PHANTOM ELEMENTS AND ITS APPLICATIONS JIANZHONG PAN AND MOO HA WOO Abstract. In our previous work[11] , a relation between Tsukiyama problem about self homotopy equivalence was found by using a generalization of phantom map. In this note , fundamental result is established for such a generalization. This is the first time one can deal with phantom maps to space not satisfying finite type condition. Application to Forgetful map is also discussed briefly. 1. Introduction The main aim of this paper is to study phantom map , its gener- alization and applications. After the discovery of the first example of phantom map by Adams and Walker[1] , theory of phantom map receives a lot of attention . The main aim of these previous studies is however to understand it , e.g., the computation and the proper- ties of phantom maps. The first application of theory of phantom maps was given by Harper and Roitberg[5],[12 ] who applied it to com- pute SNT (X) and Aut(X). Recently applications are also found by Roitberg[13 ] and Pan[10 ] where several conjectures of McGibbon were settled. On the other hand , a remarkable connection was established by Pan and Woo[11 ] between Tsukiyama problem about self homotopy equivalence and a generalization of phantom map. A byproduct of this connection is that a special case of Tsukiyama problem is almost equiv- alent to the famous Halperin conjecture in rational homotopy theory[4]. ____________ Date: Aug. 30, 2000. 1991 Mathematics Subject Classification. 55P10,55P60,55P62,55R10. Key words and phrases. Phantom map,Forgetful map, Halperin conjecture. The first author is partially supported by the NSFC project 19701032 and ZD9603 of Chinese Academy of Science and the second author wishes to acknowl- edge the financial supports of the Korea Research Foundation made in the program year of (1998) and TGRC 99. 1 2 JIANZHONG PAN AND MOO HA WOO A well known characterization of map between nilpotent spaces of finite type to be phantom map is the following Theorem 1.1. Let X, Y be nilpotent CW complexes of finite type with Y 1-connected and f : X ! Y be any map. Then the followings are equivalent o f is a phantom map o e O f ' * where e : Y ! ^Y is the profinite completion o f O ø ' * where ø : Xfi! X is the homotopy fiber of the rationalization On the other hand , in our previous paper[11 ], we generalized the concept of phantom map to that of phantom element and announced a theorem characterizing an element to be a phantom element which generalizes Theorem1.1 . In this paper we will generalize further so that we can deal with space which is not of finite type. Theorem 1.2. Let X be nilpotent CW complex of finite type , Y be 1-connected such that ßn(Y ) is reduced group for n 2 and g : X ! Y be any map.Then the followings are equivalent: o ff 2 ßj(map*(X, Y ); g) is a phantom element o (e*)# (ff) = 0 where (e*)# : ßj(map*(X, Y ); g) ! ßj(map*(X, ^Y); ^g) o (ø *)# (ff) = 0 where (ø *)# : ßj(map*(X, Y ); g) ! ßj(map*(Xfi, Y ); gf* *i) Note that the assumption that Y is 1-connected is not a real restric- tion by an observation of Zabrodsky[15 ]. We will give a complete proof of this theorem in this paper. As an application we have(Corollary of Proposition 3.3) Corollary 1.3. Let P be 1-connected finite dimensional CW complex or such that H*(P, Zp) is locally finite over Ap for each prime p and be of type F0 . Assume further that ßnBaut(P ) is reduced group for n 2. Assume further that P(0)satisfying one of the following. o P is rationally equivalent to Kähler manifold o H*(P ; Q) as an algebra has at most 3 generators o P is rationally equivalent to G=U where G is a compact Lie group and U is a closed subgroup of maximal rank PHANTOM ELEMENTS AND ITS APPLICATIONS 3 Then for all m 1 , H and every principal K(H, 2m)-bundle with total space homotopy equivalent to P , Forgetful map is injective. The organization of this paper is as follows. In section2 Theorem1.2 will be proved. The applications to Forgetful map will be discussed in section3. In this paper, We will use the following notations: o H will denote a finitely generated abelian group o map(X, Y ) is the space of continuous mappings from X to Y o map*(X, Y ) is the subspace of pointed mappings from (X, x0) to (Y, y0) o l : X ! X(0)is the rationalization o Let ø : Xfi! X be the homotopy fiber of l. Then Xfi!fiX ! X(0)is a cofibration up to homotopy o ep : Y ! ^YZp1is Bousfield-Kan's p-completion. Let Y^ = Q ^ YZp1 and e = (e2, e3, . .).: Y ! Y^. Let Yj be the homo- p topy fiber of e The readers should refer to [11 ] for all the other notations which have not been explained here. In concluding the Introduction , we 'd like to give the following Conjecture 1.4. The condition that ßnY is reduced group in this paper can be removed. 2. Phantom elements Let's begin with definition. Definition 2.1. Let spaces X be a CW complex, Y be a space and g : X ! Y any map. Then an element ff 2 ßj(map*(X, Y ); g) is called a g-phantom element if (i*n)# (ff) = 0 for all n 0 where (i*n)# : ßjmap*(X, Y ) ! ßjmap*(Xn , Y ) is the homomorphism induced by the inclusion in : Xn ! X. Denoted by P hgj(X, Y ) = {ff 2 ßj(map*(X, Y ); g)|ff is a g-phantom element } Obviously if g =constant and j = 0 , then ff is a g-phantom element iff it represents the homotopy class of a map which is a phantom map. 4 JIANZHONG PAN AND MOO HA WOO The main aim of this section is to prove Theorem1.2. Before that , let's give some results necessary to the proof . Lemma 2.2. Let Y be 1-connected such that ßn(Y ) is reduced group for n 2. Then Q o ßn(Y^) = Ext(Zp1 , ßn(Y )) p o For W a finite CW complex, e* : ßjmap*(W, Y )f ! ßjmap*(W, ^Y)f^ is injective Proof. The first statement follows from the fact that Hom(Zp1 , B) = 0 for a reduced group since otherwise there will be nontrivial divisible subgroup in B. To prove the second statement , note that the induced map ßn(Yj) ! ßn(Y ) is trivial since ßn(Y ) is reduced group and ßn(Yj) is rational thus divisible by the arithmetic square Theorem[3]. It follows that e* : ßn(Y ) ! ßn(Y^) is injective and an easy induction argument shows what we want for j 1. For j = 0, it can be proved by an argument similar to that of Theorem 2.5.3 of [6]. Proposition 2.3. Let X be nilpotent space and Y be 1-connected such that ßn(Y ) is reduced group for n 2. Then the followings hold: o [ nX(0), ^Y] = *, ~Hn(X(0), ßi(Y^)) = 0 for all n, i 0 o [ nXfi, Yj] = *, ~Hn(Xfi, ßi(Yj)) = 0 for all n, i 0 Proof. That H~n(X(0), ßi(Y^)) = 0 follows from the fact that Hom(A, B) = 0, Ext(A, B) = 0 Q for ratinal group A and B = Ext(Zp1 , B0) p Then the equation [ nX(0), ^Y] = lim [ nX(0), ^Y (n)] n implies the first statement. The equation about cohomology in second statement is true since ßn(Yj) is rational while the proof of another equation is similar to that as in the first statement. PHANTOM ELEMENTS AND ITS APPLICATIONS 5 Proposition 2.4. Let X be nilpotent space and Y be 1-connected such that ßn(Y ) is reduced group for n 2. Then the followings hold : w o ø *: map*(X, ^Y) ' map*(Xfi, ^Y) w o æ* : map*(X(0), Yj) ' map*(X(0), Y ) w o e* : map*(Xfi, Y ) ' map*(Xfi, ^Y) w o l* : map*(X(0), Yj) ' map*(X, Yj) Proof. The first and the last statements follow from the last Propo- sition and the well known Zabrodsky Lemma. The second and third statements follow from a lim1 argument for j 0 * ! lim1ßj+1map*(Z, En) ! ßjmap*(Z, E) ! lim ßjmap*(Z, En) ! * n n where in the second statement, Z = X(0)and En is the n-th term in the Postnikov-Moore tower of the map æ : Yj ! Y while in the third state- ment , Z = Xfiand En is the n-th term in the Postnikov-Moore tower of the map e : Y ! ^Y. In both case the sequence {ßjmap*(Z, En)} is a sequence consisting of isomorphisms and thus the lim1 is trivial and the wanted isomorphisms follows immediately. Proposition 2.5. Let X be nilpotent space and Y 1-connected such that ßn(Y ) is reduced group for n 2. Let g : X ! ^Y be any map. Then P hgj(X, ^Y) = * Proof. P hgj(X, ^Y) is the lim 1 of a sequence of compact groups and continuous homomorphisms which is will known to be trivial. Proposition 2.6. Let X, Y be two nilpotent spaces with Y 1-connected such that ßn(Y ) is reduced group for n 2. Then the followings hold : w o map*(Xfi, Y ) ' map*(X, ^Y) w o map*(X(0), Y ) ' map*(X, Yj) Proof. This is an easy consequence of the Proposition above. Proof of Theorem1.2. The equivalence between the last two statements follows directly from the following commutative diagram where the 6 JIANZHONG PAN AND MOO HA WOO bottom horizontal homomorphism and the right side vertical homo- morphism are isomorphisms by Proposition2.4 . (fi*)# ßjmap*(X, Y ) -- - ! ßjmap*(Xfi, Y ) ? ? (e*)#?y (e*)#?y (fi*)# ßjmap*(X, ^Y) -- - ! ßjmap*(Xfi, ^Y) Now assume the first statement, then we have (i*n)# (e*(ff)) = 0 for all n 0. It follows from Proposition 2.5 that e*(ff)) = 0. The proof of another direction is similar to that in [11 ] using Lemma2.2 instead of Sullivan's origional result which is stated only for space of finite type. Remark 2.7. It is easy to see that the above proof follows the same pattern as that given by Oda and Shitanda[9]. We give a prove here because Oda informed us that there were gaps in their proof and he don't know if the result is true or not. The similar proof applies also to the equivariant case which will be discussed in future publication. As noted in [11 ] , the natural question related to the application of phantom element to the forgetful map is Question 2.8. For two maps f, g : X ! Y ,what is the relation between P hgj(X, Y ) and P hfj(X, Y )? Proposition 2.9. Let X, Y be nilpotent CW complexes such that [ jXfi, Y ] = [ j+1Xfi, Y ] = 0 If g : X ! Y is a phantom map,then we have P hgj(X, Y ) = ßj(map*(X, Y ); g) Proof. The proof is the same as that in [11 ] . In our application we have to be able to compute P hgj(X, Y ). Before giving this kind of result, recall that a CW complex is called unstable if all the attaching maps vanish under suspension. It is Baues [2] who noted the following which is dual to Zabrodsky's integral approxima- tion. PHANTOM ELEMENTS AND ITS APPLICATIONS 7 Theorem 2.10. Let X be 1-connected CW complex. Then there is an unstable complex and a rational equivalence h : X~ ! X. Remark 2.11. Let X be an unstable CW complex. Then it is easy to prove that P hgj(X, Y ) = * for any map g : X ! Y . Proposition 2.12. Let X be a 1-connected CW complex and Y 1- connected such that ßn(Y ) is reduced group for n 2. Suppose further that the component of map*(X, ^Y) consisting constant map is weakly contractible and g : X ! Y is a phantom map. Then Y P hgj(X, Y ) = ßj(map*(X, Y ); g) = Hk(X, ßk+j+1(Yj)) k>0 Proof. As first noted by Oda and Shitanda , similar proof as in that of Theorem B of [15 ] leads to the following homotopy fibration [ map*(X, Y )g ! map*(X~, Y )* ! map*(X~, ^Y)* g where the union is over phantom maps g. On the other hand , dif- S ferent components of map*(X, Y )g are homotopy equivalent since S g map*(X, Y )g is the homotopy fiber of a map between two connected g spaces. It follows that P hgj(X, Y ) = ßj(map*(X, Y ); const) = ßj(map*(X, Yj); const) = Y = [ j-1X, Yj] = Hk(X, ßj+k+1(Yj)) k>0 We are ready to state results related to the applications. Before that we have another definition Definition 2.13. Let Ap be the modp Steenrod algebra. An unstable module M over Ap is called locally finite iff , for any x 2 M, only finite elements of Ap can acts nontrivially on M. Example 2.14. Let P be a space such that H*(P, Zp) is locally finite over Ap . Then so is P . In particular, if P is finite CW , then H*( P, Zp) is locally finite over Ap. 8 JIANZHONG PAN AND MOO HA WOO Theorem 2.15. Let X = K(H, m + 2) , Y = Baut(P ) such that ßn(Y ) is reduced group for n 2 and g : X ! Y is any map where P is 1-connected finite dimensional CW complex or such that H*(P, Zp) is locally finite over Ap for each prime p and m 1. Then for j 1 P hgj(X, Y ) = ßj(map*(X, Y ); g) = [ jX, Yj] Proof. The proof is the same as that of the corresponding result in [11 ] using results of Zabrodsky and Miller[8] or Theorem 8.8 in [14 ]. Similarly we have Theorem 2.16. Let X = BG, Y = Baut(P ) such that ßn(Y ) is re- duced group for n 2. and g : X ! Y is a phantom map where G is a connected compact Lie group and P is 1-connected finite dimensional CW complex or such that H*(P, Zp) is locally finite over Ap for each prime p. Then for j 1 we have P hgj(X, Y ) = ßj(map*(X, Y ); g) = [ jX, Yj] 3. Application to the forgetful map Given a principal G-bundle ß : P ! B, Let autG (P ) = {g|g : P ! P is a G-equivariant homotopy equivalence } and aut(P ) = {g|g : P ! P is a homotopy equivalence } There is a natural map f : autG (P ) ! aut(P ). Let AutG (P ) = ß0(autG (P )) and Aut(P ) = ß0(aut(P )) Then the map f induces a map F : AutG (P ) ! Aut(P ) which is called a Forgetful map by Tsukiyama. The question posed by Tsukiyama in [7] is the following Question 3.1. Is the forgetting map F injective? PHANTOM ELEMENTS AND ITS APPLICATIONS 9 One of the main results in [11 ] is the following Theorem 3.2. Let ß : P ! B be a principal G-bundle. Then there is an exact sequence ß1aut(P ) !ffiß1(map*(BG, Baut(P )), c) ! AutG (P ) F! Aut(P ) where c : BG ! Baut(P ) is determined by the principal bundle. Combined with results in [11 ] , we have Proposition 3.3. Let P be as in Theorem2.16 . If M ß2i(map(P(0), P(0)); id) = 0 i>1 then for all m 1 , finitely generated abelian group H and every principal K(H, 2m)-bundle with total space homotopy equivalent to P ,the associated Forgetful map is injective . We have also similar results for K(H, 2m + 1) or G bundle where G is a connected compact Lie group which will be omitted. Unlike that in [11 ], there are no complete results if group ß1(map*(BG, Baut(P )), c) is nontrivial although we know that it is still uncountable since group ß1aut(P ) itself may be uncountable too. Thus same results as in [11 ] can be obtained if ß1aut(P ) is countable. This is so if P = P 0where P 0is finite complex. An interesting question is Question 3.4. Study the map ffi in the exact sequence of Theorem 3.2. Is it possible that Image(ffi) is always countable group. _____________________- References [1]J.F.Adams and J.Walker,An example in homotopy theory,Proc. Camb. Phil. 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Math. 58(1987), pp.129-143 _____________________- Institute of Math.,Academia Sinica ,Beijing 100080, China E-mail address: pjz@math03.math.ac.cn Department of Mathematics Education , Korea University , Seoul , Korea