph(X; Y ) = Ph1 (X; Y ), which, by Theorem 10 below, means that Ph(X; Y ) = * for all finite type Y , which in turn implies that condition 1 holds for all k. To recover the full statement of McGibbon and Roitberg's result, suspend our map g to obtain a map _m _ X -! Sk+1 _ Snff: i=1 which induces a surjection in Hk+1 (-; Q). It is a simple matter to pinch off the irrelevant spheres and so obtain a map _ X -! Sk+1 which induces an isomorphism in Hk+1 (-; Q). It is reasonable to ask whether condition 3 implies condition 1 in our Theorem. If k is even, the answer is yes, because Phk(X; Sk+1) = Phk+1(X; Sk+1) = *: If k is odd, however, the implication is not generally true, becuase Phk(X; Sk+1) = Ph2k(X; Sk+1); which need not be trivial, as the next example shows. Example There are no essential phantom maps f : CP1 - ! S4 with G(f) = 3 (because H3(CP1 ; Q) = 0). However, there are essen- tial phantom maps CP1 - ! S4 with G(f) = 6, because, according to Zabrodsky [14 ], Ph(CP1 ; S4) = [CP1 ; S4] ~=[CP1o; S4] ~=H6(CP1 ; ss7(S4) R) 6= 0; where R is a rational vector space with the cardinality of the real num- bers. We end this section with the dual to Theorem 8. THE GRAY FILTRATION ON PHANTOM MAPS 9 Theorem 9. Let Y be nilpotent and of finite type. Each of the follow- ing statements implies the next: 1 Ph(K(Z; k); Y )W= *; g 2 there is a map K(Z; k) -! Y such that i_ j g* : ssk+1 K(Z; k) Q) -! ssn+1(Y ) Q is surjective; 3 Phk(X; Y ) = Phk+1(X; Y ) for all finite type nilpotent X. The deduction of McGibbon and Roitberg's Theorem 10 from this result proceeds just as above. 5. Maps with Infinite Gray Index We began this paper by observing that if there is an essential phantom map f : X -! Y between nilpotent spaces of finite type, then there must be a dimension n such that both Hn (X; Q) 6= 0 and ssn+1(Y ) Q 6= 0. We have shown above that the Gray index of f is the least dimension n which has this property and which is relevant to f. Can it happen that none of these dimensions (whose existence is guaranteed by f) is relevant to f? In other words, can it happen that an essential phantom map has infinite Gray index? In his thesis [3], Gray claims that G(f) < 1 for every essential phantom map f. However there is a flaw in his argument; in fact, McGibbon and Strom [10 ] have shown that if X is of finite type and its cohomology contains an element of infinite height under the action of the Steenrod algebra, then there are essential phantom maps out of X with infinite Gray index. In these examples, even though the domains are of finite type, the targets are definitely not. Thus, we make the following conjecture. Conjecture If X and Y are both nilpotent and of finite type, then Ph1 (X; Y ) = *. We have not been able to prove this conjecture, but we do have a useful partial result. Let Gn = [X; Y (n)] as in the second section, and recall that a map f 2 Ph(X; Y ) ~= lim1nGn has infinite Gray index if and only if it is in the kernel of each map pk : lim1nGn -! lim1nG(n)k: The maps pk fit together to give us a map p1 : Ph(X; Y ) -! limklim1nG(n)k: 10 L^E MINH HA AND JEFFREY STROM Since f has infinite Gray index if and only if it is in the kernel of p1 , we call the target of this map Ph (X; Y )=Ph 1 (X; Y ). We will also use the notation Ph_(X; Y ) = Ph (X; Y )=Ph 1 (X; Y ): Similarly, we will let Ph_k(X; Y ) = Ph k(X; Y )=Ph 1 (X; Y ). Thus our conjecture is that Ph (X; Y ) = Ph_(X; Y ) whenever both X and Y are nilpotent and of finite type. Theorem 10. Suppose X and Y are nilpotent and of finite type and that Ph_(X; Y ) = *. Then Ph (X; Y ) = *. In other words, if Ph (X; Y ) = Ph 1(X; Y ), then Ph (X; Y ) = *. In other words, the only way that Ph (X; Y ) ccan consist solely of phantom maps with infinite gray index is for Ph (X; Y ) = *. Corollary 11. If Hm (X; Q) = 0 or ssm+1 (Y ) Q = 0 for m > k, then Ph k(X; Y ) = *: We can say much more if the towers we are concerned with happen to be towers of abelian groups. Theorem 12. A map OE : A -! B induces a surjection OE* : Ph_(B; Y ) -! Ph_(A; Y ); if and only if it induces a surjection OE* : Ph (B; Y ) -! Ph (A; Y ). Dually, OE* : Ph_(X; A) -! Ph_(X; B) is surjective if and only if OE* : Ph (X; A) -! Ph (X; B) is surjective. (All spaces should be nilpotent and of finite type.) This theorem, like Theorem 10, is really a topological interpretation of an algebraic result on maps between towers. In our proof, we show that the algebaic proposition underlying Theorem 10 applies to the tower {Cn}, where Cn is the cokernel of the map [Bn; Y ] -! [An; Y ]: In order to do this, we need the tower {Cn} to be a tower of groups, which is why we require the target to be a loop space. Recently, L^e Minh Ha has succeeded in proving that a map between good towers of groups induces a surjection on lim1 if and only if the induced map on the abelianizations of the towers induces a surjection on lim1 [5]. One application of this is a proof of Theorem 12 without the assumption that the target be a loop space. THE GRAY FILTRATION ON PHANTOM MAPS 11 Finally, let us see what we can derive when we take l = 1 in Theo- rems 4 and 5. Corollary 13. If OE induces surjections OE* : Hm (B; Q) -! Hm (A; Q) for all m k, then Ph_k(A; Y ) = OE*(Ph_k(B; Y )): If Y = Y 0, then we may replace Ph_ with Ph . Dually, if OE : Y - ! Z induces surjections ssm (Y )Q -! ssm (Z)Q in the range k m, then Ph_k-1(A; Z) = OE*(Ph_k-1(A; Y )): If X = X0, then we may replace Ph_ with Ph . Of course all these spaces must be nilpotent and of finite type. If the conjecture were known to be true, we would recover Theorem 2 of [8] by setting k = 0 in this corollary. When Y is a loop space or X is a suspension, we do not need to appeal to the conjecture. Taking l = 1 in Theorem 6, we find that if OE : A -! B induces trivial maps in rational cohomology in dimensions greater than k, then OE*(Phk(B; Y )) Ph 1(A; Y ). In other words, OE*(Ph_k(B; Y )) = * Ph_(A; Y ): The (possibly) stronger conclusion OE*(Ph(X; Y )) = * is also true; it follows from Theorem 2 in [8]. 6. Proofs Proof of Proposition 1 First suppose that G(f) > k. This means that there is a phantom map f : X=Xk -! Y extending f. We will show that f lifts to a phantom map fe: X=Xk -! Y . Indeed, the map of towers {[X=Xk; Y (n)]} ____//_{[X=Xk; Y (n)]} is surjective, and so the induced map on phantom sets is surjective as well. To go in the other direction, observe that the map of towers {[X=Xk; Y (n)]} _____//{[X; Y (n)]} is surjective. || 12 L^E MINH HA AND JEFFREY STROM Proof of Theorem 2 We have a fibration Y -! Y - ! Y (k). Tak- ing the nth Postnikov section and passing to loop spaces yields the following fibration Y (n) ____//_Y (n)____//Y (k): Applying [X; -], we obtain an exact sequence [X; Y (n)] ____________//_[X; Y (n)]______________//[X; Y (k)] JJJ vv;;v JJJJ uu::u JJJ vvv JJJ uuu JJJ vv JJJ uuu J%%J vv $$(n) u In Gk in which In is the image of [X; Y (n)] in [X; Y (n)]. When we take lim1, we obtain Ph(X; Y ) OO | OOOOO | OOOO fflffl| O'' pk lim1nIn _______//_Ph(X; Y )___//_lim1nG(n)k in which the row is exact and the vertical map is surjective. This identifies Ker(pk) with Im(Ph(X; Y ) -! Ph(X; Y )). Therefore f ~k * if and only if G0(f) > k. This proves the first statement; the second statement follows from Proposition 1. || Proof of Proposition 3 We have to show that the map jk : lim1nG(n)k+1__//_lim1nG(n)k is injective. Consider the short exact sequence of towers 0 -! {Jn} -! {G(n)k+1} -! {G(n)k} -! * : The fibration K(ssk+1Y; k)_____//Y (k+1)_____//Y (k) shows us that each Jn a quotient of [X; K(ssk+1Y; k)] ~=Hk(X; ssk+1Y ). If Gray's condition is not satisfied, then this is a finite group. Applying the six term lim - lim1 exact sequence, we obtain the exact sequence lim1nJn -! lim1nG(n)k+1-!lim1G(n)k-! * : Since a tower of finite groups has trivial lim1 the proof is complete. || Proof of Theorem 4 According to [2], we can write A = colim Lk where each Lk is a subcomplex of A such that Hm (Lk) -! Hm (A) is an isomorphism for m k and Hm (Lk) = 0 for m > k. A careful THE GRAY FILTRATION ON PHANTOM MAPS 13 examination of their proof reveals that there are subcomplexes Kk-1 Mk Kk so that 1 Mk is an k-dimensional~subcomplex of A 2 Hm (Mk; Q) -=! Hm (X; Q) for m k. It follows that A -! A=Mk induces an isomorphism in rational homol- ogy in dimensions above k. We see from the commutative diagram OE A ___________//B q || || fflffl|OE |fflffl A=Mk-1 _____//B=Bk-1 | | fflffl| A=Ak-1 that Phk(A; Y ) q*(Ph(A=Mk-1; Y )). Observe also that OE induces surjections OE*: Hm (B=Bk-1; Q) -! Hm (A=Mk-1; Q) for m l. The proof of Theorem 4 now reduces to the following Lemma. Lemma 4.1. If OE : A -! B and OE* : Hm (B; Q) -! Hm (A; Q) is sur- jective for each m l, then OE induces a surjection on ~l+1-equivalence classes. Proof of Lemma Write G(n)k(A) = Im([A; Y (n)] -! [A; Y (k)]) and similarly for G(n)k(B). We have to show that OE induces a surjection lim1nG(n)l+1(B)___//lim1nG(n)l+1(A): To do this, we will show that each map G(n)l+1(B)___//G(n)l+1(A) rationalizes to a surjection. Since these are finitely generated nilpotent groups, it will follow from Lemma 1.3 of [8] that the images of these maps have finite index. These towers also have the special property that each term has finite index in the next (Proposition 0.1 of [9]). It is proved in Lemma 2.2 of [8] that a map of towers such as these in which the images have finite index induces a surjection on lim1. The map Gl+1(B) -! Gl+1(A) is by definition OE* : [B; Y (l+1)]___//[A; Y (l+1)]: 14 L^E MINH HA AND JEFFREY STROM Since these groups only depend on finite skeleta of A and B, the ratio- nalization of this map can be identified with OE* : [B; Yo(l+1)]___//[A; Yo(l+1)]: Since OE induces a surjection on rational cohomology, and loop spaces are rationally equivalent to products of Eilenberg-MacLane spaces, this map is a surjection. Because of the finite index property of these towers, we know that each map G(n)l+1-!Gl+1 is a rational isomorphism. It follows from the commutative diagram G(n)l+1(B)___//G(n)l+1(A) | | | | fflffl| fflffl| Gl+1(B) _____//Gl+1(A) that G(n)l+1(B) -! G(n)l+1(A) rationalizes to a surjection, as desired. || Proof of Theorem 5 The proof of the dual to Theorem 4 is analagous. The only difference is that the better naturality properties of the Post- nikov sections make it unnecessary to fiddle with dual versions of the Mk. || Proof of Theorem 6 Since Phl+1(A; Y ) = Ker(pl+1), we have to show that pl+1OE*(Phk(B; Y )) = *. Let OE A -! B -! C be a cofiber sequence. Then * : Hm (C; Q) -! Hm (B; Q) is surjective for k m l. In the diagram Phk(B; Y ) n n n n pl+1|| wwnn fflffl| * _*___// 1 (n) _OE_//_ 1 (n) lim1nG(n)l+1(C) limn Gl+1(B) limn Gl+1(A) the row is not exact, but the composition is trivial. By Theorem 4 * is surjective, so OE* is trivial. The proof of the dual result is similar. || Proof of Theorem 8 The groups in the tower {[X; (Sk+1)(m)]} are each countable, and the image of each term in the next has finite index by Theorem 0.1 of [9]. First we assume that lim1m[X; (Sk+1)(m)] = Ph(X; Sk) = *: THE GRAY FILTRATION ON PHANTOM MAPS 15 By Theorem 2 of [7], the tower {[X; (Sk+1)(m)]} must be Mittag- Leffler. By Lemma 3.2 in [7], this means that the index of Im([X; Sk+1] -! [X; (Sk+1)(m)]) = Im(limm [X; (Sk+1)(m)] -! [X; (Sk+1)(m)]) in [X; (Sk+1)(m)] is finite for each m. Taking m = k + 1, we see that the index of Im([X; Sk+1] -! Hk(X; Z)) in Hk(X; Z) is finite. Choose a basis u1; : :;:un of integral classes for Hk(X; Q). Then there are maps gi : X -! Sk+1 and constants i such that iui 2 g*i(Hk(Sk+1; Z)): It follows that the map Yn g : X -! Sk+1 i=1 given by gi in the ith coordinate, induces a surjection in rational coho- mology. Q Now assume that such a map g : X -! Sk+1 is given, and let f 2 Phk(X; Y ). Since g* induces a surjection in k-dimensional rational cohomology,Qthere is a phantom map f0 ~k+1 f such that f0 2 g*(Ph(Q Sk+1; Y )). But it follows easily from Example 2.3 in [4] that Ph( Sk+1; Y ) = *, which shows that f ~k+1 *; in other words, f 2 Phk+1(X; Y ). || Proof of Theorem 9 The proof that condition 1 implies condition 2 is strictly dual to that of Theorem 8. W Now assume given a map g : K(Z; k) -! Y which induces a surjection i_ j g* : ssk+1 K(Z; k) Q -! ssk+1(Y ) Q: Since the inclusion of a wedge into the corresponding product has a section after suspending, we can form the composite Y iY j i_ j g K(Z; k) -! K(Z; k) - ! K(Z; k) -! Y: This composite map, g, induces a surjection in ssk(-) Q. 16 L^E MINH HA AND JEFFREY STROM Consider the diagram G(n)k+1____________//_Gk+1 = || |=| Q fflffl| fflffl| [X; K(Z;Ok)]O______//[X; Y (n)]____//_[X; Y (k+1)] OO ~=|| |~=| Q Q| Q g* Q | ssk( K(Z; k))________________________//_ssk(Y (k+1)) Q Since the image of g* has finite index, we conclude that the index of G(n)k+1in Gk+1 has a finite lower bound for all n. This implies that the tower {G(n)k+1} is Mittag-Leffler, and so we have proved that Ph (X; Y ) = Ph k+1(X; Y ): The calculation Ph k(X; Y ) = Im(Ph (X; Y )) = Im(Ph k+1(X; Y )) Ph k+1(X; Y ) completes the proof. || Proof of Theorem 10 Write Gn = [X; Y (n)], as above. Since Ph (X; Y ) ~= lim1nGn and Ph_(X; Y ) ~= limklim1nG(n)k, Theorem 10 is a direct consequence of the following algebraic result. Proposition 10.1 If {Gn} is a tower of finitely generated nilpotent groups and limk lim1nG(n)k= *, then lim1nGn = *. Proof of Proposition As we have seen, the tower {lim1nG(n)k} is a tower of surjections. Because limk lim1nG(n)k= *, it must be that lim1nG(n)k= * for all large values of k, say for k N. By Theorem 0.1 of [9], each tower {G(n)k} consists of finitely generated nilpotent groups. Since Theorem 2 of [7] tells us that such a tower has trivial lim1if and only if it is Mittag-Leffler, we conclude that the tower {G(n)k} is Mittag-Leffler for k N. This does not quite imply that the tower {Gn} is Mittag-Leffler; but it does show that the tower {G0n}, which is defined by G0n= Gn for n N and G0n= 0 otherwise, is Mittag-Leffler. Since these towers have isomorphic lim1, we see that lim1nGn = *, as desired. || Proof of Corollary 11 Write X = colimMk, where the Mk are as in the proof of Theorem 4. To show that Phk+1(X; Y ) = *, recall that Phk+1(X; Y ) Im(Ph(X=Mk; Y ) -! Ph(X; Y )): THE GRAY FILTRATION ON PHANTOM MAPS 17 Since X=Mk is rationally trivial, Proposition 3 shows that Ph(X=Mk; Y ) = Ph1 (X=Mk; Y ) for every finite type nilpotent Y , and so Ph(X=Mk; Y ) = * by Theorem 10. The proof of the dual result is similar. || Proof of Theorem 12 In both cases, we have a map of tow- ers of finitely generated abelian groups which induces a surjection on limklim 1n, and we want to conclude that the map induces a surjection on lim1. This is a purely algebraic statement which we prove below. Proposition 12.1 If OE : {Bn} -! {An} is a map of towers of finitely generated abelian groups such that OE : limklim 1nB(n)k-! limk lim1nA(n)k is surjective, then OE : lim1nBn -! lim1nAn is also surjective. Proof of Proposition According to Bousfield and Kan ([1], page 254-255), there is a short exact sequence * -! lim1klimnG(n)k-! Ph(X; Y ) -! limklim1nG(n)k-! * : Let Cn be the cokernel of Bn -! An. From the six term exact sequence and the Bousfield-Kan short exact sequence, we obtain the following diagram * _____//lim1klimnB(n)k___//_lim1nBn____//limklim1nB(n)k___//* | | | | | | | | | | | fflffl| fflffl| fflffl| | fflffl| fflffl| * _____//lim1klimnA(n)k___//lim1nAn_____//limklim1nA(n)k___//* | | | | | | | | |* | | fflffl| fflffl| fflffl| | fflffl| fflffl| * _____//lim1klimnC(n)k___//lim1nCn_____//limklim1nC(n)k___//* | | fflffl| * in which the rows and the center column are exact, and each vertical composite is trivial. We have to show that lim1nCn = *. Since the maps lim1nAn -! lim1nCn and limk lim1nB(n)k-! limk lim1nA(n)k 18 L^E MINH HA AND JEFFREY STROM are both surjective, we find that Proposition 10.1 applies to the tower {Cn}. || References 1.A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localization* *s, SLNM 304 Springer, Berlin (1972) 2.E. H. Brown and A. H. Copeland, Jr., An homology analogue of Postnikov systems, Mich. J. Math. 6 (1959) 313-330 3.B. Gray, Operations and a problem of Heller, Ph.D.thesis, University of Chic* *ago (1965) 4.B. Gray and C. A. McGibbon, Universal phantom maps, Topology 32(1993), 371-394. 5.L^e Minh Ha, On the Gray Index of Phantom Maps, Ph. D. Thesis, Wayne State University (2000) 6.C.A. McGibbon, Phantom maps, chapter 25 in Handbook of Algebraic Topol- ogy, Elsevier(1995) 7.C. A. McGibbon and J. M. Moller, On spaces with the same n-type for all n, Topology 31 (1992) 177-201 8.C. A. McGibbon and J. Roitberg, Phantom maps and rational equivaleces, Am. J. Math. 116 (1994) 1365-1379 9.C. A. McGibbon and R. Steiner, Some questions about the first derived functor of the inverse limit, J. Pure and Applied Algebra 103 (1995) 325-340 10.C. A. McGibbon and J. Strom, Numerical invariants of phantom maps, in preparation. 11.R. J. Milgram, Surgery with coefficients, Ann. Math. 100 (1974) 194-248 12.J. Strom, Higher order phantom maps, in preparation. 13.G. W. Whitehead, Elements of Homotopy theory, Graduate texts in Mathe- matics, number 61, Springer, 14.A. Zabrodsky, On phantom maps and a theorem of H. Miller, Israel J. Math. 58 No. 2 (1987) 129-143 Wayne State University, Detroit, MI 48202 ha@math.wayne.edu Dartmouth College, Hanover, NH 03755 jeffrey.strom@dartmouth.edu