Q-SUBALGEBRAS, MILNOR BASIS, AND COHOMOLOGY OF EILENBERG-MAC LANE SPACES Hirotaka Tamanoi Department of Mathematics, University of California Santa Cruz Abstract. We describe mod p cohomology rings of Eilenberg-Mac Lane spaces* * in terms of the Milnor basis rather than in terms of admissible monomials of* * the Steen- rod algebra. We give a formula of excess for the Milnor basis elements, c* *orrecting Kraines' formula in odd prime case. Using the Milnor basis description, * *we study and characterize certain polynomial subalgebras generated by elements obt* *ained by applying maximum number of Milnor primitives on mod p fundamental classes* * of Eilenberg-Mac Lane spaces. A simple and interesting unstable pattern eme* *rges. These subalgebras are exact images of the BP-Thom map into mod p cohomolo* *gy rings. Contents 1.Introduction and summary of results 2 2.The structure of the mod p Steenrod algebra 6 3.Description of mod p cohomology rings of Eilenberg-Mac Lane spaces in terms of admissible monomials 9 4.Decomposition formulae for Milnor's Steenrod reduced powers and Steenrod squares 11 5.Mod p cohomology of Eilenberg-Mac Lane spaces in terms of the Milnor basis, and Q-subalgebras: odd prime case 17 5.1. Mod p cohomology of mod ph Eilenberg-Mac Lane spaces 18 5.2. Mod p cohomology of integral Eilenberg-Mac Lane spaces 25 6.Mod 2 cohomology of Eilenberg-Mac Lane spaces in terms of the Milnor basis, and Q-subalgebras 31 6.1. Mod 2 cohomology of mod 2h Eilenberg-Mac Lane spaces 33 6.2. Mod 2 cohomology of integral Eilenberg-Mac Lane spaces 37 References 41 ______________ 1991 Mathematics Subject Classification. 55. Key words and phrases. Admissible monomials, Bockstein map, Eilenberg-Mac La* *ne spaces, excess, Hopf algebras, invariant subalgebras, Milnor basis, Milnor primitives, * *Steenrod algebras, Steenrod reduced powers, Steenrod squares. Typeset by AM S-T* *EX 1 2 HIROTAKA TAMANOI x1. Introduction and summary of results The structure of mod p cohomology rings of Eilenberg-Mac Lane spaces was determined in the early 1950s (see [1] and [8]) in terms of admissible monomial* *s of Steenrod squares for even prime case, and of Steenrod reduced powers and Bockst* *ein operators for odd prime case. In the late 1950s, Milnor [4] determined the Hopf algebra structure of the mod p Steenrod algebra A(p)*, and he gave a new basis (Milnor basis) of the Steenrod algebra for all primes. Milnor basis elements a* *re rather complicated in terms of admissible monomials, but the Milnor basis is, in a sense, a more natural basis of the Steenrod algebra because it comes from the structure theory of the Steenrod algebra as a Hopf algebra. The first purpose of this paper is to reformulate mod p cohomology rings of various Eilenberg-Mac Lane spaces in terms of the Milnor basis for both even and odd prime cases [see Theorem 1 below]. Although a long time has passed since the publication of [4], an account of an explicit description of the above result w* *ith all details is missing from literature, except that a formula for a certain ver* *sion of excess of Milnor basis elements was discussed by Kraines [3, Definition 2, p. 3* *63]. The author was informed of [3] after finishing the original version of this pap* *er. The notion of excess is usually defined for admissible monomials in the Steenrod al* *gebra. Kraines defined a notion of excess for any element in the Steenrod algebra in t* *erms of (non)triviality of its action on mod p fundamental classes of mod p Eilenber* *g- Mac Lane spaces. However, his definition is not equivalent to the usual one in * *odd prime case, contrary to his claim. There is a very subtle but crucial differen* *ce between these two notions of excess. We calculate the correct formula of excess* * for Milnor basis elements in Lemma 5-4 for odd prime case. The excess formula for even prime case is given in Lemma 6-1. Kraines' notion of excess does not detect free algebra generators of cohomology rings of Eilenberg-Mac Lane spaces in ter* *ms of Milnor basis elements: his notion of excess only detects nontriviality of ce* *rtain elements which may not be algebra generators. Indeed, [3] does not go any furth* *er to discuss the algebra structure of these rings. But our correct excess formula* * does provide a simple description of free algebra generators of these cohomology rin* *gs in terms of Milnor basis elements, and this is the first purpose of this paper. Ou* *r paper is, in a sense, a continuation of his paper. But our approach is more systemat* *ic, detailed, and comprehensive. Our main ingredients are certain decomposition formulae of Milnor's Steenrod reduced powers and of Milnor's Steenrod squares. We note that [3] also discusses similar formulae [cf. Lemma 4-1], but our formulae are a lot more precise [Prop* *o- sition 4-6, Corollary 4-7], and directly give us what we want. As an element of* * the Steenrod algebra, any Milnor basis element is a stable cohomology operation. But usual Steenrod reduced powers and Steenrod squares also exhibit unstable proper* *ty with respect to dimension of cohomology elements being acted on. Our decompo- sition formulae serve two purposes. The first purpose is to exhibit unstable na* *ture of Milnor basis elements. The second purpose is to relate Milnor basis elements* * to certain closely associated admissible monomials. Our second purpose of this paper is to describe the action of Milnor primiti* *ves on mod p fundamental classes of Eilenberg-Mac Lane spaces [see Corollary 2]. We show that this action exhibits a surprisingly simple and regular pattern. For e* *xam- ple, on the mod p fundamental class n+1 of the mod p Eilenberg-Mac Lane space K(Z=p; n + 1), any product of k distinct Milnor primitives can act nontrivially MILNOR BASIS AND COHOMOLOGY OF EILENBERG-MAC LANE SPACES 3 as long as k n + 1. However, as soon as k > n + 1, any product of k Milnor primitives acts trivially on n+1 . The case k = n + 1 is the borderline case, * *and this is the case we are most interested in. The element of this type of the sma* *llest positive degree is Qn . .Q.0n+1 of degree 2(1 + p + . .+.pn). The Q-subalgebra * *in the title is a polynomial subalgebra generated by elements obtained by the acti* *on of maximum number of Milnor primitives, in this case n + 1, on the fundamental class n+1 . We show that this algebra can be characterized as the smallest A(p)* **- invariant subalgebra of the cohomology algebra H* K(Z=p; n + 1); Zp containing the element Qn . .Q.0n+1 [see Theorem 3]. Similar subalgebras exist in mod p cohomology rings of all Eilenberg-Mac Lane spaces. In [10], these Q-subalgebras are also characterized as the images of the Thom map from BP-cohomology to mod p cohomology of Eilenberg-Mac Lane spaces. We note that in [2] actions of products of at most two Milnor primitives on * *mod p fundamental classes of integral Eilenberg-Mac Lane spaces are discussed. We summarize our results. For any prime p, let Z=p = Zp be the ring of mod p integers. The Hopf algebra structure of the mod p Steenrod algebra A* = A(p)* was determined by Milnor for both even and odd prime p [4]. See x2 for a more detailed summary including all relevant facts needed for this paper. When p is * *odd, its dual algebra A* = A(p)* is a tensor product of a polynomial algebra and an exterior algebra of the following form: (1-1) A* = Zp(o0; o1; : :;:or; : :): Zp[ 1; 2; : :;:r; : :]:; where |or| = 2pr - 1, |r| = 2(pr - 1). For even prime case, see Theorem 2-1. Let E = ("0; "1; : :):range over all sequences of zeroes and ones which are alm* *ost all zero, and let R = (r1; r2; : :):range over all sequences of non-negative in* *tegers which are almost all zero. Then the set of elements o (E)(R) = o0"0o1"1. .r.11r* *22. . . forms an additive basis of the dual Steenrod algebra A*. The element dual to oj* * is the j-th Milnor primitive Qj 2 A* for j 0. These elements generate an exterior subalgebra of the Steenrod algebra: (1-2) QiQj + QjQi = 0; for alli; j 0: For a sequence E as above, let QE = Q"00Q"11. ...Let PR 2 A* be the element dual to (R). Elements of the form PR close under multiplication. See Theorem 2-2 for the explicit rule of multiplication. It is known that the set of elements {QE P* *R }E;R forms an additive basis of A* dual to {o (E)(R)}E;R up to sign. Elements of the form QE PR 2 A* are called Milnor basis elements. Elements of the form PR are called Milnor's Steenrod reduced powers. For the mod 2 Steenrod algebra, we have Milnor's Steenrod squares denoted by SqR [Theorem 2-3]. Let i = (0; : :;:0; 1; 0; : :):be a sequence with 1 at i-th place and zero e* *very- where else. Let R be the set of sequences R as above. For any R 2 R, we define an integer `[R] and a shifted sequence t(R) by X X (1-3) `[R] = rj; t(R) = (r2; r3; : :;:rk; : :):= rj+1j: j1 j1 Two sequences are added or subtracted componentwise. The following four cases are discussed in this paper: 4 HIROTAKA TAMANOI (1) Mod p cohomology of mod ph Eilenberg-Mac Lane spaces at an odd prime p. (2) Mod p cohomology of integral Eilenberg-Mac Lane spaces at an odd prime p. (3) Mod 2 cohomology of mod 2h Eilenberg-Mac Lane spaces at prime 2. (4) Mod 2 cohomology of integral Eilenberg-Mac Lane spaces at prime 2. Situations are rather different among the above four cases which are discussed * *in x5.1, x5.2, x6.1, and x6.2, respectively. To illustrate our results in this pa* *per, we explicitly describe our results here for the case of the mod ph Eilenberg-Mac L* *ane space K(Z=ph; n + 1). Theorem 1 [Theorem 5-2]. Let p be an odd prime, and let n 0 and h 1. Let n+1 2 Hn+1 K(Z=ph; n + 1); Zp be the mod p fundamental class. Let E and R be sequences as above. Then the following identities hold: (1-4) QE PR n+1 = 0 if `[E] + 2`[R] n + 2: t(E) t(R) p (1-5) QE PR n+1 = Q P n+1 if `[E] + 2`[R] = n + 1 and "0 = 0: The mod p cohomology ring of the mod ph Eilenberg-Mac Lane space is a free alge* *bra described in terms of the Milnor basis as follows: h E R (1-6) H* K(Z=p ; n + 1); Zp = FZp[ Q P n+1 | `[t(E)] + 2`[R] < n + 1 ]: Here we adopt a convention that Q0n+1 means ffihn+1 , where ffih is the h-th Bockstein operator. Please note that the correct formula of excess of the Milnor basis element Q* *E PR is `[t(E)]+2`[R] = `[E]+2`[R]-"0, which is defined as the excess of the associa* *ted admissible monomial [Lemma 5-4]. In [3], Kraines defines his version of excess* * of QE PR and proves that it is given by `[E]+2`[R]. This difference between his ve* *rsion and ours is more significant than it may seem. Kraines' excess detects nontrivi* *ality of elements of the form QE PR n+1 , whereas our excess detects indecomposabilit* *y of elements of the above form. We have similar descriptions of mod p cohomology rings of integral Eilenberg- Mac Lane spaces in terms of the Milnor basis [Theorem 5-10]. Since the mod p fundamental class on+2 of the integral Eilenberg-Mac Lane space K(Z; n + 2) has the property Q0on+2 = 0, we need an extra condition on pairs of sequences (E; R) to describe the mod p cohomology ring, and the proof is more involved. We are also interested in the action of Milnor primitives on mod p fundament* *al classes of Eilenberg-Mac Lane spaces. Theorem 1 implies the following corollary. Corollary 2 [Corollary 5-7]. The action of Milnor primitives on the mod p fundamental class n+1 of the mod ph Eilenberg-Mac Lane space K(Z=ph; n + 1) is described as follows: (1-7) QE n+1 = 0 if `[E] n + 2; t(E) p (1-8) QE n+1 = Q n+1 if `[E] = n + 1 and "0 = 0; (1-9) QE n+1 6= 0 if `[E] n + 1: When `[E] = n + 1 and "0 = 1, or `[E] n, the element QE n+1 is a polynomial or exterior algebra generator of the cohomology ring H* K(Z=ph; n + 1); Zp . Namely, on the fundamental class n+1 , all products of n + 2 or more Milnor primitives act trivially, and any product of n + 1 or fewer distinct Milnor pri* *mitives MILNOR BASIS AND COHOMOLOGY OF EILENBERG-MAC LANE SPACES 5 always acts nontrivially. Furthermore, except for the case `[E] = n + 1 and "0 * *= 0, any element of the form QE n+1 with `[E] n + 1 is a free algebra generator of the mod p cohomology algebra. A similar statement holds concerning the action of Milnor primitives on the * *mod p fundamental class on+2 of the integral Eilenberg-Mac Lane space K(Z; n + 2). It turns out that all products of n + 1 or more Milnor primitives act trivially* * on on+2 , and any product of n or fewer distinct Milnor primitives not including Q0 acts nontrivially on on+2 . In the second case, all of the resulting elements a* *re free algebra generators of the cohomology ring [Proposition 5-11, Proposition 5-12]. We have corresponding statements for even prime case, with one difference. When p = 2, after applying maximum number of Milnor primitives on mod 2 fundamental classes of mod 2h or integral Eilenberg-Mac Lane spaces, resulting nontrivial elements are always squares of algebra generators of mod 2 cohomology rings [Corollary 6-6, Proposition 6-9]. In Corollary 2 we are particularly interested in the borderline case: `[E] =* * n + 1 and "0 = 1. Let S+n be the set of sequences of n strictly increasing positive i* *ntegers: (1-10) S+n = {(s1; s2; : :;:sn) 2 Zn | 0 < s1 < s2 < . .<.sn}: Then for any S 2 S+n, the element QS Q0n+1 = Qsn. .Q.s1Q0n+1 has even degree 2(1 + ps1 + . .+.psn), and it is a polynomial generator of the cohomology ring * *by the last part of Corollary 2. Let Q be a polynomial subalgebra of the cohomology ring generated by these elements. That is, (1-11) Q = Q(Z=ph; n + 1) = Zp[ QS Q0n+1 | S 2 S+n]: Note that the element of the lowest positive degree in this polynomial subalgeb* *ra is Qn. .Q.1Q0n+1 of degree 2(1 + p + . .+.pn). Theorem 3 [Theorem 5-9]. The polynomial subalgebra Q(Z=ph; n + 1) of the cohomology ring H* K(Z=ph; n + 1); Zp is the smallest A(p)*-invariant subalgeb* *ra containing the element Qn . .Q.1Q0n+1 . Any Milnor primitive acts trivially on this subalgebra. We can also characterize the polynomial subalgebra Q as the image of the BP- Thom map from BP-cohomology of K(Z=ph; n + 1) to its mod p cohomology [10]. Similar polynomial subalgebras exist in mod p cohomology rings of integral Eilenberg-Mac Lane spaces. For n 0, they are given as follows: (1-12) Q(Z; n + 2) = Zp[ QS on+2 | S 2 S+n] H* K(Z; n + 2); Zp : We can show that the polynomial subalgebra Q(Z; n+2) is again the smallest A(p)* **- invariant subalgebra containing the element Qn . .Q.1on+2 of the smallest posit* *ive degree 2(1 + p + . .+.pn), and that this subalgebra is annihilated by any Milnor primitive [Theorem 5-14]. The above polynomial Q-subalgebras (1-11) and (1-12) are isomorphic as alge- bras by the homomorphism induced by the Bockstein map ffih : K(Z=ph; n + 1) -! K(Z; n + 2) [Proposition 5-15]. We also describe results corresponding to Theorem 1, Corollary 2, and Theorem 3 above for even prime case in x6. The statements are basically similar but sim* *pler, 6 HIROTAKA TAMANOI although there are some subtle differences in the details. One difference betw* *een odd prime case and even prime case is that when p = 2, all elements in the Q- subalgebras are always squares in mod 2 cohomology rings [Theorem 6-7, Theorem 6-11]. The organization of this paper is as follows. In x2, we review the Hopf alge* *bra structure of the mod p Steenrod algebra determined by Milnor. In x3, we recall the classical description of mod p cohomology rings of various Eilenberg-Mac La* *ne spaces in terms of admissible monomials. Sections 2 and 3 are included here in order to be more self-contained and for use in later sections. In x4, we presen* *t our key decomposition formulae for Milnor's Steenrod reduced powers and Steenrod squares. Although the proof is elementary, these decomposition results are esse* *ntial for the rest of the paper. These formulae are designed to make unstable nature * *of Milnor basis elements explicit. In x5 and x6, we describe mod p cohomology rings of Eilenberg-Mac Lane spaces for any prime p in terms of the Milnor basis eleme* *nts rather than in terms of admissible monomials. We also prove certain vanishing a* *nd p-th power properties of actions of certain Milnor basis elements. In the same sections, we also describe the action of products of Milnor prim* *i- tives on mod p fundamental cohomology classes of mod ph or integral Eilenberg- Mac Lane spaces and we prove the characterizing property of the Q-subalgebras. Acknowledgement. The author thanks Steve Wilson for informing him of the paper [3], and Peter Landweber for his careful reading of the manuscript. Doug Ravenel and Steve Wilson had calculated the Hopf ring of mod p homology of Eilenberg- Mac Lane spaces in [12, Theorem 8.11], and they identified homology algebra gen* *er- ators in terms of O-products, but not in terms of (co)homology operations. Fina* *lly, the author thanks hospitality and financial support from the Institut des Hautes Etudes Scientifiques during the period this work was undertaken. x2. The structure of the mod p Steenrod algebra We review those basic properties of the mod p Steenrod algebra which are rel- evant to the present paper. Basic references are [4, 5, 9]. Let p be a prime,* * even or odd, and let A(p)* be the mod p Steenrod algebra. It is well known that A(p)* has the structure of a Hopf algebra [6]. Let A(p)* denote the dual Hopf algebr* *a. To describe the structure of A*, let Zp[ . ] denote a polynomial algebra over Zp generated by elements inside of [ . ], and let Zp( . ) denote an exterior algeb* *ra over Zp generated by elements inside of ( . ). Theorem 2-1 (Milnor [4]). Let OE : A(p)* -! A(p)* A(p)* be the coalgebra map for the dual of the Steenrod algebra A(p)*. (I) The dual Hopf algebra A(2)* is a polynomial algebra described as follows: A(2)* = Z2[ i1; i2; : :;:ir; : :]:; |ii| = 2i- 1; (2-1) Xk i OE(ik) = i2k-i ii: i=0 (II) Let p be an odd prime. Then the structure of the dual Hopf algebra A(p)* ** is MILNOR BASIS AND COHOMOLOGY OF EILENBERG-MAC LANE SPACES 7 described as follows: A(p)* = Zp(o0; o1; : :;:or; : :): Zp[ 1; 2; : :;:r; : :]:; where |or| = 2pr - 1; and |r| = 2(pr - 1); (2-2) Xk k i X pi OE(k) = pk-i i; OE(ok) = k-i oi+ ok 1: i=0 i=0 Let R = (r1; r2; : :):range over all sequences of non-negative integers whic* *h are almost all zero, and let E = ("0; "1; : :):range over all sequences of zeroes a* *nd ones which are almost all zero. Let i = (0; : :;:0; 1; 0; : :):bePa sequence w* *ith 1 atPthe i-th entry and 0 everywhere else. So we can write R = i1 rii and E = i0 "ii+1. Let p be an odd prime. We let (2-3) o (E)(R) = o0"0o1"1. .r.11r22. .2.A(p)*: For i 0, let Qi 2 A(p)2pi-1 be the element dual to oi, and let PR 2 A(p)* be t* *he element dual to (R). Let QE PR = Q"00Q"11. .P.Rbe the product of elements QE and PR in A(p)*. Elements PR are complicated expressions of Steenrod reduced powers Pi = Pi1 . If any entry of the sequence R is negative, then we set PR = 0 by convention. Theorem 2-2 (Milnor [4, 5]). Let p be an odd prime. The set {QE PR }E;R forms an additive basis of the Steenrod algebra A(p)*. This basis is dual to the bas* *is {o (E)(R)}E;R of A(p)* up to sign. Elements Qi for i 0 are primitive and they generate an exterior subalgebra of A(p)*: (2-4) QiQj + QjQi = 0; i; j 0: For any sequence R, the element Qk commutes with PR by the following formula: k R-pk (2-5) PR Qk = QkPR + Qk+1 PR-p 1 + . .+.Qk+j P j + . .:. For two sequences R = (r1; r2; : :):and S = (s1; s2; : :):of non-negative integ* *ers, almost all zero, the product of PR and PS is given by X (2-6) PR PS = b(X)PT(X) ; R(X)=R S(X)=S where X range over all infinite matrices 0 * x01 x02 : :1: B x10 x11 x12 : :C: (2-7) B@ x20 x21 x22 : :C:A .. . . . . .. .. .. of non-negative integers, almost all zero, with upper left corner omitted, such* * that X X (2-8) ri = pjxij (weighted row sum ); sj = xij (column sum ): j0 i0 8 HIROTAKA TAMANOI These relations are denoted by R(X) = R and S(X) = S, in short. From such a matrix X, the sequence T (X) = (t1; t2; : :):and the coefficient b(X) are defin* *ed by X Q n1 tn! (2-9) tn = xij (diagonal sum ); b(X) = ____________Q: i+j=n i+j1 xij! The Cartan formula holds for PR : for any two cohomology elements x; y, X (2-10) PR (xy) = (PR1 x).(PR2 y); R1+R2=R where PRi = 0 if any entry in Ri is negative. The usual Steenrod reduced power * *Pm coincides with a Milnor basis element Pm1 : Hk (X; Zp) -! Hk+2(p-1)m (X; Zp) for m 1, and it has the following unstable property for any x 2 Hk (X; Zp): ae Pm1 x = 0 if 2m > k; (2-11) Pm1 x = xp if 2m = k: The above unstable property in (2-11) will play a crucial role in our calcul* *ation of mod p cohomology rings of Eilenberg-Mac Lane spaces in terms of the Milnor basis. Next, we describe the mod 2 Steenrod algebra. Let H = (h1; h2; : :):range ov* *er all sequences of non-negative integers which are almost all zero. In the dual S* *teenrod algebra, let i(H) = ih11ih22. .2.A(2)*. The set {i(H)}H forms an additive basis of A(2)*. Let SqH 2 A(2)* be the element dual to i(H) with respect to this bas* *is. For i 0, let Qi = Sqi+1 and let PR = Sq2R for any sequence R of non-negative integers which are almost all zero. Theorem 2-3 (Milnor [4, 5]). The set of elements {Sq H}H forms an additive basis of the mod 2 Steenrod algebra A(2)* dual to the basis {i(H)}H of A(2)*. * *The elements Qi for i 0 are primitive and they form an exterior subalgebra of A(2)* **: (2-12) Q2i= 0; QiQj = QjQi; i; j 0: For any sequence E = ("0; "1; : :):of zeroes and ones which are almost all zero* *, and for any sequence R of non-negative integers which are almost all zero, we have (2-13) QE PR = Q"00Q"11. .P.R= SqE+2R : For any sequence R of non-negative integers which are almost all zero, the elem* *ent Qk commutes with PR by the following formula: X k (2-14) PR Qk = QkPR + Qk+j PR-2 j : j1 For any two sequences R, S of non-negative integers, almost all zero, the produ* *ct of SqR and SqS is given by the same formula as in (2-6): X (2-15) Sq RSqS = b(X)Sq T(X): R(X)=R S(X)=S MILNOR BASIS AND COHOMOLOGY OF EILENBERG-MAC LANE SPACES 9 The Cartan formula holds for SqR : for any cohomology elements x; y, X (2-16) SqR(xy) = SqR1(x).SqR2 (y); R1+R2=R where Sq Ri = 0 if any entry of Ri is negative. The usual Steenrod square Sq m coincides with a Milnor basis element Sqm1 : Hk (X; Z2) -! Hk+m (X; Z2), and * *it has the following unstable property for any element x 2 Hk (X; Z2): ae Sqm1 x = 0 if m > k; (2-17) Sqm1 x = x2 if m = k: As we can see, we can treat Steenrod algebras for even or odd primes equally using the Milnor basis {QE PR }E;R . However, we have to keep in mind that in A(2)*, the elements PR do not close under multiplication unlike odd prime case, because PR = Sq2R for even prime case. x3. Description of mod p cohomology rings of Eilenberg-Mac Lane spaces in terms of admissible monomials In this section, we recall the well-known structure of mod p cohomology ring* *s of Eilenberg-Mac Lane spaces in terms of admissible monomials [Theorems 3-4, 3-6]. We will need this description later. First, we deal with odd prime case. Let (3-1) = Q"00Ps11 Q"10Ps21 Q"20. .Q."j-10Psj1 Q"j0Psj+11 Q"j+10. . . be a monomial in Steenrod reduced powers and the Bockstein operator Q0, where "j = 0; 1 for j 0 and (s1; s2; : :):is a sequence of non-negative integers whi* *ch are almost all zero. Put fi "j-1 fi ij = fiQ0 Psj1 fi= "j-1 + 2(p - 1)sj for j 1; and (3-2) d() = X ij = X "j + 2(p - 1) X sj: j1 j0 j1 Here d() is the degree of the operation 2 A(p)*. The next lemma is straightfor- ward. Lemma 3-1. Let p be any prime. For a monomial as in (3-1), the following conditions on are equivalent: (1) ij pij+1 for all j 1. (2) sj psj+1 + "j for all j 1. Any monomial of the form (3-1) satisfying one of the equivalent conditions in Lemma 3-1 is called an admissible monomial. Both of the above admissibility conditions are found in the literature. The condition (2) suits better for our * *purpose. Next, we discuss the notion of excess of admissible monomials of the form (3-1). By the admissibility condition, we have ij pij+1 for all j 1. We let __e p() = (i1 - pi2) + (i2 - pi3) + . .+.(ij - pij+1) + . . . (3-3) = pi1 - (p - 1)d(): 10 HIROTAKA TAMANOI This is the usual definition of excess of admissible monomials. However, for o* *dd prime case we use a slightly improved version of excess defined as follows. Fi* *rst note that __ep() - "0 is always divisible by p - 1. We then let __e() - " (3-4) ep() = _p_______0_p2-Z1; p : odd prime : We note the following relation between __ep() and ep(), which is immediate. Lemma 3-2. Let p be an odd prime and let be an admissible monomial in A(p)*. Then for any positive integer n, we have __ep() < n(p - 1) if and only if ep() * *< n. Proof. If __ep() < n(p - 1), then __ep() - "0 < n(p - 1). Dividing both sides b* *y p - 1, we_have ep() < n. Conversely, suppose ep() < n. Then, by definition, we have ep() - "0 < n(p - 1). Since __ep() - "0 is always divisible by p - 1 as we rema* *rked right before (3-4), we have __ep() - "0 = m(p - 1) for some m < n. Since "0 < p* * - 1 for any odd prime p, we have __ep() = m(p - 1) + "0 < (m + 1)(p - 1) n(p - 1). This completes the proof of Lemma 3-2. Definition 3-3. For any admissible monomial as in (3-1), the integer ep() defined in (3-4) is called (modified) excess of . Although __ep() is the one we use for even prime case, we found it more conv* *enient and simpler to use modified excess ep() for odd prime cases. Kraines gives a different and inequivalent definition of excess for any element in the Steenrod* * algebra [3]. To describe cohomology rings of Eilenberg-Mac Lane spaces, we use the follow* *ing notation. For a non-negatively and integrally graded vector space V over Zp, * *let V evenand V odd be even and odd graded parts of V . The free algebra FZp[V ] generated by the graded vector space V is the tensor product of the polynomial algebra on V evenand the exterior algebra on V odd: (3-5) FZp[V ] = Zp[V even] Zp(V odd): The well-known description of mod p cohomology rings of Eilenberg-Mac Lane spaces in terms of admissible monomials goes as follows for any odd prime p: Theorem 3-4. Let p be an odd prime. Let h 1 and n 0. Let be a monomial in Steenrod reduced powers and the Bockstein operator Q0 as in (3-1). (I) Let n+1 2 Hn+1 K(Z=ph; n + 1); Zp be the fundamental class. The mod p cohomology ring of the mod ph Eilenberg-Mac Lane space is a free algebra given * *by h (3-6) H* K(Z=p ; n + 1); Zp = FZp n+1 | is admissible andep() < n + 1 : Here if h 2 and ends with a Bockstein operator Q0, then this Bockstein should be regarded as the h-th Bockstein operator ffih. (II) Let on+2 2 Hn+2 K(Z; n + 2); Zp be the fundamental class. The mod p cohomology ring of the integral Eilenberg-Mac Lane space is a free algebra give* *n by fifi is admissible,ep() < n + 2; (3-7) H* K(Z; n + 2); Zp = FZp on+2 fifiand doesn't end with Q : 0 Next, we describe mod 2 cohomology rings of Eilenberg-Mac Lane spaces for even prime case p = 2. Let (3-8) = Sqs11 .Sqs21 . .S.qsr1 . . . be a monomial in Steenrod squares for some sequence of non-negative integers S = (s1; s2; : :;:sr; : :):which are almost all zero. MILNOR BASIS AND COHOMOLOGY OF EILENBERG-MAC LANE SPACES 11 Definition 3-5. The monomial in (3-8) is said to be admissible if sj 2sj+1 f* *or all j 1. The excess e2() of an admissible monomial is given by X (3-9) e2() = (sj - 2sj+1) = 2s1 - d(); j1 P where d() = j1 sj is the degree of the monomial . Note that the above excess e2() is the p = 2 version of __ep() in (3-3), rat* *her than (3-4). With these definitions, the well-known description of mod 2 cohomol* *ogy rings of Eilenberg-Mac Lane spaces is given as follows: Theorem 3-6. Let be a monomial in Steenrod squares as in (3-8). Let n 0. (I) Let n+1 2 Hn+1 K(Z=2h; n + 1); Z2 be the mod 2 fundamental class, where n 1 and h 1, or n = 0 and h = 1. The mod 2 cohomology ring of the mod 2h Eilenberg-Mac Lane space K(Z=2h; n + 1) is a polynomial algebra: h (3-10) H* K(Z=2 ; n + 1); Z2 = Z2[ n+1 | is admissible ande2() < n + 1 ]: Here, when h > 1 and ends with Sq1 , this last operator Sq1 should be regar* *ded as the h-th Bockstein ffih. When n = 0 and h > 1, the mod p cohomology ring of K(Z=2h; 1) has an exterior factor and it is given by h (3-100) H* K(Z=2 ; 1); Z2 = Z2[ ffih1 ] Z2(1): (II) Let on+2 2 Hn+2 K(Z; n+2); Z2 be the mod 2 fundamental class. The mod 2 cohomology ring of the integral Eilenberg-Mac Lane space is a polynomial alge* *bra fifi is admissible,e2() < n + 2; (3-11) H* K(Z; n + 2); Z2 = Z2 on+2 fifi : and does not end with Sq1 x4. Decomposition formulae for Milnor's Steenrod reduced powers and Steenrod squares In this section, we prove decomposition formulae for Milnor's Steenrod reduc* *ed powers and Steenrod squares. These decomposition formulae will play a crucial role in describing unstable action of Milnor basis elements. Although our formu* *la [Proposition 4-6] is very precise, for the purpose of this paper, Corollary 4-7* * and Proposition 4-8 are sufficient. We worked out a precise formula for future refe* *rence. Although the method of the proof is elementary, we need a very careful analysis* * to obtain an exact formula. The main point of these decomposition formulae is that they extract unstable nature of Milnor basis elements explicitly. Let R be the set of all sequences R of non-negative integers which are almos* *t all zero. We introduce a partial ordering in R as follows. Let R = (r1; r2; : :;:rk* *; : :): and S = (s1; s2; : :;:sk; : :):be two sequences in R. Then we write S R when sk rk for all k 1. We write S < R if S R and sk < rk for some k 1. For two such sequences R and S, we define their multi-binomial coefficient (R; S) by Y Y (rk + sk)! (4-1) (R; S) = (rk; sk) = __________: k1 k1 rk! sk! 12 HIROTAKA TAMANOI For R = (r1; r2; : :;:rk; : :):2 R, let a weighted sum map oe : R -! N [ {0}, a translation operation t : R -! R, and a length map ` : R -! N [ {0} be defined * *by X oe[R]= r1 + pr2 + . .+.pk-1 rk + . .=. pj-1rj; j1 X (4-2) t(R) = (r2; r3; : :;:rk; : :):= rj+1j; j1 X `[R]= r1 + r2 + . .+.rk + . .=. rj: j1 Here, i = (0; : :;:0; 1; 0; : :):with 1 at the i-th place and 0 everywhere else* *. For convenience we let 0 be the zero sequence, that is, 0 = (0; 0; : :):. Lemma 4-1. Let p be an odd prime and let R be a sequence of non-negative integ* *ers which are almost all zero. Then X (4-3) Poe[R]1.Pt(R)= PR + R - S + oe[S]1; t(S) PR-S+oe[S]1+t(S) : S2R;s1=0 0 0 and the first entry of Si is zero for 1 i k. (2) 0 < Si+1 R - (S1 + S2 + . .+.Si) + t(S1 + S2 + . .+.Si) for 0 i < k. 14 HIROTAKA TAMANOI The set of all R-admissible chains of length k is denoted by Ck(R). For any -!S= (S1; S2; : :;:Sk) 2 Ck(R), we let |-!S| = S1 + S2 + . .+.Sk 2 R. For a given k and R = (r1; r2; : :;:r`; : :):such that r` 6= 0 for some ` k* * + 1, an example of an R-admissible chain of length k is given as follows. Let Sj be* * a sequence obtained from tj-1(R) by replacing the first entry by 0, namely, Sj = (0; rj+1; : :;:r`; : :):> 0 for 1 j k. Then a sequence -!S= (S1; S2; : :;:Sk)* * is an R-admissible chain of length k. We list several properties of the set Ck(R). Lemma 4-4. Let R be a sequence of non-negative integers which are almost all zero. The set Ck(R) of R-admissible chains of length k has the following proper* *ties: (1) For any k 1, the set Ck(R) has finitely many elements. P (2) For any length k admissible chain -!S2 Ck(R), we have t(|-!S|) ti(R* *). (3) The set Ck(R) is empty if k > ` P i1 ti(R) . i1 Proof. Let -!S= (S1; S2; : :;:Sk) 2 Ck(R) be any R-admissible chain of length k. By definition, Sk satisfies 0 < Sk R - (S1 + . .+.Sk-1 ) + t(S1 + . .+.Sk-1 ). From this, we immediatelyPhave R - |-!S| + t(|-!S|) 0. By Lemma 4-2, this implies that t(|-!S|) i1 ti(R). This proves (ii). P Since there are finitely many possibilities of -!S such that |-!S| i1 ti* *(R), there can be finitely many elements in Ck(R). This proves (i). Suppose Ck(R) is not empty and let -!S= (S1; S2; : :;:Sk) 2PCk(R). Since Si > 0 for 1 i k by definition, we have `[Si] 1. So `[|-!S|] = ki=1`[Si] k. ByPapplying `[ . ] to the inequalityPin (ii), we have k `[|-!S|] = `[t(|-!S|)] `[ i1 ti(R)]. Thus, if k > `[ i1 ti(R)], then Ck(R) = ;. Let -!S= (S1; S2; : :;:Sk) 2 Ck(R) be an R-admissible chain of length k. We * *let Yk (4-6) ak(-!S) = R-(S1+. .+.Si)+oe[S1+. .+.Si]1+t(S1+. .+.Si-1); t(Si) : i=1 Here, by convention, S0 = 0. When k is understood, we simply write a(-!S) for ak(-!S). With these preparations, we can further continue our process of decom- posing PR which was started in (4-4) and (4-5). Lemma 4-5. Let p be an odd prime. Let R be a sequence of non-negative integers which are almost all zero. Then for any r 1, we have (4-7) r-1 X X -! -! -! 2 * *-! PR = Poe[R]1.Pt(R)+ (-1)k ak( S )P(oe[R]+oe[t(| S|)])1.Pt(R)-t(| S|)+t (|* * S|) k=1 -!S2Ck(R) r X -! R-|-!S|+t(|-!S|)+oe[|-!* *S|] + (-1) ar( S )P : * * 1 -!S2Cr(R) Proof. We prove (4-7) by induction on r 1. When r = 1, (4-7) is the same as (4-4) and we are done. We assume (4-7) for some r 1 and we prove the formula corresponding to r + 1. For this, we apply the partial factorization formula (* *4-4) MILNOR BASIS AND COHOMOLOGY OF EILENBERG-MAC LANE SPACES 15 to each term in the second summation on the right hand side of (4-7). For -!S= (S1; S2; : :;:Sr)-2!Cr(R),-if!we-have!t(R) - t(|-!S|) + t2(|-!S|) = 0, then Lem* *ma 4-1 applied to PR-| S|+t(| S|)+oe[| S|]1does not give anything different,-because!o* *nly-the! first entry of the exponent sequence is nontrivial. If t(R) - t(| S |) + t2(| S* * |) > 0, then (4-4) gives -! -! -! -! -! 2 -! PR-| S|+t(| S|)+oe[| S|]1= P(oe[R]+oe[t(| S|)])1.Pt(R)-t(| S|)+t (| S|) X 0 - (R0- Sr+1 + oe[Sr+1]1; t(Sr+1))PR -Sr+1+oe[Sr+1]1+t(Sr+1); 0 n + 1 + "0, then for any E2 t(E) and S 0 as above we have fi fi fiQE2 .Pt(R)-t(S)+t2(S)n+1 fi< 2(oe[R] + oe[t(S)] + oe[t(E)]): Thus by dimensional reason (2-11), we have E t(R)-t(S)+t2(S) P(oe[R]+oe[t(S)]+oe[t(E)])1Q 2 .P n+1 = 0; for any E2 and S. Hence all the terms in the above summation vanish, and we have QE PR n+1 = 0. When `[E] + 2`[R] = n + 1 + "0, the element E t(R)-t(S)+t2(S) P(oe[R]+oe[t(S)]+oe[t(E)])1Q 2 .P n+1 can be nontrivial only when S = 0 and E2 = t(E) by dimensional reason as above. There is only one such term in the double summation above, and we have t(E) t(R) " t(E) t(R) p QE PR n+1 = Q"00P(oe[R]+oe[t(E)])1Q P n+1 = Q00 Q P n+1 ; since |Qt(E)Pt(R)n+1 | = 2(oe[R] + oe[t(E)]). This element vanishes if "0 = 1 b* *y the derivation property of Q0. This proves (5-3). When "0 = 0, we get (5-4). The statement (5-5), which will be proved shortly, implicitly says that when `[E] + 2`[R] < n + 1 + "0 (note that this includes the case `[E] + 2`[R] = n + * *1 and "0 = 1), the element QE PR n+1 is nontrivial because it is an algebra generator* *. For the case `[E] + 2`[R] = n + 1 with "0 = 0, formula (5-4) does not explicitly st* *ate nontriviality of the element. But repeated use of (5-4) reveals that it is nont* *rivial, as shown in the next corollary, assuming (5-5) for a moment. This corollary can also be found in [3]. Corollary 5-3. Let sequences E and R be such that `[E] + 2`[R] = n + 1 with "0 = 0. Let k = min {j | "j 6= 0 or rj 6= 0}. Then tk(E) tk(R) pk (5-6) QE PR n+1 = Q P n+1 6= 0; where `[tk+1 (E)] + 2`[tk(R)] < n + 1, and consequently Qtk(E)Ptk(R)n+1 is an indecomposable free algebra generator of the cohomology H* K(Z=ph; n + 1); Zp . Proof. Since `[ti(E)] + 2`[ti(R)] = n + 1 and "i = 0 for 0 i k - 1, repeated k k pk use of (5-4) gives QE PR n+1 = Qt (E)Pt (R)n+1 . Since tk(E) = ("k; "k+1 ; :* * :): and tk(R) = (rk+1 ; rk+2 ; : :):, we have `[tk(E)] + 2`[tk(R)] = `[E] + 2`[R] -* * 2rk = n + 1 - 2rk. By our choice of k, we have either rk 6= 0 or "k 6= 0. If rk 6= 0,* * then n + 1 - 2rk < n + 1 n + 1 + "k. If "k 6= 0, then n + 1 - 2rk n + 1 < n + 1 + * *"k. Hence in either case we have `[tk+1 (E)] + 2`[tk(R)] < n + 1, and Qtk(E)Ptk(R)n* *+1 20 HIROTAKA TAMANOI is an indecomposable free algebra generator of the cohomology ring by (5-5). Th* *is completes the proof. The statement (5-5) is proved by reformulating the classical description of * *the cohomology algebra described in x3 in terms of admissible monomials. We compare Milnor basis elements and admissible monomials in the Steenrod algebra. To do this, we apply (4-9) and (4-12) repeatedly to QE PR n+1 . We have QE PR n+1 = QE Poe[R]1Pt[R]n+1 + (other terms ) = Q"00P(oe[R]+oe[t(E)])1Qt(E)Pt(R)n+1 + (other terms ): Repeating this procedure on Qt(E)Pt(R)n+1 again and again, we get QE PR n+1 = (E; R) + (other terms ), where (E; R) = Q"00Ps11 Q"10Ps21 Q"20. ....Q."r-10Psr1 Q"r0. .;. (5-7) with sr = oe[tr-1 (R)] + oe[tr(E)] for r 1: If we let S = (s1; s2; : :):, then after an easy calculation we find that X X R + t(E) (5-8) S = pj.tj(R) + pj.tj+1(E) = _________; j0 j0 1 - p.t where the last formal operator expression makes sense since the sequences E and R have entries which are almost all zero. Lemma 5-4. Let E be a sequence of zeroes and ones which are almost all zero, and let R be a sequence of non-negative integers which are almost all zero. Let (E; R) be a monomial in Steenrod reduced powers and the Bockstein operator Q0 as in (5-7). Then the following statements hold: (1) The monomial (E; R) is admissible. (2) The excess of the admissible monomial (E; R) is given by (5-9) ep (E; R) = `[t(E)] + 2`[R] = `[E] + 2`[R] - "0: This correspondence (E; R) -! (E; R) gives rise to the following bijection: (5-10) {(E; R)} !1:1 { admissible monomials in A(p)* }; where E ranges over all sequences of zeroes and ones which are almost all zero,* * and R ranges over all sequences of non-negative integers which are almost all zero. Proof. To check the admissibility of the monomial (E; R), we can check either (* *1) or (2) in Lemma 3-1. For our purpose, condition (2) suits better. From (5-8), we have S - p.t(S) - t(E) = R, or sj - (psj+1 + "j) = rj 0 for all j 1 in terms * *of components of the sequences. Hence condition (2) in Lemma 3-1 is satisfied, and (E; R) is an admissible monomial. Next, we calculate excess of the monomial (E; R). First we calculate __ep() * *given in (3-3). For the monomial (E; R), the integer ij in (3-2) is given by ij = "j-1 + 2(p - 1)sj = "j-1 + 2(p - 1)oe[tj-1(R)] + 2(p - 1)oe[tj(E)]: MILNOR BASIS AND COHOMOLOGY OF EILENBERG-MAC LANE SPACES 21 A simple calculation showsPthat ij - pij+1 = "j-1 + (p - 2)"j + 2(p - 1)rj for * *all j 1. Thus, __ep(E; R) = j1 (ij - pij+1) = "0 + (p - 1)`[t(E)] + 2(p - 1)`[R* *]. Hence our modified excess given in (3-4) is equal to ep (E; R) = `[t(E)] + 2`[* *R]. Since `[t(E)] = `[E] - "0, we get (5-9). To check the bijection between the set of pairs of sequences (E; R) and the * *set of admissible monomials, let = Q"00Ps11 Q"10Ps21 Q"20. ....Q."r-10Psr1 Q"r0. . . be any admissible monomial. Let E = ("0; "1; : :):and S = (s1; s2; : :):be t* *he exponent sequences associated with the monomial . Then E is a sequence of zeroes and ones which are almost all zero. By the admissibility condition, the* *se integers must satisfy sj - (psj+1 + "j) 0 for all j 1. Let this integer be rj* * and let R = (r1; r2; : :):. Then R is a sequence of non-negative integers which * *are almost all zero. From this calculation, we have R = S - p.t(S ) + t(E ) = (1 - p.t)(S ) - t(E ): Thus, given an admissible monomial in A(p)*, we have obtained a pair of se- quences (E ; R ) with the property stated in Lemma 5-4. For this (E ; R ), the S sequence for the corresponding (E ; R ) given by the formula (5-8) is (R + t(E ))=(1 - p.t) = S , which is the original S-sequence for . Thus the correspo* *n- dence -! (E ; R ) -! (E ; R ) is the identity map. It is also immediate to che* *ck that the correspondence (E; R) -! (E; R) -! (E ; R ) is also the identity map. This proves that the correspondence (5-10) between the set of pairs of sequences (E; R) and the set of admissible monomials in A(p)* is a bijection. This comple* *tes the proof. We remark that `[E] + 2`[R] is called excess in [3]. His excess differs from* * ours by 0 or 1. This difference is essential when we describe free algebra generator* *s of the cohomology rings of Eilenberg-Mac Lane spaces. Since sequences E and R terminate eventually, the associated admissible mono- mial (E; R) also terminates eventually. We examine how terminates depending on the pair of sequences (E; R). Lemma 5-5. Let E = ("0; "1; : :;:"`; 0; : :):be a sequence of zeroes and ones * *such that "` = 1, and R = (r1; r2; : :;:rk; 0; : :):be a sequence of non-negative in* *tegers such that rk > 0. (I) If ` k, then the associated admissible monomial (E; R) ends with Q0 and is of the form (5-11) (E; R) = Q"00P(oe[R]+oe[t(E)])1.Q.".`-10P(r`+"`)1 Q0: (II) If ` < k, then the associated admissible monomial (E; R) ends with a Steenrod reduced power Prk1 , and it is of the form (5-12) (E; R) = Q"00P(oe[R]+oe[t(E)])1.Q.".k-10Prk1 : Proof. This follows immediately from the formula for S given in (5-8). 22 HIROTAKA TAMANOI We want to examine the relationship between QE PR and (E; R) more closely. Given sequences E = ("0; "1; : :):and R = (r1; r2; : :):as before, let (5-13) I(E; R) = ("0; r1; "1; r2; : :;:rk; "k; rk+1 ; : :):: Note that if I(E; R) ends with an entry from E, then we are in (I) of Lemma 5-5 and (E; R) ends with Q0. If I(E; R) ends with an entry from R, then we are in (II) of Lemma 5-5 and (E; R) ends with a Steenrod reduced power. Following Milnor, we introduce a lexicographic total ordering from the right* * on the set of all such sequences I(E; R). Note that for a given pair (E; R), there* * are finitely many pairs (E0; R0) of the same degree such that I(E0; R0) < I(E; R). Proposition 5-6 [cf. 4, Lemma 8]. For any pair of sequences (E; R) as above, X 0 0 (5-14) (E; R) = QE PR + c(E0; R0)QE PR ; I(E0;R0) = c(E0; R0) implies that `[E] = `[E0]. To show this, following Milnor, for k 1, let Mk = Ppk-11 . .P.p1 P1 . Then it is well-known that <; k> is nontrivial only when = Mk, and that <; ok> 6= 0 only when = MkQ0. (See Lemma070in [4].) Now suppose <(E; R); o ER > 6= 0. We apply the diagonal map in the Steenrod algebra `[E0] + `[R0] times to the element (E; R). Each term in the resulting expression consists of `[E0] + `[R0] tensor products. For the nontriviality of* * the above pairing, in this iterated diagonal expression there must be a term such t* *hat `[E0] tensor factors among `[E0] + `[R0] factors contain exactly one Bockstein * *each, due to the above property of dual pairings for k and ok. Since Q0 is primitive, the number of Bocksteins does not change under the diagonal map, and hence it is equal to `[E]. So we must have `[E] = `[E0]. This completes the proof. We can now complete the proof of (5-5) in Theorem 5-2. Completion of the proof of Theorem 5-2. From Theorem 3-4 (I) and Lemma 5-4, the mod p cohomology ring of the mod ph Eilenberg-Mac Lane space K(Z=ph; n+1) can be described as follows in terms of admissible monomials (E; R): h (5-15) H* K(Z=p ; n + 1); Zp = FZp[ (E; R)n+1 | `[t(E)] + 2`[R] < n + 1 ]: Since QE PR n+1 2 H* K(Z=ph; n + 1); Zp for any pair of sequences (E; R), we consider a subalgebra generated by some of these elements: h (5-16) Zp{ QE PR n+1 | `[t(E)] + 2`[R] < n + 1 } H* K(Z=p ; n + 1); Zp : Here, Zp{ . } denotes a subalgebra generated by elements in { . }. There may be some algebraic relations among these generators. Our aim is to show that there * *are no extra relations among the elements QE PR n+1 other than obvious relations th* *at MILNOR BASIS AND COHOMOLOGY OF EILENBERG-MAC LANE SPACES 23 squares of odd degree elements are 0, and the inclusion relation in (5-16) is a* *ctually an identity. To see this, we use Milnor's result (5-14). For any pair of sequ* *ences (E; R) such that `[t(E)] + 2`[R] < n + 1, we have X 0 0 (E; R)n+1 = QE PR n+1 + c(E0; R0)QE PR n+1 : I(E0;R0) 1, Q0n+1 really means ffihn+1 in terms of the h-th Bockstein. But we keep using the notation Q0 in accordance with the convention stated right before Theorem 5-2. By (1) in Corollary 5-7, no more Milnor primitives can act nontrivially. However, some Steenrod reduced powers can still act nontrivially. Nontrivial actions are described as follows. Recall that 0 = (0; 0; : :;:0; : :* *):. Lemma 5-8. Let sequences E and R be as usual. Then 8 Xn >>>E = 0 and R = pjsj-j for some < j=0 QE PR QnQn-1 . .Q.0n+1 6= 0 () > set of mutually distinct non-negative >>: integers{s0; s1; : :;:sn} such that sj j: 24 HIROTAKA TAMANOI Pn When E = 0 and R = pjsj-j for a set {s0; s1; : :;:sn} as above, j=0 PR Qn . .Q.0n+1 = QsnQsn-1. .Q.s0n+1 (5-19) pk = Qsn-k Qsn-1-k . .Q.s1-kQs0-kn+1 6= 0; where k is the smallest integer among sj's, and the last element inside the par* *en- thesis is a polynomial generator of the cohomology ring H* K(Z=ph; n + 1); Zp . Proof. Suppose QE PR QnQn-1 . .Q.0n+1 6= 0. By (2-5), we have X P n k QE PR QnQn-1 . .Q.0n+1 = QE Qn+in . .Q.1+i1Qi0PR- k=0p ik n+1 : (i0;:::;in)0 Since there are n + 1 + `[E] MilnorPprimitives, an element of the above form can be nonzero only when E = 0 and R - nk=0pkik = 0 by (5-3). In this case, P n k P k=0p ik Qn . .Q.0n+1 = Qn+in . .Q.1+i1Qi0n+1 : This element is nonzero if and only if the integers n + in; : :;:1 + i1; i0 are* * mutually distinct. Letting sj = j + ij for 0 j n, we get the first part of the stateme* *nt. The second part follows from (2), (3), and (4) of Corollary 5-7. From Theorem 5-2, an element of the form Qsn. .Q.s1Qs0n+1 for some inte- gers 0 s0 < s1 < . . .< sn is an indecomposable polynomial generator of the cohomology ring if and only if s0 = 0. To deal with all such sequences, we let (5-20) S+n = {(s1; s2; : :;:sn) 2 Zn | 0 < s1 < s2 < . .<.sn}: To each sequence S = (s1; s2; : :;:sn) 2 S+n, we associate an element h (5-21) QS Q0n+1 = Qsn. .Q.s1Q0n+1 2 H* K(Z=p ; n + 1); Zp : Note that this generator has even degree 2(1 + ps1 + ps2 + . .+.psn), and hence* * it is a polynomial generator in the cohomology ring for any S 2 S+n. We consider a subring generated by these elements. Theorem 5-9. The polynomial subalgebra h (5-22) Q = Q(Z=ph; n + 1) = Zp[ QS Q0n+1 | S 2 S+n] H* K(Z=p ; n + 1); Zp is the smallest A(p)*-invariant subalgebra containing the element Qn . .Q.0n+1 * * of degree 2(1 + p + . .+.pn). Any Milnor primitive acts trivially on the subalgebr* *a Q. Proof. Since there are already n + 1 Milnor primitives in the element QS Q0n+1 , no more Milnor primitives can act nontrivially on it. By the derivation proper* *ty of Milnor primitives, they act trivially on the entire polynomial subalgebra Q.* * By Lemma 5-8 and the Cartan formula of the Steenrod reduced powers (2-10), we also see that the above subalgebra is preserved under the action of Steenrod re- duced powers. Thus Q is invariant under the action of the entire Steenrod algeb* *ra A(p)*. The algebra Q contains the element Qn . .Q.0n+1 , and all the other alge- bra generator of Q can be obtained by the action of Steenrod reduced powers as MILNOR BASIS AND COHOMOLOGY OF EILENBERG-MAC LANE SPACES 25 P n PR Qn . .Q.1Q0n+1 = QS Q0n+1 , where R = k=1 pksk-k by (5-19). Thus, any A(p)*-invariant subalgebra of the cohomology ring H* K(Z=ph; n+1); Zp contain- ing the element Qn . .Q.1Q0n+1 should also contain the subalgebra Q. Thus, Q is the smallest A(p)*-invariant subalgebra containing Qn . .Q.1Q0n+1 . This com- pletes the proof. Remark. In [10], it is shown that the Q-subalgebra (5-22) is precisely the imag* *e of the BP-Thom map for the Eilenberg-Mac Lane space: h * h ae* : BP * K(Z=p ; n + 1) -! HZp K(Z=p ; n + 1) : The mod p cohomology ring of the BP-spectrum is the following cyclic module over A(p)* generated by the BP-Thom map ae: HZ*p(BP ) = A(p)*=(Q0; Q1; : :;:Qn; : :):.ae: It follows that the image of the BP-Thom map ae* is always annihilated by Milnor primitives, whatever the space is. This "explains" the fact that the subalgebra* * Q in (5-22) is annihilated by all Milnor primitives. x5.2. Mod p cohomology rings of integral Eilenberg-Mac Lane spaces in terms of the Milnor basis, and Q-subalgebras. We consider the mod p cohomology of the integral Eilenberg-Mac Lane space K(Z; n + 2) which is related to the mod ph Eilenberg-Mac Lane space K(Z=ph; n + 1) by the Bockstein map ffih : K(Z=ph; n + 1) -! K(Z; n + 2) for n 0. The mod p cohomology rings for these spaces are closely related (see Proposition 5-15 below). Since K(Z; 1) ~* *= S1 is homotopically rather trivial, we do not deal with it. Recall that for sequences E = ("0; "1; : :):of zeroes and ones which are alm* *ost all zero, and R = (r1; r2; : :):of non-negative integers which are almost all z* *ero, we defined another sequence I(E; R) in (5-13). Proof of the next theorem is basically the same as the proof of Theorem 5-2. However, the property Q0on+2 = 0 requires extra care for the proof of (5-25). Theorem 5-10. Let n 0 and let p be odd. Let on+2 2 Hn+2 K(Z; n + 2); Zp be the mod p fundamental class. Let E, R, and I(E; R) be as above. Then the following statements hold: (5-23) QE PR on+2 = 0 if `[E] + 2`[R] n + 3; t(E) t(R) p (5-24) QE PR on+2 = Q P on+2 if `[E] + 2`[R] = n + 2 and "0 = 0: The mod p cohomology ring of the integral Eilenberg-Mac Lane space K(Z; n + 2) is a free algebra given by (5-25) fi E R fi`[t(E)] + 2`[R] < n + 2 and H* K(Z; n + 2); Zp = FZp Q P on+2 fifi : I(E; R) ends with an entry from R Proof. For the proof of (5-23) and (5-24), we can apply the same argument used to prove (5-3) and (5-4), since only dimension of fundamental classes is releva* *nt for the argument. 26 HIROTAKA TAMANOI In (3-7), the mod p cohomology ring of an integral Eilenberg-Mac Lane space was described in terms of admissible monomials. First we rewrite this descripti* *on in terms of (E; R)'s. By (5-9), Lemma 5-5, and (5-13), we have (5-26) fi fi`[t(E)] + 2`[R] < n + 2 and H* K(Z; n + 2); Zp = FZp (E; R)on+2 fifi : I(E; R) ends with an entry from R Now we consider the following subalgebra of this cohomology algebra: (5-27)ae fi oe fi`[t(E)] + 2`[R] < n + 2 and Zp QE PR on+2 fifi H* K(Z; n + 2); Zp : I(E; R) ends with an entry from R Here, as before, we are considering a subalgebra generated by elements inside of { . }, which might satisfy some nontrivial algebraic relations. Our aim is to s* *how that there are no nontrivial algebraic relations other than obvious ones coming* * from dimensional reason, and that the inclusion relation in (5-27) is actually an id* *entity. Let (E; R) be such that `[t(E)] + 2`[R] < n + 2 and I(E; R) ends with an entry from R. By Proposition 5-6, we have X 0 0 (5-28) (E; R)on+2 = QE PR on+2 + c(E0; R0)QE PR on+2 ; I(E0;R0) n + 2, then QE PR on+2 is zero by (5-23). So we may restrict the summation in (5-28) to tho* *se pairs (E0; R0) such that `[t(E0)] + 2`[R0] n + 2. If I(E0; R0) ends with an entry from E0, then (E0; R0)on+2 = 0 since (E0; R* *0) ends with Q0 by (I) in Lemma 5-5 and Q0on+2 = 0. Hence Proposition 5-6 implies 0 R0 X 00 00 E00 R00 QE P on+2 = c(E ; R )Q P on+2 ; I(E00;R00)< Qt(E)Pj-1 on+2 if "0 = 0; (5-30) QE Pj on+2 = > (-1)nQE-1 Qjon+2 if "0 = 1 and "j = 0; : 0 if " 0 = 1 and "j = 1: Proof. Suppose `[E] n + 1. If `[R] 1, then `[E] + 2`[R] n + 3 and it follows that QE PR on+2 = 0 by (5-23). If `[R] = 0, then applying Proposition 5-6, we h* *ave X 0 0 (*) QE on+2 = (E; 0)on+2 + c(E0; R0)QE PR on+2 ; I(E0;R0) 0, then this element is either trivial or a p-primary power of QS0on+2 * *for some S0 2 S+n, and hence it is not a free algebra generator of the cohomology r* *ing. If k = 0, then the element is trivial unless the set of integers {j + `j}nj=1is* * a set of mutually distinct integers, in which case the above element is an indecompos* *able polynomial generator of the cohomology ring. Similarly, for any S =P(s1; : :;:sn) 2 S+n, QE PR QS on+2 6= 0 only when E =* * 0 and R is of the form R = nj=1psj`j+ k for some non-negative integers `j and k. If k > 0, then this element is either trivial or a p-primary power of QS00on* *+2 for some S002 S+n, and hence it is not a free generator. If k = 0, then the element* * is trivial unless the set of integers {sj+`j}nj=1is a set of mutually distinct int* *egers, in which case the element PR QS on+2 = Qsn+`n . .Q.s1+`1on+2 is an indecomposable polynomial generator of the cohomology ring. Proof. Since there are already n Milnor primitives acting on on+2 in the eleme* *nt Qn . .Q.1on+2 , no more Milnor primitives can act nontrivially on this element * *by (5-29). Thus, for nontriviality of QE PR Qn . .Q.1on+2 , we must have E = 0. Now repeatedly applying (2-5), we have X n (*) PR Qn . .Q.1on+2 = Qn+`n . .Q.2+`2Q1+`1PR-p `n-...-p`1 on+2 : `1;:::;`n0 Here, PR-pn`n-...-p`1 = 0 unless R - pn`n - . .-.p`1 0. Since the term corresponding to the indices `1; `2; : :;:`n 0 has n Milnor primitives, it can* * be nontrivial only when `[R-pn`n-. .-.p`1] 1 by Proposition 5-11. This implies that the sequence R must be of the form R = k + p`1 + . .+.pn`n for some k 0. In this case, only one term in (*) is nonzero and we have (5-33). When k * * 1, MILNOR BASIS AND COHOMOLOGY OF EILENBERG-MAC LANE SPACES 29 this is a p-th power of another element by the first case in (5-30). Thus it ca* *n be indecomposable only when k = 0, and in this case the resulting element is nonze* *ro and a polynomial generator when the set of integers {1 + `1; 2 + `2; : :;:n + `* *n} is a set of n distinct integers by Proposition 5-12. The second part can be proved in a similar way. Now we can characterize the subalgebra Q in (5-32) in terms of the action of* * the Steenrod algebra A(p)*. Theorem 5-14. Let n 0. The polynomial subalgebra (5-32) Q = Q(Z; n + 2) = Zp[ QS on+2 | S 2 S+n] H* K(Z; n + 2); Zp is the smallest A(p)*-invariant subalgebra containing the element Qn. .Q.1on+2 * *. On this subalgebra Q, all Milnor primitives act trivially. Proof. Since any Milnor primitive acts trivially on any polynomial generator of* * Q by (5-29) in Proposition 5-11, Milnor primitives act trivially on the entire al* *gebra Q by their derivation property. The action of a Steenrod reduced power PR on a polynomial generator QS on+2 for S 2 S+n is described in Lemma 5-13, which shows that we have PR QS on+2 2 Q for any S 2 S+n and for any sequence R of non-negative integers which are almost all zero. From this and the Cartan formu* *la (2-10), it is clear that PR preserves the algebra Q for any R. Hence the polyno* *mial subalgebra Q is invariant under the action of the Steenrod algebra A(p)*. Since all the polynomial generator of Q can be obtained by applying certain Steenrod reduced powers to the lowest positive degree element Qn. .Q.1on+2 by Lemma 5-13, the algebra Q is contained in any A(p)*-invariant subalgebra of the mod p cohomology ring H* K(Z; n + 2); Zp containing the element Qn. .Q.1on+2 . Hence Q is the smallest A(p)*-invariant subalgebra containing Qn . .Q.1on+2 . T* *his completes the proof. Remark. In [10], we have shown that the subalgebra Q is precisely the image of * *the BP-Thom map for the integral Eilenberg-Mac Lane space: * ae* : BP * K(Z; n + 2) -! HZp K(Z; n + 2) : Since any Milnor primitives annihilate the class of the Thom map [ae] 2 HZ0p(BP* * ), it is no wonder that the image algebra Q is annihilated by Milnor primitives. We compare Q-subalgebras Q(Z=ph; n + 1) in (5-22) for various h 1 and Q(Z; n + 2) in (5-32). The relationship among these spaces is supplied by the following homotopy commutative diagram: K(Z; n + 2) ________ K(Z; n + 2) x x (5-34) ??ffih ??ffih+1 K(Z=ph; n + 1) - -ff--!K(Z=ph+1 ; n + 1); where ffih is the h-th Bockstein map and ff is the map induced by an injective * *ho- momorphism Z=ph -! Z=ph+1 , which is multiplication by p. The above diagram (5-34) commutes because ffih is essentially division by ph. The above diagram * *in- duces a commutative diagram on mod p cohomology rings containing the relevant Q-subalgebras. 30 HIROTAKA TAMANOI Proposition 5-15. We have the following isomorphisms of Q-subalgebras: Q(Z; n + 2) ________ Q(Z; n + 2) ?? ? * ~? ffi* (5-35) ~=yffih =y h+1 * Q(Z=ph; n + 1) ---ff-~ Q(Z=ph+1 ; n + 1): = Proof. To avoid possible ambiguity, we denote the fundamental class for the mod ph Eilenberg-Mac Lane space K(Z=ph; n + 1) by (h)n+1. Since ffi*h(on+2 ) = ffi* *h(h)n+1 where ffih on the right hand side is the h-th Bockstein, the commutativity of (* *5-35) implies that ff*(ffih+1 (h+1)n+1) = ffih(h)n+1. Since the cohomology operation* *s commute with induced maps, for any S 2 S+n we have ffi*h(QS on+2 ) = QS ffi*hon+2 = QS * *ffih(h)n+1. Similarly, we have ff*(QS ffih+1 (h+1)n+1) = QS ffih(h)n+1. Thus the maps ffi** *hand ff* in- duce 1:1 correspondences among the sets of polynomial generators of the above Q-subalgebras. Hence they induce isomorphisms of Q-subalgebras. In Theorems 5-9 and 5-14, we considered subalgebras of cohomology algebras i* *n- variant under the entire Steenrod algebra A(p)*. The Steenrod algebra has vario* *us interesting subalgebras. For example, for each positive integer m,mMilnor-consi* *dered1 a subalgebra A[m]* generated by elements Q0, P1 , Pp1 , : :,:Pp 1 . Proposition 5-16 [4, x8 Proposition 2]. For each m 1, the subalgebra A[m]* of the Steenrod algebra is finite dimensional, and its vector space basis over * *Zp is given by the collection of elements of thePform m r (5-36) Q"00. .Q."mmP j=1 jj ; where 0 r1 < pm , 0 r2 < pm-1 , : :,:0 rm < p, and "j = 0; 1 for 0 j m. The above form of Steenrod reduced power is very interesting. For example, we have the following result. Lemma 5-17. Suppose 0 rj < pm-j+1 for 1 j m. Then P m (5-37) [ P j=1rjj ; Q0 . .Q.m] = 0: Proof. By repeatedly applying (2-5), we have P m X P m P m j P j=1rjj Q0 . .Q.m = Qi0Qi1. .Q.imP j=1rjj- j=0p ij-j : ijj0 The term corresponding to (i0; : :;:im ) is nontrivialPif thePcorresponding exp* *onent sequence of P is non-negative, that is, when P mj=0pjij-j mj=1rjj. Since rj < pm+1-j , the onlyPway to have pm im -m mj=1rjj is im = m. Similarly, pm-1 im-1 -(m-1) mj=1rjj implies that im-1 = m - 1 or m. Continuing this process, we see that we must have j ij m for all 0 j m. Since Milnor primitives generate an exterior algebra, Qi0. .Q.im6= 0 only when intege* *rs i0; : :;:im are distinct. This can happen only when ij = j for all 0 j m. In* * the above summation, therePis only one term corresponding to this case, and we have m r P m r P j=1 jj Q0 . .Q.m = Q0 .P.Q.mP j=1 jj : m r This shows that the Steenrod reduced power P j=1 jj , where 0 rj < pm+1-j for 1 j m, commutes with the product Q0. .Q.m of Milnor primitives. This completes the proof. We can give examples of A[m]*-invariant subalgebras of the cohomology algebr* *as. MILNOR BASIS AND COHOMOLOGY OF EILENBERG-MAC LANE SPACES 31 Proposition 5-18. Let X be a topological space. Then (5-38) Q0Q1 . .Q.m.H*(X; Zp) H*(X; Zp) is an A[m]*-invariant subalgebra on which Milnor primitives Q0, : :,:Qm in A[m] act trivially. Proof. It is obvious that Milnor primitives Q0, : :,:Qm act trivially on this s* *ubspace of the cohomology ring over Zp, because they generate an exterior algebra. To s* *ee that the subspace (5-38) is actually a subalgebra, let Q0 . .Q.mx and Q0 . .Q.my be any two elements in this subspace. Then by the derivation property of Milnor primitives, we have (Q0 . .Q.mx).(Q0 . .Q.my) = Q0 . .Q.m[x.(Q0 . .Q.my)]: This shows that the subspace is closed under the cup product. It remains to be shown that the above subspace is invariant under the action of the subalgebra A[m]* of the Steenrod algebra. By Proposition 5-16, we only have to show thatPitmis invariant under the action of the Steenrod reduced power of the form P j=1rjj , where 0 rj < pm-j+1 . But by Lemma 5-17, this form of Steenrod reduced power commutes with the product Q0 . .Q.m. Since obviously the cohomology ring H*(X; Zp) is invariant under the action of Steenrod reduced powers, we see that the above subalgebra in (5-38) is invariant under the actio* *n of A[m]*. This completes the proof. Remark. Let BP be the Wilson spectrum for m 0 and let ae : BP -! HZp be the Thom map [11]. Using Sullivan exact sequences, we see that (5-39) Q0Q1 . .Q.m.H*(X; Zp) Im [ ae*: BP *(X) -! H*(X; Zp) ]; for any space X. Since the mod p cohomology of the spectrum BP is given by HZ*p(BP ) = A(p)*=(Q0; : :;:Qm ).ae, it is clear that the image of the Th* *om map ae is annihilated by the first m + 1 Milnor primitives Q0; Q1; : :;:Qm * *for any space or even for any spectrum X. x6 Mod 2 cohomology rings of Eilenberg-Mac Lane spaces in terms of the Milnor basis, and Q-subalgebras In this section, we describe mod 2 cohomology rings of Eilenberg-Mac Lane spaces in terms of Milnor's Steenrod squares SqR rather than in terms of admiss* *ible monomials as in Theorem 3-6. To describe the relationship between Milnor basis elements and admissible mo* *no- mials, we repeatedly apply the factorization formula (4-10). Let R = (r1; r2; r* *3; : :): is a sequence of non-negative integers which are almost all zero. We have SqR = Sqoe[R]1.Sqt(R)+ (other terms ) 2(R) = Sqoe[R]1.Sqoe[t(R)]1.Sqt + (other terms ) .. . = (R) + (other terms ); 32 HIROTAKA TAMANOI where (R) is a monomial in Steenrod squares given by r(R)] (R) = Sqoe[R]1.Sqoe[t(R)]1. .S.qoe[t .1. . (6-1) = Sqs11 .Sqs21 . .S.qsr1 . .;. where S = (s1; s2; : :):is a sequence of non-negative integers sr = oe[tr-1 (R* *)] for r 1 which are almost all zero. Recall from (4-2) that for a sequence of non-negative integers R = (r1; r2; r3; : :):which are almost all zero, we defin* *ed a weighted sum oe[R] 2 Z and a shifted sequence t(R) by X oe[R] = r1 + 2r2 + 22r3 + . .=. 2j-1rj; t(R) = (r2; r3; r4; . .).: j1 Recall that a monomial as in (6-1) is calledPadmissible if sj 2sj+1 for j 1, and its excess e2() is defined by e2() = j1 (sj - 2sj+1). Note that e2() is t* *he p = 2 version of __ep() given in (3-3), rather than ep() given in (3-4). Lemma 6-1. For any sequence R = (r1; r2; r3; : :):of non-negative integers whi* *ch are almost all zero, the monomial (R)Pgiven in (6-1) is admissible and the exce* *ss of (R) is given by e2 (R) = `[R] = j1 rj. The sequences R and S of non-negative integers in (6-1) correspond to each o* *ther in 1 : 1 manner, and they are related by the following formulae: ( 2 2 R S = R + 2t(R) + 2 t (R) + . .=.______1;- 2t R = S - 2t(S) = (1 - 2t)(S): Thus there is a 1 : 1 correspondence between the set of sequences R of non-nega* *tive integers which are almost all zero and the set of admissible monomials in Steen* *rod squares. Proof. To check the admissibility, we calculate sj - 2sj+1. Since sj = oe[tj-1* *(R)] for r 1, we have sj - 2sj+1 = oe[tj-1(R)] - 2oe[tj(R)] = rj, which is non-nega* *tive because R is a non-negative sequence. Hence thePmonomial (R) in (6-1)Pis admis- sible. Its excess is then given by e2 (R) = j1 (sj - 2sj+1) = j1 rj = `[R* *]. The relationship between the two sequences R and S is straightforward to prove,* * so is the 1:1 correspondence of non-negative finite sequences R and admissible mon* *o- mials in A(2)*. On the set of all sequences R of non-negative integers which are almost all * *zero, we introduce a lexicographic total ordering from the right. We can then be more precise about the relationship between admissible monomials (R) and Milnor basis elements SqR . Proposition 6-2. For any sequence R as above, we have X 0 (6-2) (R) = SqR + c(R0)Sq R : R0 n + 1; t(R) 2 (6-5) Sq Rn+1 = Sq n+1 if `[R] = n + 1: In terms of Milnor primitives and Steenrod reduced powers, equivalent statements are given as follows: (6-40) QE PR n+1 = 0 if `[E] + 2`[R] > n + 1; t(E) t(R) 2 (6-50) QE PR n+1 = Q P n+1 if `[E] + 2`[R] = n + 1: 34 HIROTAKA TAMANOI The mod 2 cohomology ring of K(Z=2h; n + 1) is a polynomial algebra described in terms of the Milnor basis as follows: h R H* K(Z=2 ; n + 1); Z2 = Z2[ Sq n+1 | `[R] < n + 1 ] (6-6) = Z2[ QE PR n+1 | `[E] + 2`[R] < n + 1 ]: Proof. To prove (6-4) and (6-5), we apply the factorization formula (4-10) of S* *qR to the fundamental class n+1 . We have X 2 (*) SqR n+1 = c(S)Sq (oe[R]+oe[t(S)])1.Sqt(R)-t(S)+tn(S)+1: S2R;s1=0P 0t(S) i1 ti(R) We compare the degree of Sq t(R)-t(S)+t2(S)n+1 and oe[R] + oe[t(S)] and use the unstable property of Steenrod squares (2-17). By Lemma 6-3, we have fi 2 fi (**) fiSqt(R)-t(S)+t (S)n+1 fi= oe[R] - oe[t(S)] + (n + 1 - `[R]): When `[R] > n + 1, the above is strictly less than oe[R] + oe[t(S)] for any S. * *Hence all terms in the summation (*) vanish and we have SqR n+1 = 0. This proves (6-4* *). If `[R] = n + 1, then we have (**) < oe[R] + oe[t(S)] except for the case S = 0. Thus only one term in (*) remains, and by degree reason this term gives t(R) t(R) 2 Sq Rn+1 = Sqoe[R]1 Sq n+1 = Sq n+1 ; fi fi since fiSqt(R)n+1 fi= oe[R]. This proves (6-5). The equivalent statements (6-40* *) and (6-50) can be derived using the identity (2-13). To show (6-6), we first rewrite the description of mod 2 cohomology rings of* * mod 2h Eilenberg-Mac Lane spaces in terms of sequences R. Since we are assuming that n 1, or n = 0 and h = 1, the cohomology ring for K(Z=2h; n + 1) is given by (3-10), and together with Lemma 6-1 we have h (6-7) H* K(Z=2 ; n + 1); Z2 = Z2[ (R)n+1 | `[R] < n + 1 ]: We consider the following subalgebra generated by some of the elements defined * *by the Milnor basis: h (6-8) Z2{ SqRn+1 | `[R] < n + 1 } H* K(Z=2 ; n + 1); Z2 : Our aim is to show that there are no algebraic relations among these generators, and that the subalgebra (6-8) coincides with the entire cohomology ring. Applyi* *ng Proposition 6-2 for R with `[R] < n + 1 to the fundamental class n+1 , we have X 0 (R)n+1 = SqR n+1 + c(R0)Sq R n+1 : R0 n + 1; t(E) 2 (6-10) QE n+1 = Q n+1 6= 0 if `[E] = n + 1: In particular, if E = (0; : :;:0; "j-1; "j; : :):with "j-1 = 1 and `[E] = n + 1* *, then tj(E) 2j (6-11) QE n+1 = Q n+1 6= 0; where the element Qtj(E)n+1 2 H* K(Z=2h; n + 1); Z2 with `[tj(E)] = n is an indecomposable polynomial generator. Proof. Straightforward from (6-40) and (6-50). Since QE n+1 with `[E] = n is a polynomial generator by (6-6), all of its 2j-th powers are nontrivial, which im* *plies nontriviality of the element in (6-10). In particular, for any sequence S = (s1; s2; : :;:sn+1 ) of n + 1 strictly i* *ncreasing non-negative integers, the corresponding element QS n+1 = Qsn+1. .Q.s1n+1 is no* *n- trivial by (6-11). The element of this type with the smallest degree is Qn . .Q* *.0n+1 of even degree 2(1 + 2 + 22 + . .+.2n). One can obtain any element of the above type QS n+1 from the element Qn . .Q.0n+1 by the action of Steenrod squares as shown in the next lemma. Lemma 6-6. Let n 1, or n = 0 and h = 1. Let R be a sequence of non-negative integers which are almost all zero.P Then PR QnQn-1 . .Q.0n+1 6= 0 only if the sequence R is of the form R = nj=02j`j for some non-negative integers `j. In this case, we have P n j (6-12) P j=02 `j QnQn-1 . .Q.0n+1 = Qn+`n. .Q.1+`1Q`0n+1 : This element is nontrivial if the set of integers {`0; 1 + `1; : :;:n + `n} is * *a set of distinct non-negative integers. Similarly, for any sequence of non-negative integers 0 s1P< . . .< sn+1 , we have PR Qsn+1 . .Q.s1n+1 6= 0 only if R is of the form R = n+1j=12sj`j for so* *me non-negative integers `j. In this case, P n+1 sj (6-120) P j=12 `j Qsn+1. .Q.s1n+1 = Q(sn+1+`n+1). .Q.(s1+`1)n+1 : 36 HIROTAKA TAMANOI Proof. From the general formula (2-14), for any sequence R we have X P n j (*) PR Qn. .Q.1Q0n+1 = Qn+`n. .Q.1+`1Q`0PR- j=02 `j .n+1 : `0;:::;`n0 By (6-40), QE PR0n+1 = 0 when `[E] = n + 1 and R0 6= 0. Thus a nontrivial term can result in the above summation (*) only when the exponent sequence ofPP on the right hand side vanishes, that is, only when R is of the form R = nj=02j`j for some `0; : :;:`n 0. In this case, all other terms in the summation on the * *right hand side of (*) vanish and we have PR Qn. .Q.1Q0n+1 = Qn+`n. .Q.1+`1Q`0n+1 : By (6-10), this element is nontrivial when the integers j + `j are all distinct* *. This proves the first half of the lemma. The second part can be proved in a similar way. Now we consider a subalgebra of H* K(Z=2h; n + 1); Z2 generated by elements obtained by applying maximum number of Milnor primitives on the fundamen- tal class. For any sequence S = (s1; s2; : :;:sn) 2 S+n of strictly increasing* * posi- tive integers, let QS Q0n+1 = Qsn . .Q.s1Q0n+1 as before. Its degree is given * *by |QS Q0n+1 | = 2(1+2s1+2s2+. .+.2sn). For S as above, let t(S) = (s1-1; : :;:sn-* *1) be a sequence of strictly increasing non-negative integers. Theorem 6-7. Let n 1, or n = 0 and h = 1. The subalgebra Q of the cohomology ring H* K(Z=2h; n + 1); Z2 given by (6-13) Q = Q(Z=2h; n + 1) = Z2[ QS Q0n+1 | S 2 S+n] is the smallest A(2)*-invariant polynomial subalgebra of [H* K(Z=2h; n + 1); Z2* * ]2 containing Qn . .Q.1Q0n+1 . Any element in this algebra is a square and we have 2 + QS Q0n+1 = Qt(S)n+1 ; for S 2 Sn : This subalgebra is annihilated by any Milnor primitive Qj for j 0. Proof. Any generator QS Q0n+1 for S 2 S+n of the subalgebra Q has already n + 1 Milnor primitives in it, so no more Milnor primitives can act nontrivially. By * *the derivation property of Milnor primitives, all Milnor primitives annihilate the * *entire subalgebra Q. By (6-11) and (6-120), any Steenrod reduced power operation acting on an el- ement QS Q0n+1 produces either a trivial element or a 2k-th power of another element QS0Q0n+1 for some k 0 and for some S0 2 S+n. Since Qj's act trivially on the subalgebra Q, the Cartan formula (2-16) for SqR reduces to a Cartan for- mula for the Steenrod reduced powers PR = Sq2R when these operators act on Q. Hence Q is preserved under the action of the entire mod 2 Steenrod algebra A(2)* **. By (6-10), all generators in the algebra Q are squares. Since the characteri* *stic is 2, the algebra Q is a subalgebra of [H* K(Z=2h; n + 1); Z2 ]2. Since all algebra generators of Q are obtained by applying Steenrod reduced powers to the element Qn. .Q.1Q0n+1 by Lemma 6-6, the subalgebra Q is con- tained in any A(2)*-invariant subalgebra containing the element Qn. .Q.1Q0n+1 . Hence Q is the smallest A(2)*-invariant subalgebra containing Qn . .Q.1Q0n+1 . MILNOR BASIS AND COHOMOLOGY OF EILENBERG-MAC LANE SPACES 37 It remains to be shown that elements QS Q0n+1 are polynomial generators of the algebra Q, that is, there are no algebraic relations among these elements i* *n the subalgebra Q. To see this, suppose we have an algebraic relation of the form P (QS1Q0n+1 ; QS2Q0n+1 ; : :;:QS`Q0n+1 ) = 0 for some nontrivial polynomial P (x1; x2; : :;:x`) over Z2 in ` variables, and * *for some sequences S1; S2; : :;:S` 2 S+n. Since QSjQ0n+1 = (Qt(Sj)n+1 )2 for 1 j `, and since the ring H* K(Z=2h; n + 1); Z2 is a polynomial algebra over the fiel* *d of characteristic 2, we can take the unique square root of the above relation: P (QS01n+1 ; QS02n+1 ; : :;:QS0`n+1 ) = 0: This means that there is a nontrivial algebraic relation among polynomial gener* *a- tors QS0jn+1 of the cohomology ring H* K(Z=2h; n+1); Z2 . This is a contradicti* *on to our description of the cohomology ring given in (6-6) which says that elemen* *ts of the form QSjn+1 where `[Sj] = n must be polynomial generators for any sequence Sj of strictly increasing n non-negative integers, and hence there cannot be any algebraic relation. Thus elements QS Q0n+1 must be algebraically independent within the ring Q. Consequently they are polynomial generators of Q. We emphasize that although the elements QS Q0n+1 for S 2 S+nare algebraically independent in the ring Q, all of these elements are squares in the cohomology * *ring H* K(Z=2h; n + 1); Z2 and hence they are decomposable in this larger ring. x6.2. Mod 2 cohomology rings of integral Eilenberg-Mac Lane spaces in terms of the Milnor basis, and Q-subalgebras. Next, we describe the mod 2 cohomology ring of the integral Eilenberg-Mac Lane space K(Z; n + 2) for n 0 in terms of Milnor basis elements, and we describe its various properties. Rec* *all that I(E; R) was defined in (5-13). Theorem 6-8. Let n 0 and let on+2 2 Hn+2 K(Z; n + 2); Z2 be the mod 2 fundamental class. Then the following identities hold: (6-14) SqR on+2 = 0 if `[R] > n + 2; t(R) 2 (6-15) SqR on+2 = Sq on+2 if `[R] = n + 2: These identities can be restated in terms of Milnor primitives and Steenrod red* *uced powers as follows: (6-140) QE PR on+2 = 0 if `[E] + 2`[R] > n + 2; t(E) t(R) 2 (6-150) QE PR on+2 = Q P on+2 if `[E] + 2`[R] = n + 2: The mod 2 cohomology ring of K(Z; n + 2) is a polynomial algebra given by (6-16) fifi`[R] < n + 2 and rk > 1 if H* K(Z; n + 2); Z2 = Z2 SqR on+2 fifi R = (r1; : :;:rk; 0; : :):with rk 6= 0 fifi`[E] + 2`[R] < n + 2 and I(E; R) = Z2 QE PR on+2 fifi : ends with an entry from R 38 HIROTAKA TAMANOI Proof. Identities (6-14) and (6-15) can be proved in a way similar to the proof* * of (6-4) and (6-5), since only dimension of cohomology classes are relevant for the argument. In (3-11), a description of the cohomology ring H* K(Z; n + 2); Z2 is given* * in terms of admissible monomials in the mod 2 Steenrod algebra. We first rewrite t* *his description in terms of (R) given in (6-1). Let R = (r1; r2; : :;:rk; 0; : :):* *be any sequence of non-negative integers with rk 6= 0. Then (R) is of the form k-1(R)] (R) = Sqoe[R]1.Sqoe[t(R)]1. .S.qoe[t ;1 where Sqoe[tk-1(R)]1= Sqrk1 . Since (R) is always admissible for any sequence R and its excess e2 (R) is given by `[R] by Lemma 6-1, the description of (3-11)* * in terms of admissible monomials can be rewritten as follows: (6-17) fifi`[R] < n + 2 and rk > 1 if H* K(Z; n + 2); Z2 = Z2 (R)on+2 fifi : R = (r1; : :;:rk; 0; : :):with rk 6= 0 We consider the following subalgebra of the cohomology ring: (*) Z2{ SqRon+2 | `[R] < n + 2; R does not end with 1 } H* K(Z; n + 2); Z2 : Our objective is to show that this subalgebra in fact coincides with the entire cohomology ring, and that the generators are algebraically independent. Let R be a sequence with `[R] < n + 2 and ending with an integer greater than 1. Applying the identity in Proposition 6-2 to the fundamental class on+2 and using (6-14),* * we can write X 0 (R)on+2 = SqR on+2 + c(R0)Sq R on+2 : R0< Qs2-s1 . .Q.sn-s1Qj-s1on+2 ifj > s1; (6-19) QE Pj on+2 = > 0 ifj = s1; : 2j Qs1-jQs2-j . .Q.sn-jon+2 ifj < s1: For the same E as above, suppose `[R] 2. Then QE PR on+2 = 0. (II) Suppose `[E] n + 1. Then for any sequence R, we have QE PR on+2 = 0. Proof.PBy (2-12), we have Qsn = Psn Q0 - Q0Psn . For E as in (I), let E0 = n-1 E E0 sn j=1sj+1. Since Q0on+2 = 0, we have Q on+2 = Q Q0P on+2 . Since the excess of this element is given by `[E0] + 1 + 2`[R] = n + 2, by (6-150) this is further equal to Qt(E0)Psn-1 on+2 2. Since the element inside of the parenth* *esis is a polynomial generator of the cohomology ring by (6-16), its square is nonze* *ro. This proves (6-18). For (6-19), we can apply (6-150) as long as the excess of * *the cohomology operation is n + 2. Suppose s1 < j. Then we have 2s1 QE Pj on+2 = Q0Qs2-s1 . .Q.sn-s1Pj-s1 on+2 2s1 = Qs2-s1 . .Q.sn-s1Qj-s1on+2 : Here we used Q0Pj-s1 on+2 = Qj-s1on+2 . When j = s1, we do not have the reduced power term in the above calculation and due to the identity Q0on+2 = 0, the above vanishes. When s1 > j, applying (6-150) j times, we have 2j QE Pj on+2 = Qs1-j . .Q.sn-jon+2 ; since P0 = 1. This proves (6-19). When `[E] = n and `[R] 2, it follows that `[E] + 2`[R] n + 4. Hence by (6-140) we have QE PR on+2 = 0.P For (II), when `[E] = n + 1 and R = 0, we let E = s0+1 + nj=1sj+1 for 0 s0 < . .<.sn. Then using (6-18), we have 2 QE on+2 = Qs0 Qs1-1 . .Q.sn-1-1Psn-1 on+2 = 0; because Qs0 acts as a derivation. When `[E] = n + 1 and `[R] 1, we have `[E] + 2`[R] n + 3. Hence by (6-140) we have QE PR on+2 = 0. This completes the proof of Proposition 6-9. Note that the elements inside of the parenthesis on the right hand side of (* *6-19) is a square by (6-18), if it is nontrivial. From (II) of Proposition 6-9, no n + 1 products of Milnor primitives can act* * non- trivially on the fundamental class on+2 , whereas any product of n distinct Mil* *nor 40 HIROTAKA TAMANOI primitives can act nontrivially on on+2 by (6-18) of Proposition 6-9. The colle* *ction of such elements {QS on+2 | S 2 S+n} is a set of algebraically independent ele- ments, which can be seen in a similar way as in the proof of Theorem 6-7, altho* *ugh such elements are always squares of some other elements in the cohomology ring H* K(Z; n + 2); Z2 by (6-18). So these elements generate a polynomial subalge- bra of the cohomology ring. The lowest degree element among such elements is Qn. .Q.2Q1on+2 of degree 2(1 + 2 + 22 + . .+.2n). We examine the action of the Steenrod algebra A(2)* on this element. Lemma 6-10. Let n 0 and let on+2 2 Hn+2 K(Z; n+2); Z2 be the fundamental class. Let R be a sequence of non-negative integers which are almostPall zero. * *Then PR Qn . .Q.2Q1on+2 6= 0 only when the sequence R is of the form R = nj=02j`j for some non-negative integers `j for 0 j n. In this case, P n j P j=02 `j Qn. .Q.2Q1on+2 = Qn+`n . .Q.2+`2Q1+`1P`0 on+2 ; whose actual value in terms only of Milnor primitives is determined by (6-19). Consequently, for any sequence R, (6-20) PR Qn. .Q.1on+2 2 Q = Z2[ QS on+2 | S 2 S+n]: In particular, for any sequence S = (s1; s2; : :;:sn) 2 S+n of strictly increas* *ing n positive integers, we have P n j (6-21) QS on+2 = P j=12 sj-j Qn. .Q.1on+2 : Proof. Applying (2-14) repeatedly, we have X P n j (*) PR Qn . .Q.2Q1on+2 = Qn+`n. .Q.2+`2Q1+`1PR- j=12 `j on+2 : `1;:::;`n0 Since there are n Milnor primitives in each term of the right handPside, for no* *ntriv- iality of the term corresponding to `1; : :;:`n we must haveP|R - nj=12j`j| 1 by (6-140). Thus the sequence R must be of the form R = nj=12j`j+ `0 for some `0 0. In this case, all other terms on the right hand side of (*) vanish * *and P n j (**) P j=02 `j Qn . .Q.2Q1on+2 = Qn+`n. .Q.2+`2Q1+`1P`0 on+2 : Suppose `0 = 0. The element (**) is trivial unless the set of integers {j + `j}* *nj=1is a set of mutually distinct integers, in which case this element is in the subal* *gebra Q in (6-20). If `0 > 0, then by (6-19) this element is either trivial or a 2k-t* *h power of an element of the form QS on+2 for some S 2 S+n and for some k. In either ca* *se, elements of the form (*) are in the subalgebra Z2[ QS on+2 | S 2 S+n]. This pro* *ves (6-20). The formula (6-21) is a straightforward consequence of (**). The subalgebra in (6-20) has a very special property and it is of particular interest. MILNOR BASIS AND COHOMOLOGY OF EILENBERG-MAC LANE SPACES 41 Theorem 6-11. Let n 0 and let on+2 2 Hn+2 K(Z; n + 2); Z2 be the funda- mental class. Then the subalgebra + * (6-22) Q = Q K(Z; n + 2) = Z2[ QS on+2 | S 2 Sn ] H K(Z; n + 2); Z2 is the smallest A(2)*-invariant polynomial subalgebra of [H* K(Z; n+2); Z2 ]2 c* *on- taining the element Qn. .Q.2Q1on+2 . If S = (0 < s1 < s2 < . .<.sn), then 2 (6-23) QS on+2 = Qs1-1. .Q.sn-1-1Psn-1 on+2 : The element inside of the parenthesis on the right hand side is a polynomial ge* *nera- tor of the cohomology ring. The subalgebra Q is annihilated by the Milnor primi* *tive Qj for any j 1. Proof. Since there are already n Milnor primitives in QS on+2 , no more Milnor primitives can act nontrivially on QS on+2 for any S 2 S+n by (II) of Proposit* *ion 6-9. Since Milnor primitives act as derivations, they act trivially on the ent* *ire subalgebra (6-22). (This can also be concluded from the fact that every element* * of the form QS on+2 for S 2 S+n is a square and Milnor primitives act as derivatio* *ns.) As for the action of Steenrod reduced powers, we recall that for each S 2 S+n, there is a sequence RS such that PRS Qn. .Q.1on+2 = QS on+2 by (6-21). Thus for any sequence R, we have PR QS on+2 = PR PRS Qn .0.Q.1on+20. We can rewrite the element PR PRS 2 A(2)* in terms of a basis {QE PR } of A(2)*. Using (6-20), we then conclude that the element PR QS on+2 is in the algebra Q. This shows that elements resulting from the action of A(2)* on any generator QS on+2 are in the algebra Q. Using the Cartan formula (2-16), we see that A(2)* preserves the ent* *ire subalgebra Q. Since all algebra generators of Q are obtained from Qn . .Q.1on+2 by applying certain Steenrod reduced powers in A(2)* as in (6-21), Q is contain* *ed in any A(2)*-invariant subalgebra containing the element Qn . .Q.1on+2 . Hence Q is the smallest A(2)*-invariant subalgebra containing Qn . .Q.1on+2 . The form* *ula (6-23) is obtained by rewriting (6-18). This completes the proof. As in odd prime case, the Q-subalgebras in (6-13) and (6-22) are isomorphic * *to each other. Proposition 6-12. Let ffih : K(Z=2h; n + 1) -! K(Z; n + 2) be the h-th Bockste* *in map for h 1 and n 0. Then ffih induces an isomorphism between Q-subalgebras: ~= h (6-24) ffi*h: Q(Z; n + 2) -! Q(Z=2 ; n + 1): Proof. The proof is the same as the proof of Proposition 5-15. References [1] H. Cartan, Sur les groupes d'Eilenberg-Mac Lane I, II, Proc. Nat. Acad. Sci* *. USA 40 (1954), 467-471 and 704-707. [2] D. C. Johnson and W. S. Wilson, The projective dimension of the complex bor* *dism of Eilenberg-Mac Lane spaces, Osaka J. Math. 14 (1977), 533-536. [3] D. Kraines, On excess in Milnor basis, Bull. London Math. Soc. 3 (1971), 36* *3-365. [4] J. W. Milnor, The Steenrod algebra and its dual, Ann. of Math. 67 (1958), 1* *50-171. [5] J. W. Milnor, On the cobordism ring * and a complex analogue, Amer. J. Math* *. 82 (1960), 505-521. 42 HIROTAKA TAMANOI [6] J W. Milnor and J. C. Moore, On the structure of Hopf algebras, Ann. of Mat* *h. 81 (1965), 211-264. [7] D. C. Ravenel and W.S.Wilson, The Morava K theories of Eilenberg Mac Lane s* *paces and the Conner-Floyd conjecture, Amer. J. Math. 102 (1980), 691-748. [8] J. P. Serre, Homologie singuliere des espace fibres, Ann. of Math. 54 (1951* *), 425-505. [9] N. E. Steenrod and D. B. A. Epstein, Cohomology Operations, Annals of Math.* * Studies 50, Princeton Univ. Press, Princeton NJ, 1974. [10]H. Tamanoi, The image of BP Thom map for Eilenberg-Mac Lane spaces, Transac* *tions of AMS 349 (1997), 1209-1237. [11]W. S. Wilson, The -spectrum for Brown-Peterson cohomology, Part I, Comment.* * Math. Helv. 48 (1973), 45-55; Part II, Amer. J. Math. 97 (1975), 101-123. [12]W. S. Wilson, Brown-Peterson Homology: An Introduction and Sampler, CBMS Re* *gional Conference Series in Math., no 48, AMS, Providence, Rhode Island, 1982. Santa Cruz, CA 95064, USA E-mail address: tamanoi@math.ucsc.edu