HAVING THE H-SPACE STRUCTURE IS NOT A GENERIC PROPERTY JIANZHONG PAN Abstract. In this note, we answer in negative a question posed by McGibbon[7] about the generic property of H-space structure. In fact we verify the conjecture of Roitberg [12]. Incidentally, the same example also answers in negative the open problem 10 in McGibbon[7] 1. Introduction Let X be a connected CW complex, the L-S category of X, cat(X), of X is the least integer k 0 such that X can be covered by k + 1 open subsets which are contractible in X. Of course the condition that X is a CW complex is unnecessary for the above definition to have a meaning . However it is in this context that a rich theory of category exists. Recent works in rational homotopy theory gave rise to a theory of rational L-S category. This makes it possible to calculate the rational L-S category and thus attack the rational Ganea Conjecture[1]. On the other hand works by N.Iwase, H.Scheerer, and D.Stanley provided the method to determine the L-S category itself in some case which lead to the construction of counterexamples to the Ganea Conjecture, see, ____________ Date: June,5,2000. 1991 Mathematics Subject Classification. 55P60,55P45. Key words and phrases. H-space, Phantom map,Mislin genus. The author is partially supported by the NSFC project 19701032 and ZD9603 of Chinese Academy of Science . 1 2 JIANZHONG PAN e.g., [14 ]. Besides the application to the Ganea Conjecture , another interesting application of these ideas was given by Roitberg[12 ]. To explain this we have to state one problem posed by McGibbon[7]: Question 1.1. If X and Y have the same Mislin genus ,i.e. X(p)' Y(p)for all p where X(p)is the p-localization of X, does it follow that cat(X) = cat(Y )? In his paper[12 ], applying some results about category by Iwase[4] and results about phantom maps , Roitberg was able to answer the above question negatively. His main result can be stated as follows Theorem 1.2. Let OE : K(Z, 5) ! S4 be an essential, special, phan- tom map and X be the mapping cone of OE .Then cat(X) = 2. Remark 1.3. It is well known that cat(X) = 1 iff X is a co-H-space . It follows from the above theorem that X is not a co-H-space . On the other hand it easy to know that S4 _ 2K(Z, 5) has the same Mislin genus with X and is a co-H-space . From this point of view , the theorem above answers in negative the following Question 1.4. If X and Y have the same Mislin genus and X is co- H-space , does it follow that Y is also a co-H-space? Question1.1 has an obvious Eckmann-Hilton dual . However it may not be a good question at present time since the dual L-S category is not well developed. A manageable problem is the obvious dual of Question1.4 which has been posed by McGibbon in [7]. A more precise conjecture was given by Roitberg in his paper[12 ]. The purpose of this paper is to establish Roitberg's conjecture and thus also answers in negative McGibbon's problem. HAVING THE H-SPACE STRUCTURE IS NOT A GENERIC PROPERTY 3 Theorem 1.5. Let _ : K(Z, 2) ! S6 be an essential, special, phan- tom map and Z be the homotopy fiber of _. Then Z and K(Z, 2)x 2S6 have the same Mislin genus and Z is not an H-space. Since the dual L-S category is not well developed and Iwase's paper[4] is unavailable to the author, method from the well developed theory of H-space will be used in stead. Another feature is the application of the newly developed Gray index of phantom map [5],[9]. Actually , by duality , our method also gives an alternative proof of the main results in [12 ]. Incidentally , the example constructed above combined with Theo- rem 3.4 in [3] also provides a negative answer to the open problem 10 in [7]: Is X an H-space if each of its Postnikov approximations X(n) is ? Actually we have the following Theorem 1.6. Let _ : K(Z, 2) ! S6 be an essential, special, phan- tom map and Z be the homotopy fiber of _. Then Z(n) is an H-space for each n but Z is not. Proof. It follows immediately from Theorem 3.4 in [3] that Z(n) and (K(Z, 2) x 2S6)(n)have the same homotopy type and thus Z(n) is an H-space for each n. In this paper all spaces involved are assumed to be 1-connected CW complexes with finite type. The author would like to thank Prof. Roitberg for his interest in this work and for pointing out a fatal error in the earlier version of this paper . It is to correct that error that we find the application of Gray index to this work. 4 JIANZHONG PAN 2. Background about H-spaces and Phantom maps First we will recall some backgrouds about H-spaces , see [17 ] for details. An H-space is a space X with a map ~ : X x X ! X such W that ~|X_X = F where F : X X ! X is the natural folding map. An H-map between H-spaces is a map of spaces f : X ! Y such that the following diagram commutes up to homotopy. fxf X x X --- ! Y x Y ? ? ~X ?y ~Y?y f X --- ! Y In this case we say that f is a ~X - ~Y H-map. Two elementary but important results are the followings Proposition 2.1. Let (X, ~X ) be an H-space. Then, for any space M, [M, X] is an algebraic loop, i.e., for any f, g 2 [X, Y ] there exists a unique Df,g2 [M, X] such that ~*(Df,g, g) = f Proposition 2.2. If f : (X, ~X ) ! (Y, ~Y ) is an H-map , then the homotopy fiber of f is an H-space. Thus it is important to know when is a map an H-map or what is the obstruction for a map to be an H-map. Definition 2.3. Let (X, ~) and (Y, ~0) be H-spaces and f : X ! Y be a map of spaces. H-derivation of f is the map HD(f) 2 [X ^ X, Y ] which is defined by HD(f) = Df~,~0(fxf) where : X x X ! X ^ X is the natural quotient map. HAVING THE H-SPACE STRUCTURE IS NOT A GENERIC PROPERTY 5 Remark 2.4. The definition of H-derivation depends on the H-space structures on both X and Y . Remark 2.5. Let (X, ~) and (Y, ~0) be H-spaces and f : X ! Y be a map of spaces. It is well known that f : (X, ~X ) ! (Y, ~Y ) is an H-map iff HD(f) = *. An easy but crucial corollary of this last remark is the following Corollary 2.6. Let (X, ~) and (X0, ~0) be H-spaces and f : X ! X0 be a map of spaces. Assume further that ßiX0 = 0 for i 2d and X is (d - 1)-connected . Then f is a ~ - ~0 H-map. Following is one of the fundamental properties of H-derivation Proposition 2.7. Let (Xi, ~i) be H-spaces, i = 0, 1, 2 and f : X0 ! X1, g : X1 ! X2 be maps of spaces. (a)If f : X0 ! X1 is a ~0 - ~1 H-map, then HD(gf) = HD(g)(f ^ f) (b)If g : X1 ! X2 is a ~1 - ~2 H-map, then HD(gf) = gHD(f) Another ingredient for the main result is the phantom map . Recall that a map f from a CW complex X is called an phantom map if its restriction to the n-th skeleton is inessential for any integer n. Let P h(X, Y ) denote the set of homotopy classes of phantom maps from X to Y . The following result which follows from the Sullivan conjecture provides us many examples of phantom maps. Theorem 2.8. [8] Let Y = iK and X = jZ such that i, j 0 ,K is a 1-connected finite CW complex. Then every map from X to Y is a phantom map if Z is as follows: o Z is the classifying space of a 1-connected compact Lie group o Z is an infinite loop space with torsion fundamental group o Z has only finitely number of nontrivial homotopy groups 6 JIANZHONG PAN and in this case we have Y O P h(X, Y ) = [X, Y ] = [X(0), Y ] = Hn (X, ßn+1(Y ) R) n>0 Let P h(X, Y ) denote the set of homotopy classes of phantom maps from X to Y . The p-localization lp induces a natural map l*p: P h(X, Y ) ! P h(X, Y(p)) It follows that there is a natural map Y l : P h(X, Y ) ! P h(X, Y(p)) p It is well known that l is an epimorphism [15 ] and Ker(l) is nontrivial iff P h(X, Y ) is nontrivial[7], see also [3],[11 ] . The phantom map in Ker(l) is called special , following Roitberg[12 ], see also,[6] where it is called the clone of constant map. On the other hand , Gray, Le Minh Ha , McGibbon and Strom [2], [5],[9] introduced the notion of Gray index which is defined as follows: Definition 2.9. Let f : X ! Y be a phantom map. Then f can be ~f factorized as the composition X ! X=Xk ! Y for each k . The Gray index of f , denoted by G(f), is the largest integer k such that the f~ can be chosen to be a phantom map. G(f) = 1 if no such k exists. Remark 2.10. Let f : X ! Y be a phantom map. Then f can be lifted to the k-th connected covering for each k and G(f) + 1 is the largest integer k such that the kth lifting can be chosen to be a phantom map. A useful fact we need is Proposition 2.11. Let f : X ! Y be a phantom map. Then (i)G(f) n if X is n-connected or Y is n + 1-connected. N N (ii) G(f) 2 {k|Hn (X, ßn+1(Y ) Q) 6= 0} if Hn (X, ßn+1(Y ) Q) = 0 for n sufficiently large. HAVING THE H-SPACE STRUCTURE IS NOT A GENERIC PROPERTY 7 For the proof of the Proposition above, see [5] and [9]. An immediate corollary of the above Proposition which is crucial to our purpose is Corollary 2.12. Let f : K(Z, 2m) ^ K(Z, 2m) ! 1S4m+2 be any essential map. Then G(f) = 4m where m 1. Now we are ready to prove the main result. 3. Proof of Theorem1.5 First is a preliminary lemma needed later. Lemma 3.1. Let X be an H-space which is (e - 1)-connected and ßi(X) = 0 for i 2e and Y = jK where K is a (d + j - 1)-connected finite CW-complex with j 1 and d > e 2. Let _ : X ! Y be any essential map. Then Z(=homotopy fiber of _) is not an H-space if HD(_) O (i ^ i) is essential where i : Z ! X is the homotopy fiber of _ . Proof. If Z is an H-space , then * = _ O i : Z ! Y is an H-map and HD(*) = 0. Since i is an H-map by Corollary2.6 , it follows by Proposition2.7 that * = HD(*) = HD(_ O i) = HD(_) O (i ^ i) which is in contradiction to the condition. Remark 3.2. To apply the above lemma it suffices to discuss when _ is not an H-map and when i ^ i induces an injective. Theorem 3.3. Let X = K(Z, 2m) and Y = 1S4m+2 with m 1. Then there is no essential H-map from X to Y . 8 JIANZHONG PAN Proof. By Proposition2.8, the rationalization r : X ! X(0)which is an H-map induces an isomorphism of groups r* : [X(0), Y ] ! [X, Y ] It follows from Proposition2.7 that it suffices to prove that there is no essential H-map from X(0)to Y . On the other hand the map h : S4m+2 ! K(Z, 4m + 2) which repre- sents a generator of H4m+2 (S4m+2 ; Z) ~= Z induces an isomorphism of groups ( 3h)* : [X(0), Y ] ! [X(0), K(Z, 4m + 1)] Again the Proposition2.7 implies that it suffices to prove that there is no essential H-map from X(0)to K(Z, 4m + 1) which is well known to be equivalent to the injectivity of the following homomorphism `* : H4m+2 ( X(0); Z) ! H4m+2 ( X(0)^ X(0); Z) where ` is defined as follows: Let X * X = X x I x X={(x, 0, y) ~ (x, 0, y0), (x, 1, y) ~ (x0, 1, y)} be the join . There is a well defined map k : X * X ! X ^ X by k[x,t,y]=(x,y,t). It is well known that X * X is homotopy equivalent to X ^ X. If X is an H-space with multiplication ~ , then ` is the composite map X ^ X ' X * X !k (X x X) ! X where the last map is the map - ß1 + ~ - ß2 and ß1, ß2 are the projection of X x X to the first and second factors respectively . HAVING THE H-SPACE STRUCTURE IS NOT A GENERIC PROPERTY 9 Consider the following commutative diagram where the horizontal maps which are isomorphisms come from the universal coefficient The- orem H4m+2 ( X(0); Z) --- ! Ext (H4m+1 ( X(0); Z), Z) ? ? `*?y Ext(`*,Z)?y H4m+2 ( X(0)^ X(0); Z) --- ! Ext(H4m+1 ( X(0)^ X(0); Z), Z) It follows that it suffices to prove that the map `* : H4m+1 ( X(0)^ X(0); Z) ! H4m+1 ( X(0); Z) or equivalently the map `* : H4m (X ^ X; Q) ! H4m (X; Q) is injective . On the other hand, it is well known that the map `* is dual to the reduced coproduct which is an isomorphism in this case and thus completes the proof. The Theorem1.5 is actually a corollary of the following more general Theorem Theorem 3.4. Let X = K(Z, 2m) and Y = S4m+2 with m 1. Let _ : X ! Y be any essential, special, phantom map. Then Z(=homotopy fiber of _) is not an H-space and has the same Mislin genus with X x Y . Proof. That Z and X x Y have the same Mislin genus follows from the condition that _ : X ! Y is a special phantom map. On the other hand Lemma3.1 and Theorem3.3 apply here. Thus the Theorem above follows from the following Proposition. Proposition 3.5. Let X = K(Z, 2m) and Y = S4m+2 with m 1. Let _ : X ! Y be any essential, special, phantom map which exists by 10 JIANZHONG PAN Proposition2.8 and the remark after it. Then (i ^ i)* : [X ^ X, Y ] ! [Z ^ Z, Y ] is injective where i : Z ! X is the homotopy fiber of _ . Proof. Let f : X ^ X ! Y be any essential map . If f O (i ^ i) ~ * we will prove that this leads to a contradiction which concludes the proof. Since f is a phantom map , f O (i ^ i) ~ * is also a phantom map.Thus we have the following commutative diagram up to homotopy. f Z ^ Z --i^i-! X ^ X --- ! Y ? ? ? ? ? ? y y idy i(0)^i(0) f~ Z(0)^ Z(0) --- - ! X(0)^ X(0) --- ! Y If f O (i ^ i) ~ * , then f~O (i(0)^ i(0)) is the composite Z(0)^ Z(0)! Zfi^ Zfi!h Y where Zfiis the homotopy fiber of the rationalization X ! X(0). On the other hand we claim that Claim 3.6. Any map h : Zfi^ Zfi! Y factors through a map Zfi^ Zfi! Ffi^ Ffiwhere F = 2S4m+2 . Assuming this , note that i(0)^ i(0)admits a right inverse, we have that f is the composite X ^ X ! X(0)^ X(0)! Ffi^ Ffi! Y . It is easy to know that Ffiis 4m - 1-connected and thus Ffi^ Ffi is 8m - 2-connected. It follows that f is the composite X ^ X ! X(0)^ X(0)! Y < 8m - 2 >! Y . By Remark2.10, G(f) 8m - 3 which contradicts Corollary2.12. Remark 3.7. Roitberg has shown us how the use of Gray index can be avoided. It remains to prove the Claim3.6 which follows from the following Lemma 3.8. There is a map g ^ g : Zfi^ Zfi! Ffi^ Ffisuch that the following map is a weak homotopy equivalence ( g ^ g)* : map*( Ffi^ Ffi, Y ) ! map*( Zfi^ Zfi, Y ) HAVING THE H-SPACE STRUCTURE IS NOT A GENERIC PROPERTY 11 Proof. Roitberg and P. Touhey proved in [13 ] that , if X, Y have the same Mislin genus, then Xfi' Yfi. So we have a map g which is a composite Zfi' (X x F )fiiø!Ffi where ß : X x F ! F is the projection. To prove g ^ g induces a homotopy equivalence it suffices to prove that ßfi: (X x F )fi! (F )fiinduces a homotopy equivalence ( ßfi^ ßfi)* : map*( Ffi^ Ffi, Y ) ! map*( (X x F )fi^ (X x F )fi, Y ) which follows directly from the fact that map*(Xfi, Y ) is weakly con- tractible and the fact map*( Xfi, Y ) ' map*( X, ^Y) which can be found in [16 ] , for a stronger result , see Pan and Woo [10 ]. _____________________- References [1]T.Ganea, Some problems on numerical homotopy invariants, Lecture Notes in Math., v.249, Springer , Berlin,1971, pp.23-30 [2]B.Gray, Operations and a problem of Heller, PhD thesis, University of Chicago,1965 [3]J.R.Harper, J.Roitberg, Phantom maps and spaces of the same n-type for all n , J. of Pure and Applied Algebra 80(1992), pp.123-137 [4]N.Iwase, A1 method in L-S category, Preprint. [5]L^eMinh H`aand J. Strom , The Gray filtration of phantom maps, Preprint, Hopf Archive ,Nov., 1999 [6]C.A.McGibbon, Clones of spaces and maps in homotopy theory, Comment. Math. Helv. , 68(1993), pp.263-277 12 JIANZHONG PAN [7]C.A.McGibbon, Mislin genus of a space, CRM Proceedings and Lecture Notes, 6(1994), pp.75-102 [8]C.A.McGibbon, Phantom maps, Handbook of algebraic topology, pp.1209- 1257 ( I. M. James,ed.), North-Holland, 1995 [9]C.A.McGibbon and J.Strom, Numerical invariants of phantom maps, Preprint, Hopf Archive ,Feb., 2000 [10]Pan Jianzhong, Woo Mooha , On the Phantom elements, to appear in Con- temp. Math. [11]J.Roitberg, Computing homotopy classes of phantom maps, CRM Proceedings and Lecture Notes, 6(1994), pp. 141-168 [12]J.Roitberg, The Lusternik-Schnirelmann category of certain infinite CW- complexes, Topology 39(2000), pp.95-101 [13]J.Roitberg, P. Touhey, The homotopy fiber of profinite completion, to appear in Topology and its Applications. [14]D.Stanley, Spaces with Lusternik-Schnirelmann category n and cone length n+ 1, Preprint in Hopf Archive ,1998 [15]R.J.Steiner, Localization, completion and Infinite complexes, Mathematika. 24(1977), pp.1-15 [16]P.Touhey, A Phantom Dissertation , CUNY , (1997). [17]A.Zabrodsky, Hopf spaces, Math. Studies 22, North-Holland, 1976 _____________________- Institute of Math.,Academia Sinica ,Beijing 100080, China E-mail address: pjz@math03.math.ac.cn