The E (1; 2) Cohomology of the Eilenberg-Mac Lane Space K (Z; 3)

Transcription

1 The E (; 2) Cohomology of the Eilenberg-Mac Lane Space K (Z; 3) by Hsin-hao Su A dissertation submitted to the Johns Hopkins University in conformity with the requirements for the degree of Doctor of Philosophy. Baltimore, Maryland May, 2006.

2 Abstract In this paper, we give new bases of E order to de ne a basis of E new bases, we have a short exact sequence of Hopf algebras and E BP hi 6 in and E BP hi 6. With these two E E (K (Z; 3)) f E y (i) : i 0 v E x (i) : i 0 E. The action of v is calculated by using formal group laws and used to compute E (K (Z; 3)) as a completion of E z (0). Readers: Dr. J. Michael Boardman (Advisor) and Dr. Jack Morava ii

3 Acknowledgement I would like to thank my advisor, Dr. W. Stephen Wilson, for all of his help over the past few years. His guidance and patience have been exceptional. None of this project would have been possible without his vision, guidance and patience during my study. I would also like to thank my other advisor, Dr. J. Michael Boardman, for his helpful advice and reading through my thesis. This project could never have been nished without his assistance. I would also like to thank the faculty at Johns Hopkins University and National Tsing Hua University. I owe special thanks to Dr. Dung Yung Yan and Dr. Jer-shyong Lin for their constant encouragement and support. Additionally, I would like to thank my o ce sta and fellow graduate students in the Department of Mathematics at Johns Hopkins University. They have my appreciation for their friendship and support. Our daily discussion and chats together contributed immeasurably to my successful completion of this degree. Finally, I would especially like to thank my family for their love and support. I am very grateful to my family-in-law for their love and prayer. I dedicate this dissertation to my wife, Hsin-I Alice Fu, without whose unwavering love, encouragement and support it would never have been completed. iii

4 Contents Introduction 2 Preliminary 3 2. Formal Group Laws Wilson s Splitting Theorem The Formal Group Law for E (; 2) Appendix: Low Degree Examples BP Homology and Cohomology of and BP hi Generators of Cohomology The Module QE (K (Z; 3)) Bases of Homology and Cohomology E (; 2) (K (Z; 3)) The Bockstein Spectral Sequence Topology of Cohomology in E (; 2) (K (Z; 3)) Relations in E (; 2) (K (Z; 3)) iv

5 Introduction The Eilenberg-Mac Lane Space K (Z; 2) is readily recognized as complex projective space CP. Given any multiplicative complex-oriented generalized cohomology theory E ( ), a classical result of Dold states that E (CP ) = E [[x]], the graded ring of formal power series over the coe cient ring E of E, where x denotes the Chern class of the Hopf line bundle over CP. This paper studies the E-cohomology of the next Eilenberg-Mac Lane space K (Z; 3), which may be viewed as the classifying space of K (Z; 2), for certain complexoriented E. Note that, for any space X, a cohomology class in E k (X) corresponds to a map X! E k, de ned up to homotopy, where E k comes from the Brown Representation Theorem, which can be stated as: for any cohomology theory E, for all k, there exists a classifying space E k such that E k (X) = [X; E k ], the set of homotopy classes of unbased maps X! E k. For a prime p, the Johnson-Wilson spectrum BP hqi is a BP -module spectrum with (BP hqi) = Z (p) [v ; ; v q ]. For q > 0, there is a stable co bre sequence 2(pq ) BP hqi vq! BP hqi! BP hq i! 2pq BP hqi. Unstably, the boundary map is and the iteration is BP hk i j! BP hki j+2p k K Zp i ; q +! K Z (p) ; q + 2! BP hi q+2p+!! BP hqi g(q) where g (q) = 2 (p q + p q + + p + ) and BP h0i q+2 = K Z (p) ; q + 2. In [RWY], there is an exact sequence in the category of K (n) -Hopf algebras: K (n)! K (n) K Z (p) ; q + 2! K (n) BP hqi g(q) vq! K (n) BP hqi g(q) jvqj.

6 This induces, by Theorem.9 in [RWY], BP K Z (p) ; q + 2 BP BP hqi g(q) =, vq where vq denotes the ideal generated by the image under v q of the augmentation ideal in BP BP hqi g(q) jvqj. In 999, Tamanoi [T], who was motivated by the above result, created new generators and wrote down a very simple relation in terms of his new generators. Let hqi g(q)+t be the inclusion map hqi g(q)+t : BP hqi g(q)+t! BP g(q)+t for t 0. Recall that by Wilson s Splitting Theorem [Wil75], BP hqi g(q)+t is a factor space of BP g(q)+t. Thus, hqi g(q)+t exists. This map de nes a cohomology class hqi g(q)+t 2 BP g(q)+t BP hqi g(q)+t, namely the BP fundamental class for BP g(q)+t BP hqi g(q)+t. Then, Tamanoi de- ned the v q -series [v q ] = v q which comes from the composition map Also, he showed that where the class hqi q+j hqi g(q) 2 BP BP g(q) jvqj hqi g(q), BP hqi g(q) v q! BP hqig(q) jvqj [v q ] = v q hqi q hqi g(q) jvqj! BP g(q) jvqj. + v q+ hqi q+ + + v q+j hqi q+j +, with j is de ned as the following composition: BP hqi g(q)! BP g(q) v q! BP g(q) jvqj! BP g(q)+2p q+j 2p q and the projection is provided by Wilson s theorem. projhq+ji! BP hq + ji g(q)+2p q+j 2p q 2

7 We take p = 2. By the short exact sequence of Hopf algebras BP BP (K (Z; 3)) f BP BP hi 6 v BP BP, we have BP (K (Z; 3)) BP BP hi 6 =, (v) where (v) denotes the image under v of the augmentation ideal of BP. The relation Tamanoi found in this case is v x (0) = v y (0) + v 2 y () + + v i y (i ) +. By tensoring with E (; 2), we have E (; 2) (K (Z; 3)) = E (; 2) BP hi 6 (v ) where v x (0) = v y (0) + v 2 y (). We use the formal group law to create a new set of generators y(i) 0, for which the relations in E (; 2) BP hi 6 take the simpli ed form v x (i) = v 2 i y 0 (i) + v 2i 2 y 0 (i+) + D (i) for i 0, where D (i) is decomposable. This paper is organized as follows: In section 2, we collect some basic notation and theorems for the cohomology theories BP and E (k; n). We calculate some formal group laws in section 3. In section 4, a basis of BP and BP BP hi 6 will be given so that we can write down a dual basis for BP and BP BP hi 6. These are used to prove the main results. 2 Preliminary The coe cient ring for Brown-Peterson cohomology is BP = Z(p) [v ; v 2 ; ] where the degree of v n is 2 (p n ). In homology, we often write the coe cient ring as BP, with BP n = BP n (see [RW77]). We have the Hopf algebroid (BP ; BP BP ) which satis es: 3

8 . BP = Z(p) [v ; v 2 ; ], where deg v n = 2 (p n ). 2. BP BP = BP [t ; t 2 ; ], where deg t n = 2 (p n ). 3. R : BP! BP BP is a ring homomorphism, where R v is written as [v]. Let I n be the ideal (p; v ; v 2 ; ; v n ). And, v 0 = p. o Let BP hni = Z (p) [v ; v 2 ; ; v n ] and nbp hni be the -spectrum of BP hni. Let P (n) be the theory with coe cient ring P (n) = BP I n. And, P (0) = BP or BP ^p where BP ^p is the p-adic completion of BP. We have stable co brations 2(pn ) P (n) vn! P (n)! P (n + ) which lead to long exact sequences in cohomology. There are spectra E (k; n) with coe cient rings with similar stable co brations E (k; n) = v n BP hni I k 2(pk ) E (k; n) v k! E (k; n)! E (k + ; n) and long exact sequences. In particular, for k n, E (k; n) = v n BP hni I k = F p vk ; v k+ ; ; v n ; vn. A special case, when k = n > 0, is the n th Morava K-theory, K (n) (X), with K (n) = Z=p vn ; vn. Also, when k = 0, E (n) = E (0; n) = v n BP hni = Z (p) [v ; v 2 ; ; v n ; v n ]. Let E denote any of BP ^p, P (n) or E (k; n) where 0 k n and 0 < n. 4

9 2. Formal Group Laws We summarize the properties of formal group laws. We are working with the Brown- Peterson spectrum with prime 2. De nition 2. A formal group law is a power series F (y; z) = X a ij y i z j i;j0 such that. F (y; 0) = y and F (0; z) = z, 2. F (y; F (z; w)) = F (F (y; z) ; w), where a ij is in a commutative ring with unit. Moreover, we say a formal group law is commutative if F (y; z) = F (z; y). We write y + F z for F (y; z). Example 2.2 We have two trivial examples of commutative formal group laws: F (y; z) = y + z and F (y; z) = y + z + yz. By the rst condition in the formal group law de nition, we have 8 <, if i = a i0 = : 0, if i 6= and 8 <, if j = a 0j = : 0, if j 6=. Therefore, we may write our formal group law in the following form F (y; z) = y + z + X i;j a ij y i z j Let x 2 BP 2 (CP ) be the Conner-Floyd Chern class. 5

10 Proposition 2.3 (In [RW77] Lemma 3.3 on Page 255) We have. BP (CP ) = BP [[x]], the ring of formal power series on x = x CP over BP. 2. BP (CP CP ) = BP (CP ) b BP BP (CP ). 3. BP (CP ) is BP -free on i 2 BP 2i (CP ) for i 0, dual to x CP i, i.e. D x CP i ; j E = ij. 4. BP (CP CP ) ' BP (CP ) BP BP (CP ). 5. The diagonal CP! CP CP induces a coproduct on BP (CP ) with P ( n ) = n i n i. i=0 6. The H-space product m CP : CP CP! CP induces a coproduct m CP P on BP (CP ) with m CP xcp = a ij x CP i x CP j where aij 2 BP 2(i+j ) = BP 2(i+j ). 7. F (y; z) = y + FBP z = y + F z = P group law over BP. i;j0 i;j0 Let (s) = P i0 i s i where s is a formal indeterminate. a ij y i z j is a commutative associative formal Proposition 2.4 (In [RW77] Theorem 3.4 on Page 255) In the power series ring BP (CP ) [[s; t]], we have (s) (t) = (s + F t) where the product is that induced by the H-space structure of CP. Moreover, for a natural number n, (s) n = ([n] F (s)), where [n] F (s) = s + F s + F + + F s for n > 0. Consider x 2 BP 2 (CP ) = [CP ; BP 2 ] as a map CP x! BP 2. This map induces a map x : BP (CP )! BP (BP 2 ). 6

11 De nition 2.5 We de ne b i = x ( i ) 2 BP 2i (BP 2 ). As with (s), we have x ( (s)) = x! X i s i = X x ( i ) s i = X b i s i. i0 i0 i0 Therefore, we de ne b (s) = P i0 b i s i 2 BP (BP 2 ) [[s]], that is, b (s) = x ( (s)). The H-space product m : BP 2 BP 2! BP 2 induces a coproduct m on BP (BP ) with m (x) = P a ij x i x j where a ij 2 BP 2(i+j ) = BP 2(i+j ). i;j0 According to [RW80], BP (BP ) is a Hopf ring, where the maps of coalgebras : BP (BP k ) BP (BP k ) = BP (BP k BP k )! BP (BP k ) are induced by the maps BP k BP k! BP k that represent addition in cohomology and the maps of coalgebras : BP (BP k ) BP (BP m ) = BP (BP k BP m )! BP BP k+m are induced by the maps BP k BP m! BP k+m that represent multiplication in cohomology. De nition 2.6 In BP (BP ) [[s; t]], we de ne b (s) + [F ] b (t) = b (s) + [F ]BP b (t) = i;j0 [a ij] b (s) i b (t) j. Proposition 2.7 (In [RW77] Theorem 3.8 on Page 256) In BP (BP ) [[s; t]], we have relations:. b (s + F t) = b (s) + [F ] b (t), 2. b ([2] F (s)) = [2] [F ] (b (s)). We can rewrite the relation b (s + F t) = b (s) + [F ] b (t) as b X i;j0 a ij s i t j! = i;j0 [a ij] b (s) i b (t) j, 7

12 that is, X n0 b n! n X a ij s i t j = [a ij] i;j0 i;j0! i X b k s k k0! j X b l s l. l0 Proposition 2.8 (In [RW77] Theorem 3.(b) on Page 257) In BP for the prime 2, we have [2] F (t) = X F BP v nt 2n mod (2). n>0 In BP with prime 2, we have, see [RW77], [2] F (t) = 2t v t 2 +2vt 2 3 7v 2 + 8v 3 t v 2 v + 26v 4 t 5 v 2 v v 5 t 6 +. Remark 2.9 From Lemma.2 in [RW77],. a 0 = a 0 = and a i0 = a 0i = 0 if i 6=, 2. b 0 = [0 2 ], 3. b 0 b i = 0 for all i > 0, 4. b 0 [a] = [0 2 2i ] where a 2 BP 2i, 5. 2b = [2] b, 6. b 2 = 2b 2 v b + [v ] b 2 in BP 2 (BP 2 ) from page Wilson s Splitting Theorem We study the commutative diagram of H-spaces and canonical H-maps in which the bottom row is a bration. BP 6 v! BP 4 # # () K Z (2) ; 3 f v! BP hi6! By Wilson [Wil73], splits o BP 4 as an H-space. BP hi 6 splits o BP 6, but not as an H-space. Both maps v represent multiplication by v in cohomology. 8

13 (The bre of an H-map is automatically an H-space, and by uniqueness, the resulting H-space structure on K Z (2) ; 3 must be the standard one, up to homotopy.) We apply homology theories E ( ) and cohomology theories E ( ) to this diagram, for various E = BP or E (; 2). Note that since all the E we consider are 2-local, we may replace E K Z (2) ; 3 by E (K (Z; 3)). BP (BP k ) was computed by Ravenel-Wilson [RW77] and is a free BP -module. Since each E we use comes with a multiplicative map BP! E that induces a surjection BP! E, comparison of the Atiyah-Hirzebruch spectral sequences shows that E (BP k ) and E BP hi k are free E -modules for k = 4; 6. More precisely, E BP BP (BP k ) = E (BP k ). In view of the above splittings, the same holds for BP hi k. By dualizing, we have in cohomology. E b BP BP (BP k ) = E (BP k ) Ravenel-Wilson-Yagita [RWY] established the short exact sequence of bicommutative Hopf algebras E! E v! E BP hi 6 f! E (K (Z; 3)) for various E. We wish to use this to compute E (; 2) (K (Z; 3)).! E 2.3 The Formal Group Law for E (; 2) Consider the spectrum E (; 2) associated with the prime 2 where the coe cient ring is E (; 2) = Z=2 v ; v 2 ; v2. We study its formal group law F (y; z). For convenience, we de ne A = v and A 2 = v 2. Also, for k 3, we de ne A k = Lemma 2.0 v t 2 + F v 2 t 4 = X k A k t 2k. X m;n 2m+2 2 n=2k a mn v m v n 2. 9

14 Proof. By the formal group law, we have v t 2 + F v 2 t 4 = v t 2 + v 2 t 22 + X a mn v m v2 n t 2m+22n. m;n Take the coe cient of t 2k. A lower degree calculation (2.) is in the Appendix. We note that (a + b) 2 a 2 + b 2 mod 2. Thus, we have v t 2 + F v 2 t 4 2 q = X k A 2q k t 2q+k. In general, we de ne C q;k as the coe cient of t 2k in the equation v t 2 + F v 2 t 4 X q = C q;k t 2k. k It s easy to see that and C q;k = 0, if k < q C 2 q ;k = A 2q 2 q k. for all q and k 2 q, interpreted as 0 if k is not divisible by 2 q. 2.4 Appendix: Low Degree Examples We collect some precise low degree calculations here for reference. Example 2. The rst few terms of v t 2 + F v 2 t 4 = v t 2 + v 2 t 4 + a v v 2 t 6 + a 2 vv 2 2 t 8 + a 3 vv a 2 v v2 2 t 0 + a 4 vv a 22 vv2 2 2 t 2 + a 5 vv a 32 vv a 3 v v2 3 t 4 + a 6 vv a 42 vv a 23 vv2 2 3 t 8 + = A t 2 + A 2 t 4 + A 3 t 6 + A 4 t 8 + A 5 t 0 + A 6 t 2 + A 7 t 4 + A 8 t

15 Example 2.2 The rst few terms of v t 2 + F v 2 t 4 2! 2 X = A k t 2k k = A t 2 + A 2 t 4 + A 3 t 6 + A 4 t 8 + A t 2 + A 2 t 4 + A 3 t 6 + A 4 t 8 + = A A t 4 + (A A 2 + A 2 A ) t 6 + (A A 3 + A 2 A 2 + A 3 A ) t 8 + (A A 4 + A 2 A 3 + A 3 A 2 + A 4 A ) t 0 + = A 2 t 4 + 2A A 2 t 6 + A A A 3 t (A A 4 + A 2 A 3 ) t 0 +. or v t 2 + F v 2 t 4 2 = vt v v 2 t 6 + v a vv 2 2 t a 2 vv a v v2 2 t 0 + a 2 vv a 3 vv a 2 vv a 2 vv t 2 +2 a 2 v v2 3 + a 4 vv a 22 vv a 3 vv a a 2 vv2 3 2 t 4 + a 2 2vv t 6 +2 a 5 vv a 3 vv a 22 vv a 32 vv a 4 vv a a 2 vv a a 3 vv t 6 +2 a 32 vv a 5 vv a 3 v v2 4 + a 6 vv a 42 vv a 23 vv t 8 +2 a a 22 vv a 2 a 2 vv a a 4 vv a 2 a 3 vv2 5 2 t 8 + Example 2.3 For p = 2, the rst few terms of v t 2 + F v 2 t 4 2 2! 22 X = A k t 2k k = A t 2 + A 2 t 4 + A 3 t = A 4 t A 3 A 2 t A 3 A 3 + 3A 2 A 2 2 t 2 + A A 2 A A A 2 2A 3 + 2A 2 A 2 A 4 + 4A 3 A 5 t 6 +.

16 3 BP Homology and Cohomology of and BP hi 6 We focus on the prime 2. Let E be a 2-local cohomology theory. We borrow the notations and results from [RW77]: De nition 3. [v i ] is de ned in E 0 BP 2(2 i ) using vi 2 2(2 i ) (BP ). De nition 3.2 k = (0; ; 0; ; 0; ) where is in the k-th position. De nition 3.3 b (m) = b 2 m 2 E 2 m+ (BP 2 ). De nition 3.4 In the ring of indecomposables QE (BP ), de ne v I b J = v i v i 2 2 b j 0 (0) bj () where I = (i ; i 2 ; ) and J = (j 0 ; j ; ) are sequences of non-negative integers almost all zero. Note that Proposition 2.7 implies that the other b n are redundant. De nition 3.5 For J = (j 0 ; j ; ), we de ne the length of J as ` (J) = j 0 +j +. De nition 3.6 We call v I b J allowable if J = 2 k k n kn + J 0, where k k 2 k n and J 0 is non-negative, implies i n = 0. Proposition 3.7 The allowable v I b J (J 6= 0) form a basis of the free BP -module QBP BP 0, where BP 0 denotes the base components of BP. (BP 0 k = BP k for k > 0.) Proposition 3.8 The v I b J b (0) with v I b J allowable (J possibly zero) form a basis of the submodule of primitives P BP BP 0. Remark 3.9 A basis of QBP 2m (BP 2n ) ; n > 0 is given by all allowable v I b J with m = X j k 2 k and n = X X j k i k 2 k. k0 k0 k>0 2

17 We need to compute v : E! E BP hi 6. These are power series rings. We compute on the indecomposables, Qv : QE which is dual to P v : P E BP hi 6 multiplication by [v ]. The double suspension factors as b (0) : E! QE! QE BP hi 6,! P E. The latter is simply =! P E BP hi 6,! E BP hi 6 and similarly for P E. We therefore compute the homomorphism of free E -modules v : QE! QE BP hi 2. We temporarily omit here, as it is the only multiplication in QE BP hi. The Ravenel-Wilson basis is not the most useful here. Theorem 3.0 The elements b i for i > 0 form a basis of QE BP hi 2 E -module. as a free To show that a set of elements is a basis, it is su cient to work mod I = (2; v ; v 2 ; ) and check that we have a basis over F p, according to the Nakayama Lemma. We introduce the Bendersky generators h i of QE (BP ) (see [B82]). They may be de ned by b (x) = h 0 x + [F ] h x 2 + [F ] h 2 x 4 + [F ] (2) where [F ] denotes the formal group law using the right BP -action by the elements [v i ]. If we replace each b (i) by h i in the Ravenel-Wilson basis of QE (BP ), we obtain another basis (since h i b (i) mod I, see below.) When we map to BP hi 2, the resulting basis of E BP hi 2 consists of the elements v k hi0 h i h i2 h ik where k 0 and i 0 < i < i 2 < < i k. The theorem follows from the following lemma. 3

18 Lemma 3. Given n > 0, write n in binary form as n = 2 i i i k, with i 0 < i < < i k. Then in QE (BP 2 ), b n v k hi0 h i h i2 h ik mod (I + ([v 2 ] ; [v 3 ] ; )) : Proof. The main relation (see Proposition 2.7) reduces to 0 b ([2] (x)) [v ] b (x) 2 X i [v ] b 2 i x 2i. The coe cient of x 2i+ yields [v ] b 2 (i) 0. We deduce by induction on i that [v ] h 2 i 0, starting from b (0) = h 0. When we expand (2) and take the coe cients of x 2i, we get b (i) = h i +terms with a factor [v ] h 2 j for some j < i. Hence b (i) h i and [v ] h 2 i 0. If [v ] x 2 0 and [v ] y 2 0, the right formal group law takes the form x + [F ] y x + y + [v ] xy. Then the coe cient of x n in (2) yields b n on the left, and the only surviving term on the right is v k hi0 h i h i2 h ik. Lemma 3.2 In QE BP hi 2, each b n can be expressed as a polynomial in the elements b (i) and [v ], with no v or v 2. Proof. The coe cient of x n in (2) expresses b n as a polynomial in the h i and [v ]. If n = 2 i, we can use b (i) = h i + to express h i as a polynomial in b (i), the h j for j < i, and [v ], and hence inductively as a polynomial in the b (j) and [v ]. Remark 3.3 If n is not a power of 2, this polynomial is divisible by [v ]. Next, we consider QE. Lemma 3.4 If n is not a power of 2, there is an element ^b n 2 QE such that [v ] ^b n = b n in QE BP hi 2. Proof. When we expand (2) and take the coe cient of x n to nd b n, every term on the right (working mod [v 2 ] ; [v 3 ] ; ) contains a factor [v ]. In particular, by Lemma 3., ^bn v k hi0 h i h i2 h ik mod I. 4

19 The only missing elements from the modi ed Ravenel-Wilson basis of QE are the elements h 2 i b 2 (i) for i 0. We therefore take as a basis of QE, the elements ^b n for n not a power of 2 and the elements b 2 (i). We now compute v. The main relation, mod 2, is X b q ([2] (x)) q = b ([2] (x)) [v ] b (x) 2 X q i [v ] b 2 i x 2i. Recall that ([2] (x)) q = X k C q;k x 2k. Take coe cients of x 2i+. Then 8 < : v b 2 (i) = [v ] b 2 (i) = P q2 i C q;2 ib q, for i 0; v ^bn = [v ] ^b n = b n, for n not a power of 2 (3) gives v precisely in terms of our bases. We summarize the relationship between E, QE and P E in a diagram: v E BP hi! E 4 BP hi 2 # # Qv QE! QE BP hi 2 =# b(0) =# P v P E BP hi 6! P E \ \ E BP hi 6 v! E hx (i) ; i! E # # Qv QE BP hi 6! QE (4) 3. Generators of Cohomology We return to the primitives. Our basis of P E = P M 4 P R 4 consists of two parts:. Elements b (0) b n for n not a power of 2, which span the submodule P M Elements b (0) b (i) for i 0, which span the submodule P R 4. Our basis of P E BP hi 6 = P M 6 P R 6 consists of two parts: 5

20 . Elements b (0) ^b n for n not a power of 2, which span the submodule P M Elements b (0) b 2 (i) for i 0, which span the submodule P R 6. In terms of these bases, v is given by 8 >< v b (0) ^b n = [v ] b (0) ^b n = b (0) b n, for n not a power of 2 >: v b (0) b 2 (i) = [v ] b (0) b 2 (i) = X C q;2 ib (0) b q. q2 i (5) Note that v b (0) b 2 (i) contains terms in P M 4. Now, we can consider the duals QE and QE BP hi 6, the indecomposables of the algebras E and E BP hi 6. Proposition 3.5 (In [RWY] Page 6) Since, for k 2 (2 q + 2 q ), the spaces BP hqi k are all torsion-free for ordinary homology, we know that they are BP - free and the Brown-Peterson cohomology is just the BP -dual. Likewise, the induced cohomology homomorphisms are just the dual homomorphisms. Proposition 3.6 (In [RWY] Section 8.5 at Page 39) Both BP BP hi 6 and BP are power series rings on generators dual to the primitives in the BP -homology. Dualizing to cohomology, we write E = AbM, where A and M are power series rings generated by the duals of P R 4 and P M 4, respectively, and E BP hi 6 = B bn, where B and N are power series rings generated by the duals of P R 6 and P M 6, respectively. Since v : P M 6 = P M4 is an isomorphism, so is Qv : QM! QN (considered as quotient modules). So coker [Qv : QA QM! QB QN] = coker [Qv : QA! QB]. We therefore limit attention to A and B. De ne generators x (i) 2 A dual to b (0) b (i) and y (i) 2 B dual to b (0) b 2 (i), so that A = E x (i) : i 0 6

21 and B = E y (i) : i 0. It is to be understood that all such rings allow any in nite linear combination of monomials with coe cients in E of appropriate degrees. Note that the degree of x (i) is 2 ( + 2 i ) and the degree of y (i) is 2 ( + 2 i+ ). Then dualizing (5), in Qv : QE! QE BP hi 6, we have Recall that C 2 i ;2 j v X x (i) = C 2 i ;2 jy (j). ji = A2i 2 j i = A 2i (j i), where we write A (i) = A 2 i, and that A (0) = A = v and A () = A 2 = v 2. Then, for i 0, Qv : QA! QB is given by v x (i) = v 2 i y (i) + v2 2i y (i+) + X A 2i (k)y (i+k). (6) k2 Also, v (u n ) = w n + X j C n;2 jy (j), where u n is dual to b (0) b n and w n is dual to b (0) ^b n. 3.2 The Module QE (K (Z; 3)) We consider the indecomposables of the algebra E (K (Z; 3)), where E = E (; 2). Put z (i) = f y (i) 2 Q = QE (K (Z; 3)); these elements generate the E -module topologically, with respect to the z- ltration in which F n = F n Q consists of all elements P d k z (k), with d k 2 E. From (6), we have the relations kn v 2i z (i) + v2 2i z (i+) + X A 2i (k)z (i+k) = 0 k2 for i 0. Note that for k 2, A (k) is divisible by v v 2 ; write A (k) = v v 2 A 0 (k). We also have the v - ltration, de ned by the submodules v k Q. Since v 2 is invertible, we may rewrite the above relation as z (i+) = v 2i v 2i 2 z (i) + X k2! A 0 2 i (k) z(i+k). 7

22 Lemma 3.7 Qv 2n Q is a free E v 2n -module generated by z(0). Proof. It is easy to see that z (0) generates Qv 2n Q. We have Q = E z (0) + v Q. Multiply by v i ; then E z (0) + v i Q = E z (0) + v i E z (0) + v i+ Q = E z (0) + v i+ Q. By induction, Q = E z (0) + v Q = E z (0) + v 2n Q. To see that there are no relations, let F be the free R-module on generators y (0), y (),, y (n), considered as a quotient module of QE BP hi 6 via q : QE BP hi 6! F, where for convenience we write R = E v 2n. We note that Qv 2 n Q is a quotient of F. Suppose cz (0) = 0 in Qv 2n Q, where c 2 R. Since qv x (i) = 0 for i n, this lifts to a relation Xn Xn cy (0) = c i qv x (i) = i=0 i=0 c i v 2i y (i) + v 2i 2 y (i+) + Xn i k=2! v 2i v2 2i A 0 2 i (k) y(i+k) in F, for some coe cients c i 2 R. The coe cient of y (0) gives c = v c 0. The coe cient of y () gives 0 = v 2 c + v 2 c 0, or c 0 = v 2 v 2 c. We show by induction on i that c i = v 2i v 2i 2 ( + a i ) c i for 0 < i n, for some a i 2 (v ) R. Assume this holds for i < m. The coe cient of y (m), where 0 < m n relation gives 0 = v 2m c m + v 2m 2 c m + mx = v 2m c m + v 2m 2 ( + v A) c m v 2m k v 2m k 2 A 0 2 m k (k) cm k k=2, in the above where by the induction hypothesis we express each c m k in terms of c m and A 2 R. Now + v A is invertible in R by ( + v A) = + a m, where a m = 2n P va i i, so that c m = v 2m v 2m 2 ( + a m ) c m. The same argument applies when m = n, except that there is no c n, to yield c n = 0. Then c i = 0 for all i and c = v c 0 = 0. Theorem 3.8 The z- ltration and the v - ltration de ne the same topology on Q = QE (K (Z; 3)). Q is the free F 2 v2 ; v2 [[v ]]-module generated by z (0). i= 8

23 Proof. We have already seen that z (i+) 2 v 2i Q for all i 0 so that F n v 2n Q. Conversely, v 2n Q F n. This follows from v 2n 2 i z (i) 2 F n for 0 i n, by downward induction on i, starting from i = n. For the induction step, we multiply the above relation by v 2n 2 i+. We know the z- ltration is complete, being a quotient of the complete ltered E -module QE BP hi 6. By the above, the v - ltration is also complete, and Q = lim n Qv 2n Q = lim F 2 v2 ; v2 n [v ] v 2n = F2 v2 ; v2 [[v ]]. We would like to extend this result to the whole algebra E (K (Z; 3)). 3.3 Bases of Homology and Cohomology We need to extend bases of P E BP hi 6 and E and P E to E BP hi 6 in order to de ne x (i) and y (i) as elements of E E BP hi 6. There is a standard way to do this. Given a monomial c 2 E BP hi r in the elements [v ], b (i) and b n, de ne V c by replacing each b (i) by b (i and ) and each b n by b n=2, taking V c = 0 if any i = 0 or any n is odd. (This depends on how c is written as a monomial, but the choice does not matter; the Verschiebung V is only de ned naturally mod I.) Lemma 3.9 For any y 2 E BP hi r and monomial c as above, y 2 ; c = hy; V ci 2. Proof. hy 2 ; ci = h (y y) ; ci = hy y; ci = P hy; c 0 i hy; c 00 i, where c = c 0 c 00. Suppose that c = [v m ] b i b i2 b ik. Then c = X [v m ] b j b j2 b jk [v m ] b i j b i2 j 2 b ik j k, summing over all j ; j 2 ; ; j k, 0 j r i r, and y 2 ; c = X hy; [v m ] b j b j2 b jk i hy; [v m ] b i j b i2 j 2 b ik j k i. 9

24 Since we are working mod 2, all terms cancel in pairs except for the desired middle term hy; V ci hy; V ci. The same proof works for -products of such monomials c. We construct a basis of E BP hi 6 as follows. We have a basis of P E BP hi 6 consisting of the elements b (0) b 2 (i) (for i 0) and b (0) ^b n (for n not a power of 2). We adjoin new basis elements b (k) b 2 (k+i) (for i 0 and k > 0) and b (k) ^b n;k (for n not a power of 2 and k > 0), where V k^bn;k = ^b n. The desired basis of E BP hi 6 consists of all -products of these basis elements (including the empty product ), without repetition of factors. We de ne y (i) all i; k 0, y 2k (i) as dual to b (0) b 2 (i) is dual to b (k) b 2 (k+i). (for i 0). The lemma tells us that, for (The duality for other monomials in the y s is far more obscure.) We also de ne w n as dual to b (0) ^b n (for n not a power of 2). Then we have E BP hi 6 = B bn, where B = E y (i) : i 0 and N = E [[w n : n 6= 2 r ]]. A similar process extends the basis of P E to a basis of E. We de ne x (i) as dual to b (0) b (i) and u n as dual to b (0) b n. Then we have E = AbM, where A = E x (i) : i 0 and M = E [[u n : n 6= 2 r ]]. Proposition 3.20 We have a short exact sequence of (completed) Hopf algebras E E (K (Z; 3)) f E y (i) : i 0 v E x (i) : i 0 E. 4 E (; 2) (K (Z; 3)) In this chapter, we assume that E is E (; 2). We wish to compute v : AbM! B bn. This is more complicated than Qv because when we take decomposables into account, A no longer maps into B. We restate (6), this time including the decomposables. Lemma 4. In E (; 2) BP hi 6, v x (i) = v 2 i y (i) + v2 2i y (i+) + X A 2i (k)y (i+k) + D (i) k2 and v (u n ) = w n + X j C n;2 jy (j) + D n 20

25 where D (i) and D n are decomposable. It is easy to simplify the second equation. Because we are working in formal power series algebras, to make a valid change of generators, it is only necessary to check that it is invertible on the indecomposables. Here we simply replace w n by wn 0 = v (u n ), so that the second equation becomes v (u n ) = wn. 0 Then in the quotient algebra E (; 2) (K (Z; 3)), any occurrence of wn 0 in D (i) is replaced by zero. First, we easily recover a well-known result. Proposition 4.2 K (2) (K (Z; 3)) = K (2) y (0). Proof. In this case, (6) simpli es to v x (i) = v 2 i 2 y (i+). But, v 2 is invertible. 4. The Bockstein Spectral Sequence We have stable co brations E (; 2) v! E (; 2)! K (2). The induced long exact sequence is! E (; 2) (K (Z; 3)) v! E (; 2) (K (Z; 3))! K (2) (K (Z; E (; 2) (K (Z; 3))!. Proposition 4.3 (In [RWY] Lemma 5. at Page 74) Let X be a space. Let 0 k n and n > 0. If K (n) (X) is even-dimensional, then E (k; n) (X) is evendimensional and has no v k torsion (v 0 = p). According to the above section, we know that (ignoring the invertible element v 2 ) K (2) (K (Z; 3)) = F 2 y(0) is even-dimensional. This implies K (2) (K (Z; 3)) is even-dimensional. Therefore, E (; 2) (K (Z; 3)) is even-dimensional and has no v torsion. Hence, the long exact sequence breaks into the short exact sequence of graded groups, 0! E (; 2) (K (Z; 3)) v! E (; 2) (K (Z; 3))! K (2) (K (Z; 0. 2

26 The Bockstein spectral sequence degenerates. The ltration of the Bockstein spectral sequence provides a natural topology. We have the ideal in E (; 2) (K (Z; 3)) v i = im E (; 2) (K (Z; 3)) v! E (; 2) (K (Z; 3)) v! where we multiply by v i. This gives us a decreasing ltration by ideals E (; 2) (K (Z; 3)) = v 0 v v 2, v! E (; 2) (K (Z; 3)) where each quotient is a copy of K (2) (K (Z; 3)) = K (2) y (0). In e ect, this gives a lower bound for E (; 2) (K (Z; 3)). It gives no information on (v ) = T (v n ). n De nition 4.4 The topology on E (; 2) (K (Z; 3)) from the Bockstein spectral sequence has a basis consisting of the cosets x + (v i ) for all x 2 E (; 2) (K (Z; 3)) and all i. 4.2 Topology of Cohomology in E (; 2) (K (Z; 3)) We take E = E (; 2) in this section. De nition 4.5 De ne z (i) = f y (i) in E (K (Z; 3)) for all i 0. We treat E (K (Z; 3)) as the cokernel (in the sense of algebras) of the algebra homomorphism v : A! B, considered as a quotient of v : E! E BP hi 6. Thus, the elements z (i) generate E (K (Z; 3)) topologically. The formula for v where D (i) is decomposable. x (i) provides the relation v 2i z (i) + v 2i 2 z (i+) + X ji+2 A 2i (j i)z (j) = D (i), We have two topologies on E (K (Z; 3)). The rst, from the ltration by powers of the ideal (v ), was discussed in Section 4.. This is the v -topology. We have good information on the quotients by (v n ), but no information on (v ). The second is inherited from the topology on E BP hi 6 as dual to the algebra E BP hi 6. 22

27 In the quotient by a basic neighborhood of 0, we retain only nitely many of the monomials in the z (i). We call this the z-topology. We compare these two topologies on E (K (Z; 3)). They are not the same. Let Z be the augmentation ideal generated topologically by all the z (i). Below, we display schematically the v - ltration and the ltration by (completed) powers of Z, where we treat F 2 v2 ; v2 as the graded ground eld and suppress it. z 3 (0) z 2 (0) z (0) z () v z 2 () v 2 z 3 () z (2) v 3 As with any path-connected space, there is a canonical E -module decomposition E (K (Z; 3)) = E Z. It is obvious that no power of v lies in Z, and Section 4. shows that no power of z (0) lies in (v ). We need another coarser ltration that we call the z + - ltration. We lter E (K (Z; 3)) by ideals generated topologically by all except nitely many monomials in the z (i) for i > 0. This is again a complete ltration, being a quotient of the corresponding y + - ltration on B indicated by writing B = E y (0) y(i) : i > 0. Lemma 4.6 The v - ltration and the z + - ltration de ne the same topology on the augmentation ideal Z. Proof. A basic neighborhood U if 0 in the z + - ltration is the ideal generated (topologically) by the elements z (i) for i k and Z m, for some k and m. We saw already in the proof of Theorem 3.8 that v 2k z (i) 2 U + Z 2 for all i, i.e., v 2k Z U + Z 2. Then v 2k Z j U + Z j+ for all j, and v (m )(2k ) Z U + v (m 2)(2k ) Z 2 U + v (m 3)(2k ) Z 3 U + v 2k Z m U + Z m = U. Conversely, given n, we have seen that z (i) 2 (v n ) if 2 i n and that z n (i) 2 (vn ) for all i > 0. Thus the z + -neighborhood U lies in (v n ) if we take m = n and choose k such that 2 k n... 23

28 Remark 4.7 The v - ltration is clearly not complete in the summand E. Both the z- and z + - ltrations are complete, but make the E summand discrete. Corollary 4.8 The z- ltration and the ltration consisting of the ideals ((v) i \ Z) + z j (0) for i; j 0 de ne the same topology on E (K (Z; 3)). Proof. A basic z-neighborhood of 0 has the form U + z j (0), where U is a z + - neighborhood of 0. Choose i such that (v) i \ Z U. Then ((v) i \ Z) + U + z j (0). z j (0) z j (0) Conversely, given an ideal ((v) i \ Z) +, we can nd a z + -neighborhood U of 0 such that U (v) i \ Z. Then we have a z-neighborhood U + z j (0) ((v) i \ Z) + z j (0). Theorem 4.9 As a topological ring, E (; 2) (K (Z; 3)) = E (; 2) z (0) ^, where the completion indicates v -completion except on the E -module summand E generated by. Precisely, E (; 2) (K (Z; 3)) = E F 2 v2 ; v2 v ; z (0) z(0). Proof. By Lemma 4.6, Z is complete under the v - ltration. Section 4. shows that E (K (Z; 3)) (v n ) = F 2 v2 ; v2 z(0) [v ] (v n ). Hence Z = lim Zv n Z = lim F 2 v2 ; v2 z(0) [v ] (v ) n z (0) n n = F 2 v2 ; v2 z(0) [[v ]] z (0). The topology on E (K (Z; 3)) is given by the z- ltration, or, in view of Corollary 4.8, by the ideals ((v) i \ Z) + z j (0). We can now rewrite (v) i \ Z as vz i (0). The summand E is simply a copy of E, with no completion. 4.3 Relations in E (; 2) (K (Z; 3)) Furthermore, we try to generalize Tamanoi s result to higher degree. Therefore, we change our generators a little to get a better looking formula. We note that we do not use the invertibility of v 2. Again, we write E = E (; 2). 24

29 Earlier, we showed that in E BP hi 6, v x (i) = v 2 i y (i) + v 2i 2 y (i+) + X ji+2 A 2i (j i)y (j) + D (i) for i 0, with D (i) decomposable. It would be desirable to change generators to eliminate the unwanted terms. We can do the following: Theorem 4.0 There are alternate generators y hh ii (i) 0 E y(i) 0 : i 0 and of the algebra B such that B = v x (i) = v 2 i y 0 (i) + v 2i 2 y 0 (i+) + D (i) for all i 0, where D (i) is decomposable. Instead of changing generators directly, it is more convenient to change the basis of QE BP hi 6 before dualizing; then the change of generators will automatically be continuous. First, we need to improve a previous observation. Lemma 4. For k 2, A (k) is divisible by v 2. Proof. Recall from Lemma 2.0 that A (k) is the coe cient of x 2k+ in [2] (x) = v x 2 + F v 2 x 4 = v x 2 + v 2 x 4 + X i;j a ij v i v j 2x 2i+4j. Modulo v 2, the only terms that survive are v x 2 + v 2 x 4 + P j a j v v j 2x 4j+2. There is no term in x 2n for n 3. We consider Qv : QE suppress here, as it is the only multiplication.! QE BP hi 2, as before. Again, we Lemma 4.2 In QE, there are elements d (i) and e (i) such that: (a) v d (i) = v 2 i+ b (i) for all i 0. (b) The elements e (i) for all i 0 form a basis, and satisfy v e (i) = v 2 i b (i) + v 2i 2 b (i ). 25

30 Proof. We construct d (i) and e (i) inductively, starting from d (0) = e (0) = b 2 (0). Suppose we have d (i) for i < k. We recall from (6) that v b 2 (k) = v 2 k b (k) + v 2k 2 y (k ) + X i2 A 2k i (i) y (k i). By induction, v v 2k i+ + A 2k i (i) d (k i) = A 2k i (i) b (k i), where we note that the coe cient of d (k i) lies in E. We set e (k) = b 2 (k) + X v 2k i+ + A 2k i (i) d (k i), i2 to cancel out the unwanted terms. To remove the b (k further v 2k and set d (k) = v 2k e (k) + v 2k 2 d (k ), so that ) term also, we multiply by a v d (k) = v 2 k +2 k b (k) + v 2k v 2k 2 b (k ) + v 2k v 2k 2 b (k ) = v 2k+ b (k). Remark 4.3 Closer examination of the proof shows that the elements e (i) uniquely determined by property (b). are Proof of Theorem. We proceed as before. We use the elements b (0) e (i) as a basis of P E BP hi 6, instead of the b (0) b 2 (i), and extend to a basis of the whole of E BP hi 6. We de ne y(i) 0 as dual to b (0) e (i). The simpli ed formula for v e (i) leads to the simpli ed formula for v y(i) 0. Remark 4.4 This leads to a more transparent interpretation of Theorem 4.9. Put z(i) y 0 = f (i) 0 for each i. In terms of these generators of E (K (Z; 3)), the relations become v 2i z 0 (i) + v 2i 2 z 0 (i+) = D (i). Modulo decomposables, z(i+) 0 v2i v2 2i z(i) 0, so by induction for all n. z 0 (n) v 2n v 2n + 2 z 0 (0) Remark 4.5 It would be desirable to nd a nonlinear change of generators that would remove the terms D (i) as well, but this appears far more di cult. 26

31 References [A7] J.F. Adams, Stable Homotopy and Generalized Homology, University of Chicago Press, 97. [Brown] E.H. Brown, Jr., Cohomology Theories, Ann. of Math., 75(962), ; with a correction, Ann. of Math., 78(963), 20. [BP66] [B82] [B95] [BJW] E.H. Brown, Jr. and F.P. Peterson, A Spectrum whose Z p Cohomology Is the Algebra of Reduced p th Powers, Topology, 5(966), J. Michael Boardman, The Eightfold Way to BP -operations or E E and All That, Canadian Math. Soc. Conf. Proc., 2(982), Part, J. Michael Boardman, Stable Operations in Generalized Cohomology, In I.M. James, editor, The Handbook of Algebraic Topology, Chapter 4, Elsevier, Amsterdam, 995, J. Michael Boardman, David C. Johnson and W. Stephen Wilson, Unstable Operations in Generalized Cohomology, In I.M. James, editor, The Handbook of Algebraic Topology, Chapter 5, Elsevier, Amsterdam, 995, [G] P.G. Goerss, Hopf Rings, Dieudonné Module, and E 2 S 3, In J. -P. Meyer, J. Morava and W.S. Wilson, editors, Homotopy Invariant Algebraic Structures: a Conference in Honor of J. Michael Boardman, Contemporary Mathematics, Providence, R.I., 999, AMS. [HRW] [HT] M.J. Hopkins, D.C. Ravenel and W.S. Wilson, Morava Hopf Algebra and Spaces K (n) Equivalent to Finite Postnikov Systems, In S.O. Kochman and P. Selick, editors, Stable and Unstable Homotopy, Fields Institute Communications, Providence, R.I., 997, AMS. J.R. Hunton and P.R. Turner, Coalgebraic Algebra, J. of Pure and Applied Algebra, 29(998),

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34 Vita Hsin-hao Su was born in November 972 and was raised in Taipei, Taiwan. In 995, he received a Bachelor of Arts and Science degree in applied mathematics from Feng Chia University, Taichung, Taiwan. In 998, he received a Master of Science degree in mathematics from National Tsing Hua University, Hsinchu, Taiwan. He enrolled in the graduate program at Johns Hopkins University in the fall of He defended this thesis on March 2,