THE BP -THEORY OF TWO-FOLD PRODUCTS OF PROJECTIVE SPACES JES'US GONZ'ALEZ AND W. STEPHEN WILSON 1.Introduction In [KWa , KWb ] the need for the Brown-Peterson cohomology (for p = 2) of a product of two real projective spaces arose. In particular, they needed to unde* *rstand the elements not in the tensor product and how they behaved under maps. Although quick computations with the Adams spectral sequence or the Atiyah- Hirzebruch spectral sequence suggest the answer, there seemed to be nothing ex- plicit enough in the literature, but much of what we do is well known. We always use reduced cohomology. Recall that BP *' Z(2)[v1, v2, . .].. Let x 2 BP 2(RP 2k) be the standard generator coming from BP *(CP 1). The required theorems are as follows: Theorem 1.1. Let m n, then (in reduced cohomology) BP *(RP 2m^ RP 2n) ' BP *(RP 2m) BP* BP *(RP 2n) 2n-1BP *(RP 2m) BP *(RP 2m^ RP 2n+1) ' BP *(RP 2m) BP* BP *(RP 2n) 2n-1BP *(RP 2m) 2n+1BP *(RP 2m) The inclusion map RP 2m-2k! RP 2minduces the corresponding map in Brown- Peterson cohomology on each factor. All of the groups are finite 2-torsion. A 2-adic basis for BP *(RP 2k) is giv* *en by all vxi, 0 < i k, with v a basis element of BP *=(2). A 2-adic basis for BP *(RP 2m) BP* BP *(RP 2n) is given by all vxi x, 0 < i m, v a basis element of BP *=(2), and vxi xj with 0 < i m, 1 < j n, and v a basis element of BP *=(2, v1). A variety of comments are in order here. We can, of course, handle the BP (co)homology of any RPba^ RPdcfor any a > b and c > d. (RPbais the cofiber of the map RP b-1! RP a.) What we state in the above theorem is precisely what is needed in [KWa , KWb ]. We actually go much further and look at the situati* *on when one of the spaces is a 2r lens space. Some of the hard work here was done long ago by Conner and Floyd in Chapter 8 of [CF64 ] where they computed the tensor product part of MSO*(BZ=p x BZ=p). They didn't have BP and MU wasn't in common usage yet, so their work is at odd primes, but it shows the way. There is more to this than just the tensor product. Peter Landweber set up a general short exact sequence in [Lan66] that gives, among other things: BP*(X) BP* BP*(Y ) -! BP*(X ^ Y ) -! T orBP*(BP*(X), BP*(Y )) when X is such that BP*(X) surjects to H*(X), in particular, when X is RP nor a lens space. Our main contribution to the above theorem is to make the Tor term 1 2 JES'US GONZ'ALEZ AND W. STEPHEN WILSON explicit algebraically (there is no topology involved) and to show how it behav* *es under the map we describe. In particular, we show: Theorem 1.2. Let m n, then * * 2m * 2n 2n * 2m T orBP (BP (RP ), BP (RP )) ' BP (RP ) The inclusion map RP 2m-2k! RP 2minduces the corresponding map in Brown- Peterson cohomology. If m < n, the map RP 2n-2! RP 2ninduces the map 2nBP *(RP 2m) -! 2n-2BP *(RP 2m) that takes xi to xi+1. Note that the map to Tor in the cohomology Landweber short exact sequence raises degree by 1. Note that to compute what happens on the Tor term for a map of RP 2k! RP 2n with k < m < n we use the composition RP 2k! RP 2m ! RP 2nand use the second form of the mapping for RP 2m! RP 2nand the first for the RP 2k! RP 2m. In [JW85 ], where a lot of work similar to this is done, credit is given to B* *ob Stong for knowing the Tor term when both n and m are infinity in the homology case, so even this is not entirely new. However, the applications in [KWa , KWb ] are significant and are used to g* *ive new non-immersions of real projective spaces in fairly low dimensions. Since we could find nothing like the above theorem in the literature we felt it necessar* *y to write this up to support the applications. In the first part of Theorem 1.1, the two parts coming from the Landweber sho* *rt exact sequence are even and odd degree so there can be no extension problems to consider. In the second part there could be, and when we look at the general ca* *se of RPba^ RPdcfor any a > b and c > d there could, in principle, be several poss* *ible extension problems. None of these occur. This next result tells how to compute the BP cohomology of all such products by combining several known facts. Recall that since we are using reduced cohomology, BP *(RP n) = BP *(RP1n). Theorem 1.3. BP *(RP22ab+1) ' 2bBP *(RP 2(a-b)) BP *(RPb2a+1) ' BP *(RPb2a) 2a+1BP * BP *(RP2ab) ' BP *(RP2ab+1) 2bBP * The Landweber short exact sequence 0 -! BP *(RPba) BP* BP *(RPdc) -! BP *(RPba^ RPdc) * * a * c - ! T orBP (BP (RPb), BP (RPd)) -! 0 always splits. Combined, this allows us to compute the BP cohomology of any such product. Since there is no vi torsion for i 2 we can really use this for any BP *(-* *) for n > 0 and, of course, we can always get E(n)*(-) from BP *(-) by just tensoring, E(n)*(-) ' E(n)* BP* BP *(-), [JW73 , Remark 5.13, p 347]. In particular, what is used in [KWa , KWb ] is the case of E(2). Since there is no vi torsion for* * i 2 we know that BP *(-) injects to E(n)*(-), n > 1, for these spaces. BP homology computations can be done independently purely algebraically, mimicking the way they are done in cohomology, or, one can just use S-duality PRODUCTS OF PROJECTIVE SPACES 3 k-b-1 where we have from [Ati61] that the S-dual of RPbais RP22k-a-1(for some large k). Our computations for cohomology are immediate for homology. For BP*(RP 2n) we have generators fii 2 BP2i-1(RP 2n) for 0 < i n. The basic facts for homology are collected as a theorem: Theorem 1.4. The Landweber short exact sequence for BP*(RPba^ RPdc) always splits. The map to Tor decreases degree by 1. Any top and bottom `integral' cells split off in BP homology. Let m n, then T orBP*(BP*(RP 2m), BP*(RP 2n)) ' BP*(RP 2m) A 2-adic basis for BP*(RP 2k) is given by all vfii, 0 < i k, v a basis elem* *ent of BP*=(2). A 2-adic basis for BP*(RP 2m) BP* BP*(RP 2n) is given by vfii fin, 0 < i m, v a basis element of BP*=(2), and vfii fij, 0 < i m, 0 < j < n, with v a bas* *is element of BP*=(2, v1). The splittings associated with the `integral' cells are a consequence of Don * *Davis's result from [Dav78 ] that proves they really do split off topologically when sm* *ashed with BP . We can generalize these results to the case of L(e)ab^ RPdcwhere L(e)abis the truncated lens space for 2e (when e = 1 it is just the case RPbawe have already described). Let ffi 2 BP2i-1(L(e)a) to distinguish it from our fii and let xe 2 BP 2(L(e)a) come from BP *(CP 1). Some facts we'll need (let v0 = 2): Theorem 1.5. A 2-adic basis for BP *(L(e)2k) is given by all vj0vxie, 0 j < e, 0 < i k, with v a basis element of BP *=(2). Let n m + e - 1 and m > 1, then a 2-adic basis for BP *(L(e)2m) BP* BP *(RP 2n) is given by all vxie xj, 0 < i m, 0 < j e, v a basis element of BP *=(2), and vxie xj with 0 < i m, e < j n, and v a basis element of BP *=(2, ve1). A 2-adic basis for BP*(L(e)2k) is given by all vj0vffi, 0 j < e, 0 < i k,* * with v a basis element of BP *=(2). Let n m + e - 1 and m > 1, then a 2-adic basis for BP*(L(e)2m) BP* BP*(RP 2n) is given by all vffi fij, 0 < i m, n - e < j n, v a basis eleme* *nt of BP*=(2), and vffi fij with 0 < i m, 0 < j n - e, and v a basis element * *of BP*=(2, ve1). The Landweber short exact sequence for L(e)ab^ RPdcalways splits. Similar identities to those in Theorem 1.3 hold for BP *(L(e)ab) and BP*(L(e)* *ab). Remark 1.6. When m = 1 this is just the mod 2e BP -(co)homology of RP 2n. The proof works here for n > e as well and the result is as stated. The case of n * * e is even easier. This allows us to compute the BP (co)homology of L(e)ab^ RPdcwith some restrictions just like we did in the e = 1 case with one significant difference* *: we have lost our elegance when describing our Tor term. Consequently we bury our description in the section with the proofs. We will also describe why we need t* *he extra bit in our inequality. 4 JES'US GONZ'ALEZ AND W. STEPHEN WILSON In order to prove this we rely on the result of G. Nakos, [Nak85 ], see also * *[Gon03 ], that says that the annihilator ideal for the bottom class ff1 fi1 in BP*(BZ=(2* *e) ^ BZ=(2)) is (2, ve1). The BP cohomology has been understood for a long time, [Lan70]. We also compute BP*(L(e)2m ^ RP 2). RP 2is justethe mod 2 Moore space and when m 2e we get an annihilator ideal of (2,ev21-1). As n goes from 1 to m+e-1 this annihilator ideal must grow from (2, v21-1) to (2, ve1). Things get quite * *complex in this range. Part of this work was completed during a sabbatical visit of the first author to the University of Rochester. The visit was financially supported by a grant * *of Professor Douglas C. Ravenel and CONACyT grant 54987-E. It is a pleasure to thank Professor Ravenel for his kind help and academic motivation during this visit. 2.Proofs of Theorems 1.1 and 1.2 We recall the formal group law for Brown-Peterson cohomology, x +F y, and the corresponding 2-series (where v0 = 2): X X n x +F x = [2](x) = aixi+1= Fvnx2 . i 0 n 0 Note that this immediately implies that a0 = 2 and a1 = v1 (we are using Araki's generators here, [Ara73]). The maps RP 1 -! CP 1 -2!CP 1 give us: * BP *(RP 1) - BP *(CP 1) -2 BP *(CP 1) and * BP *[[x]]=([2](x)) - BP *[[x]] -2 BP *[[x]]. The Atiyah-Hirzebruch spectral sequence for BP *(RP 1) collapses because it is even degree and the 2-series shows how to solve all the extension problems. The same is true for BP *(RP 2n) and now we inherit, from CP 1 and CP n, BP *(RP 2n) ' BP *[x]=([2](x), xn+1). The Atiyah-Hirzebruch spectral sequence givesPour 2-adic basis for BP *(RP 2n) and we seePthat our relations are given by i 0aixj+i. (In homology they are given by i 0aifij-i.) We can show how to reduce any element in the tensor product, BP *(RP 2m) BP* BP *(RP 2n), with m n to the 2-adic basis of the Theorem 1.1. We need to filt* *er the tensor product to make this easy. First we filter on the sum, i + j, for xi* * xj. Next, if xa xb has a + b = i + j, we let it have higher filtration if b < j. * *We set up an algorithm for reduction. If we have an element that is divisible by 2, i.* *e. if we have a 2xi xj, we replace the 2x in 2xxi-1using the 2-series. All terms are* * of a higher filtration. If we have a v1xi xj with j > 1, we replace the v1x2 = a1* *x2 in v1x2xj-2 using the 2-series. All of the terms with ai, i > 1 will be of hig* *her filtration, but we will be left with -2xi xj-1. We can now replace the 2x in 2xxi-1 using the 2-series and all of our terms will be of higher filtration. PRODUCTS OF PROJECTIVE SPACES 5 This shows that we can reduce all terms in the tensor product to the 2-adic b* *asis in our first theorem. It does not prove they form a basis though, so beware. The tensor product could be smaller than this until we prove otherwise. We have pro* *ven that this is the largest the tensor product could possibly be though. The Landweber short exact sequence applies to any X and Y where BP*(X) ! H*(X) is surjective, or, in other words, the Atiyah-Hirzebruch spectral sequence collapses. In such a case there is a free BP* resolution: 0 -! A1 -! A0 -! BP*(X) -! 0. To see much more of this type of thing go to [CS69 ]. The Landweber short exact sequence now comes from this resolution by tensoring with BP*(Y ). The tensor product is just the cokernel of A1 BP* BP*(Y ) -! A0 BP* BP*(Y ) and the Tor term is the kernel. For finite complexes we can switch to cohomology and, in the case of RP 2m write down the resolution explicitly. We let A0 be free on generators di, 0 < i* * m of degree 2i. The map A0 ! BP *(RP 2m) is given by di ! xi. A1 is freePon ci, 0 < i m of degree 2i and the map @ : A1 ! A0 is given by @(ci) = j 0ajdi+j. Our Tor of interest is the kernel of: A1 BP* BP *(RP 2n) -! A0 BP* BP*(RP 2n). We start by finding an injection 2nBP *(RP 2m) -! A1 BP* BP *(RP 2n). Let X 2nxj -! ci+j xn-i. i 0 First we have to show this is well defined by showing that the relations go to * *zero: X X X 2n( ajxk+j) -! aj ci+j+k xn-i. j 0 j 0 i 0 Fix i + j = b and look at the coefficient of cb+k. We have X Xb ajxn-i = ajxn-b+j = 0. i+j=b j=0 To see that this map is an injection all we have to do is map to the quotient* * of A1 BP *(RP 2n) obtained by setting all ci= 0 except for cm . This gives us a m* *ap 2nBP *(RP 2m) -! 2mBP *(RP 2n) that takes 2nxj to 2mxn-m+j . This injects on the 2-adic basis. Our next step is to show our image is in the kernel. We have: X X X 2nxj -! ci+j xn-i -! akdi+j+k xn-i. i 0 i 0k 0 Again, fix i + k = b and find the coefficient of db+j: X Xb akxn-i = akxn-b+k = 0. i+k=b k=0 6 JES'US GONZ'ALEZ AND W. STEPHEN WILSON So far we have shown that the tensor product can be no bigger than Theorem 1.1 states and that the Tor term in Theorem 1.2 can be no smaller than what we have already found is in the kernel. Each of the Ai BP *(RP 2n) is a finite abelian 2-group. Furthermore, the i =* * 0 and 1 groups are isomorphic. Thus the kernel and the cokernel must be exactly t* *he same size in each degree. Thus if the elements we have found so far in the kern* *el are exactly the same size as our proposed tensor product then we are done becau* *se our tensor product cannot be smaller than what we already know is in the kernel. This is now just a simple counting argument. The 2-adic bases for both what we have already in the kernel and what we propose for the tensor product are free on Z(2)[v2, v3, . .]., so we can ignore* * all of the vi, i > 1 in our counting argument. We just give a 1-1 correspondence for w* *hat is left. For 0 < j m map 2nvi1xj to xj xn-i for 0 i < n and to vi-n+11xj * * x for i n. We must take care of the naturality since that is one of the motivating facto* *rs for this paper. If k < m and we have RP 2k! RP 2m, we get the obvious surjection of resolutions Ami! Akiand the map of 2nxj is preserved except that it is zero when j > k. This shows the first part of the naturality. If m < n and we map RP 2n-2to RP 2n, the map of 2nxj to A1 BP *(RP 2n) to A1 BP *(RP 2n-2) goes m-jX m-jX 2nxj -! ci+j xn-i -! ci+j xn-i (xn = 0 here). i=0 i=1 If we go m-j-1X 2nxj -! 2n-2xj+1 -! ci+j+1 xn-1-i i=0 we see we have the same thing and this shows the second part of the naturality * *on Tor. It is elementary that BP *(RP 2n+1) ' BP *(RP 2n) BP *(S2n+1) so the tensor product and Tor can be computed from this fact. The only thing left to do is show that there can be no extension problems, i.e. that Landweber's short exact sequence splits. This problem is solved in BP homology using the result from [Dav78 ] that says BP ^ RPb2a+1' BP ^ RPb2a_ 2a+1BP and BP ^ RP2ab' BP ^ RP2ab+1_ 2bBP. Since this splits topologically there can be no algebra extensions. By S-dualit* *y the same is true for cohomology. It should be noted that the above splittings are proved in [Dav78 ] for spaces with only one integral cell, but having one at each end presents no serious pro* *b- lem: Davis' topological argument relies solely on knowing the surjectivity in B* *P - homology of the pinch map RP22a+1b! S2a+1. But this is assured by the corre- sponding situation for RP22a+1b-1! S2a+1. PRODUCTS OF PROJECTIVE SPACES 7 3. Proof of Theorem 1.5 To describe BP *(L(e)2k) we need to take the formal group sum of xe 2e times to get: X [2e](xe) = ai,exi+1e. i 0 We need some facts about these elements: Lemma 3.1 ([Gon01 ]). as,eis divisible by 2~(s), where X ~(s) = bi(e - i) 0 i e, the first non-trivial differential in the spectral * *se- quence under consideration is de(ci fij) = ve1di fij-e. Apologies for the dual use of di, an historical accident. Corollary 3.4. For any n > 0 the Ee+1 term of our spectral sequence is described as follows: (i)In homological degree 1, it is a free BP*=(2)-module on generators ci f* *ij satisfying 0 < i m and 0 < j min{n, e}. (ii)In homological degree 0, it is free over BP*=(2) on di fij with n - e < j n, and over BP*=(2, ve1) on di fij with 0 < j n - e. Proposition 3.5. For n m + e - 1 and m > 1 the spectral sequence collapses after the de-differential in Proposition 3.3. In particular, Corollary 3.4 desc* *ribes a filtered version of tensor, Tor, and BP*(L(e)2m ^ RP 2n). Remark 3.6. The same description and proof works when m = 1 provided n > e. On the other hand, when n e, multiplication by 2e is trivial on the BP -(co)homo* *logy of RP 2n, so that the considerations above Proposition 3.3 show that the first * *non- trivial differential dt(if any) will hold for t > e. As an extreme case of this* * situation we note that the whole spectral sequence collapses for m = 1 and n e. In our proof we will need to use the Smith homomorphism ~: Aei! Aeideter- mined by ~(zi) = zi-1. Since this works on the quotient BP*(L(e)2m) and also for e = 1, we have ~r,s= ~r ~s:Aei BP*P 2n! Aei BP*P 2nis compatible with the filtered chain complex giving our spectral sequence and therefore produces a spectral sequence (graded) endomorphism. Proof.We proceed by contradiction. Assume that one of the generators ci fij in Corollary 3.4 (i) supports a non-trivial differential (3.7) dm (ci fij) = cdr fis + . ... Choose m mimimal with cdr fis non-zero. Of all the possible (r, s) pairs in t* *his filtration, we choose the one with r + s maximal, i.e. with s maximal. Using the spectral sequence morphism ~r-1,s-1we can pull down (3.7) to a differential dm (ci-r+1 fij-s+1) = cd1 fi1. From [Nak85 , Col85, Gon03] we know that c must be zero in BP*=(2, ve1) because the annihilator ideal of ff1 fi1 cannot be bigger than (2, ve1). We know that the only elements left that could have a differential are the ci* * fij with 0 < j e and we know that the target must be some cdr fis + . .w.ith n - e < s n and c = ve1a. Thus the degree of the target must be at least 2e + 2r - 1 + 2(n - e) + 1 = 2n + 2r. The degree of the source is, at most, 2i - 1 + 2e - 1. There can be no differential if the maximum possible degree of* * a potential source is less than the minimum possible degree of a potential target* *, i.e., i + e - 1 < n + r. Since i - r must be less than or equal to m - 1, this follow* *s from m + e - 2 < n, which was our assumption. PRODUCTS OF PROJECTIVE SPACES 9 The only thing left to do is show that there can be no tensor-Tor extension p* *rob- lems in a general product L(e)ab^RPdcinvolving integral cells, i.e. that Landwe* *ber's short exact sequence splits. As in Section 2, this problem is solved in BP homo* *logy using the same techniques for lens spaces that [Dav78 ] uses for truncated proj* *ective spaces. The BP cohomology situation is handled using the fact that truncated le* *ns spaces have S-duals just like the real projective spaces, [Kob94 , Lemma 2.2]. * * Of course we have plenty of unsolved extension problems anyway. 4.two examples Example 4.1. BP*(RP 2) = BP*=(2) on fi1. The firsteterm of the [2e](xe) series that is non-zero mod 2 is a2e-1,eand it is v21-1 mod 2 (see [Gon01 ]). The firs* *t, and only, differential in our spectral sequence comes from the chain map e-1 ci fi1 -! v21 di-2e+1 fi1 + low where low stands for "lower filtration elements". This means that the only dif- ferential in theen = 1 homology version of the spectral sequence is given by ci fi1 7! v21-1di-2e+1 fi1. The tensor and Tor products in this case (n = 1) * *can now be read off from the resulting E1 term. For instance, when m < 2e, one has @ = 0, so that tensor and Tor products are both isomorphic to Ai BP*(RP 2). However, when m 2e, the Tor product has a BP*=(2) free 2-adic basis given by the elements ci fi1, for 0 < i < 2e, whereas the tensor factor has a graded associated object generated by all die fi1 (0 < i m), free over BP*=(2) when m - 2e+ 1 < i m and over BP*=(2, v21-1) when 0 < i m - 2e+ 1. In any case, since the bottom class ff1 fi1 in the tensor product is the lowest possible fi* *ltration generator, we see that its BP*-annihilator ideal does not depend on whether we consider this class as an element in the actual tensor product or as an element* * in the associated graded E1 term. For instance,ewhen m 2e, this common annihi- lator ideal is generated by 2 and v21-1. As n increases from 1 to m + e - 1, the corresponding ideal increases to (2, ve1) which is the (constant) annihilator i* *deal of ff1 fi1 for all n m + e - 1. Example 4.2. Consider the case with e = 2 and m = n = 3. The first round of differentials (identified in Proposition 3.3) are given by ci fi3 7! v21di * * fi1 for i = 1, 2, 3. However, a straightforward calculation shows that in our chain com* *plex there holds the relation @(c3 fi2 + c2 fi3) = v21d1 fi2. This means that* * the spectral sequence has the extra d4-differential c3 fi2 7! v21d1 fi2. One e* *asily verifies that all other elements left in homological degree 1 are in fact perma* *nent cycles, so that the spectral sequence collapses from its fifth stage. References [Ara73]S. Araki. Typical formal groups in complex cobordism and K-theory. Lectu* *res in Math- ematics, Department of Mathematics, Kyoto University. Number 6. Kinokuniy* *a Book- Store Co., Ltd, Tokyo, 1973. [Ati61]M.F. Atiyah. Thom complexes. Proceedings of the London Mathematical Soci* *ety (3), 11:291-310, 1961. [CF64]P. E. Conner and E. E. Floyd. Differentiable periodic maps. Academic Pres* *s Inc., Pub- lishers, New York, 1964. Ergebnisse der Mathematik und ihrer Grenzgebiete* *, N. F., Band 33. 10 JES'US GONZ'ALEZ AND W. STEPHEN WILSON [Col85]R.A. Coley. Projective dimension of BP*BG for finite groups. PhD thesis,* * Massachusetts Institute of Technology, 1985. [CS69]P.E. Conner and L. Smith. On the complex bordism of finite complexes. Ins* *t. Hautes 'Etudes Sci. Publ. Math., (37):117-221, 1969. [Dav78]D.M. Davis. The BP-coaction for projective spaces. Canadian Journal of M* *athematics, 30:45-53, 1978. [Gon01]J. Gonz'alez. 2-divisibility in the Brown-Peterson [2k]-series. Journal * *of Pure and Applied Algebra, 157(1):57-68, 2001. [Gon03]J. Gonz'alez. A generalized Conner-Floyd conjecture and the immersion pr* *oblem for low 2-torsion lens spaces. Topology, 42(4):907-927, 2003. [JW73]D. C. Johnson and W. S. Wilson. Projective dimension and Brown-Peterson h* *omology. Topology, 12:327-353, 1973. [JW85]D. C. Johnson and W. S. Wilson. The Brown-Peterson homology of elementary* * p-groups. American Journal of Mathematics, 107:427-454, 1985. [Kob94]T. Kobayashi. Stable homotopy types of stunted lens spaces mod pr. Memoi* *rs of the Faculty of Science, Kochi University, Ser. A Math, 15:9-14, 1994. [KWa] N. Kitchloo and W.S. Wilson. The second real Johnson-Wilson theory and no* *n- immersions of RPn. Manuscript. [KWb] N. Kitchloo and W.S. Wilson. The second real Johnson-Wilson theory and no* *n- immersions of RPn, Part 2. In preparation. [Lan66]P.S. Landweber. K"unneth formulas for bordism theories. Transactions of * *the American Mathematical Society, 121:242-256, 1966. [Lan70]P. S. Landweber. Coherence, flatness and cobordism of classifying spaces* *. In Proceedings of Advanced Study Institute on Algebraic Topology, pages 256-269, Aarhus,* * 1970. [Nak85]G. Nakos. On the Brown-Peterson homology of certain classifying spaces. * *PhD thesis, Johns Hopkins University, 1985. Departamento de Matem'aticas, Centro de Investigaci'on y de Estudios Avanzado* *s del IPN, M'exico City 07000 E-mail address: jesus@math.cinvestav.mx Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218 E-mail address: wsw@math.jhu.edu