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
*