GLOBAL STRUCTURE OF THE MOD TWO
SYMMETRIC ALGEBRA, H*(BO; F2), OVER THE
STEENROD ALGEBRA
DAVID J. PENGELLEY AND FRANK WILLIAMS
The first author dedicates this paper to his parents,
Daphne M. and Eric T. Pengelley, in memoriam.
Abstract. The algebra S of symmetric invariants over the field
with two elements is an unstable algebra over the Steenrod alge-
bra A, and is isomorphic to the mod two cohomology of BO, the
classifying space for vector bundles. We provide a minimal presen-
tation for S in the category of unstable A-algebras, i.e., minimal
generators and minimal relations.
From this we produce minimal presentations for various unstable
A-algebras associated with the cohomology of related spaces, such
as the BO(2m - 1) that classify finite dimensional vector bundles,
and the connected covers of BO. The presentations then show that
certain of these unstable A-algebras coalesce to produce the Dick-
son algebras of general linear group invariants, and we speculate
about possible related topological realizability.
Our methods also produce a related simple minimal A-module
presentation of the cohomology of infinite dimensional real projec- __
tive space, with filtered quotients the unstable modules F (2p- 1)=AA p-2,
as described in an independent appendix.
December 1, 2003
1. Introduction
We continue our study [9] of invariant algebras as unstable algebras
over the Steenrod algebra A by proving a structure theorem for the
algebra S of symmetric invariants over the field F2. The algebra S is
isomorphic to the mod two cohomology of BO, the classifying space
for vector bundles [8], and we identify the two. We also make several
applications to the cohomology of related spaces, which then reveal a
relationship between S and the Dickson algebras [13].
___________
1991 Mathematics Subject Classification. Primary 55R45; Secondary 13A50,
16W22, 16W50, 55R40, 55S05, 55S10.
Key words and phrases. Symmetric algebra, Steenrod algebra, unstable algebra,
classifying space, Dickson algebra, BO, real projective space.
1
2 DAVID J. PENGELLEY AND FRANK WILLIAMS
Our goal is to provide a minimal presentation for S = H*(BO; F2) in
the category of unstable A-algebras [11], beginning with a minimally
presented generating A-module and then introducing a minimal set
of A-algebra relations. This reveals how a minimal set of A-module
building blocks for S fit together in its A-algebra structure. In brief,
our main result (Theorem 3.5) is that S = H*(BO; F2) is minimally
presented in the category of unstable A-algebras as the free unstable
A-algebra on the two-power Stiefel-Whitney classes w2k modulo rela-
i
tions expressing the fact that, for each i k - 2, Sq2 w2k differs from
k-1 2i
Sq2 Sq w2k-1by a decomposable. (By contrast, and at first seem-
ingly paradoxically, we shall also see (Theorem 2.3) that while S is
generated as an A-algebra by {w2k: k 0}, with relations linking the
resulting algebra generators, in fact the A-submodule of S generated
by {wm : m 0} is a free unstable A-module on all the Stiefel-Whitney
classes.)
We apply this structure theorem to characterize similarly the coho-
mology images B*(n) for the connected covers of BO (Theorem 4.2)
[3], which include the full cohomology algebras of BSO, BSpin, and
BO <8>. We likewise characterize the quotients H*(BO(q); F2) for the
classifying spaces of finite dimensional vector bundles [8], and in par-
ticular (Theorem 4.3) we analyze H*(BO(2n+1 - 1); F2).
Finally, we shall produce an A-algebra epimorphism from S = H*(BO; F2)
to each of the mod two Dickson algebras (Theorem 4.4), which we
characterized in [9] as unstable A-algebras. In fact we shall show that
the (n + 1)-st Dickson algebra has the role of capturing precisely the
quotient of S = H*(BO; F2) common to the cohomology of the n-th
distinct connected cover BO and to BO(2n+1-1). We speculate
about how this phenomenon may relate to spaces beyond the range in
which Dickson algebras are directly realizable topologically.
Our minimal A-algebra presentations for all the above objects will
devolve naturally from our main presentation of S, and in that sense
these A-algebras are all äp rallel" to the main presentation.
In Appendix I, which can be read independently of the rest of the
paper, we present_a related result, in which the unstable A-modules
F (2p - 1)=AA p-2appear as the filtered quotients of a simple minimal
A-presentation for H*(RP 1; F2). We thank Don Davis, Kathryn Lesh,
and Haynes Miller for useful conversations regarding these modules.
2. Motivation, first steps, and a plan
The unstable A-algebra of symmetric invariants S = H*(BO; F2) is
a polynomial algebra F2[wm : m 0, w0 = 1], with each elementary
STRUCTURE OF MOD TWO SYMMETRIC ALGEBRA 3
symmetric function (Stiefel-Whitney class) wm having degree m [8].
The action of the Steenrod algebra is completely determined from the
Wu formulas [3, 12, 14]
Xj ` '
m - j + l - 1
Sqjwm = wj-lwm+l
l=0 l
and the Cartan formula on products [11].
To ease into our categorical point of view, and to illustrate our
approach and methods, let us begin by seeing that abstract Stiefel-
Whitney classes, taken all together as free unstable A-algebra genera-
tors, along with imposed üW formulas", actually "present" S. This is
something one might easily take for granted, but should actually prove,
since in principle there might be ö ther" relations lurking in S beyond
those inherent in the Wu formulas. To avoid confusion from notational
abuse, we build from abstract classes tm which will correspond to the
actual Stiefel-Whitney classes under an isomorphism.
Proposition 2.1 (Wu formulas present S). The unstable A-algebra S =
H*(BO; F2) is isomorphic to the quotient of the abstract free unstable
A-algebra on classes tm in each degree m 1, modulo the left A-ideal
generated by abstract üW formulas" formed by writing t's in place of
w's in the Wu formulas above.
Proof.Iterating the abstract Wu formulas via the Cartan formula shows
that the abstract classes {tm : m 1} actually generate the abstract
A-algebra quotient considered merely as an algebra, i.e., its (algebra)
indecomposable quotient has rank at most one in each degree. On the
other hand, by its construction the abstract A-algebra quotient must
map onto S by sending each tm to wm , since the respective Wu formu-
las correspond. Thus the two must be isomorphic, since S is free_as_a
commutative algebra. |__|
Notice, however, that this presentation of S is far from minimal in the
category of unstable A-algebras, since it used vastly more generators
than needed. What we seek instead is to achieve three features for a
minimal presentation:
Step 1: Find a minimal A-submodule of S that will generate S as an
A-algebra.
Step 2: Find a minimal presentation of this A-submodule, i.e., with min-
imal generators and minimal relations.
Step 3: Form the free unstable A-algebra U on this module, and find
minimal relations on U so that its A-algebra quotient produces S.
4 DAVID J. PENGELLEY AND FRANK WILLIAMS
To begin, let us find a minimal set of A-algebra generators for S.
Consider the (algebra) indecomposable quotient QS, i.e., the vector
space with basis {wm : m 1} and induced A-action
` '
m - 1
Sqjwm = wm+j .
j
m-1
Since j is always zero mod two when m + j is a two-power, and
never zero when m is a two-power and j is less than m, we see that the
A-module indecomposables of QS have basis exactly {w2k: k 0}.
Since our philosophy is to begin the presentation at the A-module
level, with minimal A-algebra generators and minimal module rela-
tions, we thus start with
Definition 2.2. Let M be the free unstable A-module on abstract
classes {t2k : k 0}, where subscripts indicate the topological degree
of each class.
We wish to map M to S via t2k! w2k, and need first to ask whether
M injects. In other words, is the A-submodule of S = H*(BO; F2)
generated by {w2k : k 0} free? Or are there, to the contrary, A-
relations amongst the two-power Stiefel-Whitney classes, which will
compel us to introduce module relations on M in order to complete
steps 1 and 2 above? The Wu formulas appear to suggest that no such
relations exist. In fact we can prove something even stronger.
Theorem 2.3 (Stiefel-Whitney classes inject freely).The A-submodule
of S = H*(BO; F2) generated by {wm : m 0} is free unstable on these
classes.
The proof is in Section 5.
Remark 2.4. The proof also shows that in
H*(BO(q); F2) ~=H*(BO; F2)= (wm : m > q),
the A-submodule generated by {wm : 0 m q} is free unstable on
these classes.
Remark 2.5. The fact that the free unstable A-module Fm on a sin-
gle class in degree m injects into H*(BO(m); F2) on the class wm is
clear from the already known result [4, p. 55] that Fm is isomor-
m
phic to the invariants F1m , which clearly inject naturally into
(H*(RP 1; F2) m ) m ~=H*(BO(m); F2) on wm . Theorem 2.3 general-
izes this by handling all Fm simultaneously, showing that they do not
interfere when simultaneously perched on the Stiefel-Whitney classes
in the symmetric algebra S = H*(BO; F2).
STRUCTURE OF MOD TWO SYMMETRIC ALGEBRA 5
Corollary 2.6. The A-submodule of S = H*(BO; F2) generated by
{w2k: k 0} is free unstable, so M injects naturally into S.
This completes steps 1 and 2 of our goal, and we can begin step 3.
Definition 2.7. Let U be the free unstable A-algebra on M, in other
words, U is the free unstable A-algebra on abstract classes {t2k: k 0}.
Clearly U maps via t2k ! w2k onto the desired A-algebra S, but
the map has an enormous kernel, since QS is the vector space F2{wm :
m 1}, while QU is much larger. Our goal in step 3 is to describe a
minimal set of A-algebra relations producing S from U, i.e., minimal
generators for the kernel as an A-ideal.
Let us explore a prototype example in degree five, which is the first
place a difference occurs. There QS has only w5, whereas Sq1t4 and
Sq2Sq1t2 are distinct indecomposables in QU (recall that QU ~= M,
and that a basis for M consists of the unstable admissible monomials
on the A-generators t2k[11]). A few calculations with the Wu formulas
show that in S we have
Sq1w4 = w5 + w1w4 and
Sq2Sq1w2 = w5 + w1w4 + w2w3 + w1w22+ w21w3 + w31w2.
Thus to imitate S abstractly via U, we must impose an algebra relation
on U decreeing that
Sq1t4 = Sq2Sq1t2 + some decomposable,
per the calculations above. One challenge in doing even this, though, is
that it is not clear how to describe that needed decomposable difference
in U, since there we have no name as yet for the element corresponding
to w3. To remedy this, and to describe general formulas for relation-
ships like the one we have just discovered, we wish to use the Wu
formulas to focus our understanding as much as possible on both two-
power Steenrod squares and two-power Stiefel-Whitney classes. Thus
one of our formulas in the next section will express each Stiefel-Whitney
class purely in this way (Lemma 3.2).
While the plethora of algebra relations, such as the one above, needed
to obtain S from U may appear intractable to specify, recall that
our chosen task is actually somewhat different. Since we are work-
ing in the category of A-algebras, we seek relations in U whose A-
algebra consequences, not just their algebra consequences, will pro-
duce S. We shall show that this requires only a much smaller and
more tractable set of relations, for which our illustration in degree
five serves as perfect prototype. Specifically, the relationship between
i 2k-1 2i
Sq2 w2k and Sq Sq w2k-1for every i k - 2 will be the key place
6 DAVID J. PENGELLEY AND FRANK WILLIAMS
to focus attention. We shall impose one abstract relation on U for each
such pair (k, i), and prove that these are precisely the minimal relations
producing S = H*(BO; F2) in the category of A-algebras.
Our general plan is as follows. Form our abstract presentation candi-
date as just outlined; call it G. The construction of G will immediately
provide a natural A-algebra epimorphism to S. The hard part now is
showing that our (k, i)-indexed family of A-algebra relations leaves no
remaining kernel, i.e., that we have put in enough relations to generate
the kernel as an A-ideal. To achieve this we show that the epimorphism
G ! S induces a monomorphism QG ! QS, on the indecomposable
quotients, by computing a basis for QG. For this we appeal to our
earlier understanding [9], via the Kudo-Araki-May algebra K [10] (see
Appendix II), of bases for the unstable cyclic A-modules arising in the
analogous structure theorem for the Dickson algebras. With QG ! QS
an isomorphism, G ! S must be an isomorphism also, since S is a free
commutative algebra. The minimality of the (k, i)-family of relations
is then not hard to see by appropriate filtering.
3. Main theorem
We first identify the key A-algebra relations in S = H*(BO; F2).
Analysis of the binomial coefficients in the Wu formulas shows that
if r 1, then
j-1
(3.1) Sq2 wr2j= w2j-1wr2j+ w2j-1+r2j.
This formula will serve two purposes. It will guide us below in how to
specify any Stiefel-Whitney class from just the two-power ones, which
is needed for creating our abstract presentation. But before this it will
lead us to the key relations needed from S.
To find these, recall from the previous section that we seek a relation
i 2k-1 2i
involving a decomposable difference between Sq2 w2kand Sq Sq w2k-1
for every i k - 2. We begin with a special case of equation (3.1): For
i k - 2, we have
i
Sq2 w2k-1= w2iw2k-1+ w2k-1+2i.
k-1
Applying Sq2 , we get
k-1 2i 2k-1 2k-1
Sq2 Sq w2k-1= Sq (w2iw2k-1)+ Sq (w2k-1+2i).
Using a Wu formula on the last term, analyzing the binomial coeffi-
cients, and using (3.1) again, the reader may check that we obtain the
following relations.
STRUCTURE OF MOD TWO SYMMETRIC ALGEBRA 7
Proposition 3.1 (Key relations in S).For i k - 2,
k-1 2i 2i
(3.2) Sq2 Sq w2k-1= Sq w2k+
2k-i-1-2X
k-1
Sq2 (w2iw2k-1)+ w2k-1-2ilw2k-1+2i+2il,
l=0
i 2k-1 2i
These show explicitly how the elements Sq2 w2kand Sq Sq w2k-1
differ by a decomposable, and will guide us to the corresponding ab-
stract relations needed in G. However, the relations we have found here
involve non-two-power Stiefel-Whitney classes, which still have as yet
no analogs in U. We remedy this problem now by extending equation
(3.1).
Mixing notations, we write (3.1) as
j-1
w2j-1+r2j= (Sq2 + w2j-1)wr2j
(i.e., (Sqm + wm )x means Sqm x + wm . x). The following lemma is
then immediate.
Lemma 3.2 (Expressing Stiefel-Whitney classes).Every Stiefel-Whitney
class can be expressed in terms of two-power classes and two-power
squares as follows: If we write any m = 2n1 + . .+.2ns, where n1 >
. .>.ns, we have
2ns 2n2
(3.3) wm = Sq + w2ns . .S.q + w2n2 w2n1.
We are now ready to define formally the abstract presentation G.
Definition 3.3. In U, extend the set of generators {t2k, k 0}, to
define elements tm for all m 1, by first writing m = 2n1+ . .+.2ns,
where n1 > . .>.ns . Then by analogy with equation (3.3) set
ns 2n2
tm = (Sq2 + t2ns) . .S.q + t2n2t2n1.
Definition 3.4 (Abstract key relations).Imitating equation (3.2), let
G be the the A-algebra quotient of U by the left A-ideal generated by
the elements
i 2k-1 2i
(3.4) ` (k, i)= Sq2 t2k+ Sq Sq t2k-1+
2k-i-1-2X
k-1
Sq2 (t2k-1t2i)+ t2k-1-2ilt2k-1+2i+2il
l=0
for i k - 2.
Theorem 3.5 (Structure of S). The symmetric algebra S = H*(BO; F2)
is isomorphic to G as an algebra over the Steenrod algebra. Moreover,
8 DAVID J. PENGELLEY AND FRANK WILLIAMS
the relations (3.4) generating the A-ideal are minimal, i.e., nonredun-
dant.
The proof is in Section 5.
4. Applications and speculation
We apply the main structure theorem to the cohomology images
from the connected covers of BO, and to the cohomology of the spaces
BO(q) for classifying finite dimensional vector bundles. Finally we
shall see how these descriptions naturally converge into the Dickson
invariant algebras.
First we consider cohomology images from the connected covers.
Definition 4.1. Following [3], let B*(n) be the cohomology image of
the map induced by the projection
BO ! BO,
where BO is the n-th distinct connected cover of BO. That is,
BO is (OE(n) - 1)-connected, where n = 4s + t, 0 t 3, and
OE(n) = 8s + 2t.
In particular, for n = 0, 1, 2, 3 the projections are surjective in co-
homology, so the unstable A-algebras B*(n) are isomorphic to the co-
homologies of BO, BSO, BSpin, and BO <8>[3]. In general, B*(n) is
(2n - 1)-connected, and is the quotient of B*(0) = H*BO = S by the
A-ideal generated by {w2k: k < n} [3].
Theorem 4.2 (Structure of connected cover images).An abstract pre-
sentation of B*(n) is obtained from that of B*(0) = H*BO = S (The-
orem 3.5) as the quotient by the A-ideal generated by {t2k : k < n}.
This produces a minimal presentation as follows.
Let Kn denote the direct sum of the A-module M(n, 0) on t2n with
the free unstable A-module on the t2k, k n + 1. Here M(n, 0) is as
defined in [9], namely the free unstable A-module on one generator t2n
i
modulo the left A-submodule generated by Sq2 t2n, i n - 2.
Then B*(n) is isomorphic to the quotient of the free unstable A-
algebra on Kn by the left A-ideal generated by the elements ` (k, i),
k n + 1, i k - 2, subject to the requirement that all appearances in
` (k, i)of tm , 0 < m < 2n, are replaced by zero.
The proof is in Section 5.
For our second application, we note that the presentation for H*BO
in our main theorem will immediately produce presentations for the
cohomologies of the classifying spaces H*BO(q), since each is just the
algebra quotient (actually also A-algebra quotient) of H*BO by the
STRUCTURE OF MOD TWO SYMMETRIC ALGEBRA 9
ideal generated by {wm : m > q} [8], and wm corresponds to tm , which
we defined in the presentation of H*BO. The resulting presentation
becomes both tractable and useful for H*BO(2n+1 - 1).
Theorem 4.3 (Structure of H*BO(2n+1 - 1)). An abstract presenta-
tion of H*BO(2n+1 - 1) is obtained from that of B*(0) = H*BO = S
(Theorem 3.5) as the quotient by the A-ideal generated by {t2k : k
n + 1}. This produces a minimal presentation as follows.
H*BO(2n+1 - 1) is presented by the free unstable A-algebra on ab-
stract classes {t2k : 0 k n}, modulo the left A-ideal generated by
the elements ` (k, i)for k n + 1, i k - 2, (using Definition 3.3 of
tm for m < 2n+1), subject to the requirement that when k = n + 1, the
i
term Sq2 t2n+1 is replaced by zero for each i (all other terms involve
only t's in degrees less than 2n+1).
The proof is in Section 5.
Finally, combining the relations on S = H*(BO; F2) from the two
theorems above will produce the common A-algebra quotient of B*(n)
and H*BO(2n+1 - 1). Since the first of these is (2n - 1)-connected,
while the second is decomposable beyond degree 2n+1-1, we will obtain
an A-algebra with algebra generators in the range 2n through 2n+1 -
1. Surprisingly, this much smaller quotient of S = H*BO turns out
to be already familiar. We will show now that as an A-algebra it
is isomorphic to the n-th Dickson algebra Wn+1 (see Figure 1). In
this sense one can say that the Dickson algebra captures precisely the
cohomology common to BO and BO(2n+1-1) from H*BO, i.e.,
it is the A-algebra pushout.
Wn+1 H*BO(2n+1 - 1)
" "
B*(n) H*BO
Figure 1
Theorem 4.4 (Convergence to Dickson algebras). The quotient of the
symmetric algebra S by the left A-ideal generated by {w2k: k 6= n}is
isomorphic to the n + 1-st mod 2 Dickson algebra, Wn+1. Specifically,
using the notation of the presentation of Theorem 3.5, as an A-algebra
it is minimally presented by the free unstable A-algebra on the mod-
ule M(n, 0) (defined in Theorem 4.2), subject to the single A-algebra
relation
n 2n-1 2n-1
Sq2 Sq t2n = t2nSq t2n.
10 DAVID J. PENGELLEY AND FRANK WILLIAMS
We proved in [9] that this precisely characterizes the Dickson algebra
Wn+1.
The proof is in Section 5.
Let us speculate on how Figure 1 might fit in with something topo-
logically realizable. It is known that Wn+1 is realizable precisely for
n 3 [6], and that B*(n) H*BO is an isomorphism also pre-
cisely in this range [3]. Thus for n 3 it is reasonable to expect that
Figure 1 be realizable. For general n it is perhaps reasonable to hope
for the existence of a space Xn and a homotopy commutative square
(Figure 2) whose cohomology is compatible with Figure 1 in the sense
of combining to produce the commutative diagram of Figure 3. Addi-
tionally we would like Xn to have the property that the outer square in
Figure 3 is also a pushout of unstable A-algebras. In other words, Xn
does its best to realize a Dickson algebra, even when this is no longer
possible.
Xn ! BO(2n+1 - 1)
# #
BO ! BO
Figure 2
H*Xn Wn+1 H*BO(2n+1 - 1)
" " "
H*BO B*(n) H*BO
Figure 3
5. Proofs
Proof of Theorem 2.3.Let Fm be the free unstable A-module (equiva-
lently K module) on a generator tm in degree m. We shall show that
the A-module map f : m 0 Fm ! H*BO determined by f(tm ) = wm
is injective.
From [10], basis elements for the domain of f consist of DJtm where
J = (j1, . .,.js)and 0 j1 . . .js < m. (Appendix II recalls the
features of the elements DJ in the Kudo-Araki-May algebra K essential
to what follows.)
STRUCTURE OF MOD TWO SYMMETRIC ALGEBRA 11
On the other side of f, basis monomials of the range H*BO can
be written as . .w.n2wn1 with nondecreasing indices, i.e., labeled by
finitely nonzero tuples (. .,.n2, n1)with 0 . . .n2 n1. We order
the latter reverse lexicographically.
Now for each basis element DJtm , we consider its image f (DJtm )=
DJwm , and we claim that this element of H*BO has a "leading" mono-
mial term, i.e., that
s-1 2s-2 2
DJwm = w2m-jswm-js-1. .w.m-j2wm-j1wm + higher order terms.
______________-z_____________"
z
This will complete the proof, since distinct DJwm clearly produce dis-
tinct leading monomials, with remaining terms always of higher order;
so the DJwm are all linearly independent, and thus f is injective.
We will use the following notation: As a subscript, "> k" (resp.
"< k") denotes any index greater (resp. less) than k, each occurrence
of an unsubscripted w denotes any element of H*BO, and expressions
involving any of these mean any sum of expressions of such form.
We prove our claim by induction on s, based on the Wu formula
Djwm = Sqm-j wm = wm-j wm + higher order terms of formww>m .
Clearly the claim holds for lengths 0 and 1. For the inductive
step, consider DJbof length s + 1, and note that application of any
nontrivially-acting DJ always increases the order of a monomial in
H*BO. Now calculate, using the K-Cartan formula [10] as needed,
and recalling that the leading term z was defined above:
DJbwm
= Dj1Dj2. .D.js+1wm = Dj1 Dj2. .D.js+1wm
0 1
s-1
= Dj1@ w2m-js+1. .w.m-j2wm + higher order terms thanxwm A
_______-z______"
x
= x2wm-j1wm + x2ww>m + wDm + higher_order_terms_thanx-z__________"wm A
v
= z + ww>(m-j1)wm + higher order terms thanz
2
+ ww>m + v wm-j1wm + ww>(m-j1)wm
= z + higher order terms thanz,
__
since the terms of v2 have higher order than x2. |__|
12 DAVID J. PENGELLEY AND FRANK WILLIAMS
Proof of Theorem 3.5.There is a map of A-algebras U ! S obtained
by taking t2k to w2k, and from Lemma 3.2 and Definition 3.3 this map
takes each tm to wm . Since the relations (3.4) that define G map to
those also satisfied in S (3.2), there is an induced A-algebra epimor-
phism G ! S. We shall show that this map is monic by showing that
the induced map on the indecomposable quotients is monic, essentially
a counting argument.
To start with, note that the indecomposables are
I kff
QU = Sq t2k: k 0, I admissible, of excess< 2 .
Then QG is QU modulo the A-relations (degenerate versions of `(k, i) =
0)
i 2k-1 2i
Sq2 t2k= Sq Sq t2k-1, i k - 2.
There is an A-module filtration
I kff
FpQU = Sq t2k: 0 k p, I admissible, of excess< 2,
which induces an A-module filtration FpQG. Then
FpQG=Fp-1QG =
I pffn 2i o
Sq t2p: I admissible, of excess< 2=A Sq t2p: i p - 2.
This is the suspension of the module M(p, 1) analyzed in [9, Theorem
2.11]1, and the basis described there suspends to
{DIt2p: I = (2a1, . .,.2al), where 0 a1 . . .al<.p}
(As in the proof of Theorem 2.3, we refer the reader to Appendix II for
essentials concerning the elements DI in the Kudo-Araki-May algebra
K.)
We shall finish the proof of isomorphism by showing that the above
basis elements for p 0FpQG=Fp-1QG are in distinct degrees; in fact
we claim there is exactly one in each positive degree (The appendix
discusses the modules M(p, 1) in relation to the literature, and points
out an alternative path for substantiating our claim.). Let m be a
positive integer. Then m may be written uniquely in the form
Xs
m = 2r - 2bj,
j=1
where s 0 and 0 b1 < . .<.bs < r - 1. The reader may check by
induction on s that the unique basis element in degree m is DIt2p, where
___________
1M(p, 1) is defined in [9] as the quotient of the free unstable A-module on a
class in degree 2p - 1 modulo the action_of Sq2ifor i p - 2; in other words, *
*in
usual notation, M(p, 1) = F (2p- 1)=AA p-2.
STRUCTURE OF MOD TWO SYMMETRIC ALGEBRA 13
p = r - s and I = (2a1, . .,.2as), with aj = bj - j + 1. With both QG
and QS having rank one in each degree, QG ! QS is an isomorphism.
Then since S is a free commutative algebra, the epimorphism G ! S
must be an isomorphism also.
That the relations are minimal (nonredundant) is clear from the fact
that in FpQU=Fp-1QU, which is the suspension of the free unstable
module on a class in degree 2p - 1, the induced relations are simply
i __
Sq2 t2p= 0, for i p - 2, and these are all nonredundant. |__|
Proof of Theorem 4.2.We have already mentioned that according to
[3], B*(n) is isomorphic to the quotient of S by the A-ideal generated
by {w2k: k n-1}. Hence the images under the projection S ! B*(n)
of all wm , 1 m 2n-1, are certainly zero from Lemma 3.2. From [3]
we also have that B*(n) is a polynomial algebra generated by certain
remaining wm (see below). We denote the images of the wm in B*(n)
by the same symbols wm .
Let Hn denote the quotient of the free unstable A-algebra on Kn
by the left A-ideal generated by the elements ` (k, i), for k n + 1,
subject to the requirement that all appearances of tm , 0 < m < 2n, are
replaced by zero, as in the statement of the theorem.
We begin by defining a map from Kn to B*(n) by, as in the preced-
ing proof, assigning t2k to w2k for k n. Since the defining relations
for Kn are clearly satisfied in B*(n) (from equation (3.2)), this assign-
ment extends to the desired map. And since the defining relations for
the algebra Hn are also clearly satisfied in B*(n), this extends to an
A-algebra map Hn ! B*(n). This map is epimorphic (since B*(n) is
generated by certain wm with i 2n), so as in the preceding proof, we
need only show the the induced map on indecomposables is monomor-
phic.
According to [3]2, the polynomial generators of B*(n) are the wm
for which ff(m - 1), the number of ones in the binary representation
of m - 1, is at least n. We filter QHn as in the proof of the previous
theorem,
I ff
FpQHn= Sq t2k2 QHn : k p ,
and as in the previous proof the filtered quotient FpQHn=Fp-1QHn is
the suspension of the module M(p, 1) for p n, and 0 for p < n. It is
straightforward to check that the alpha numbers of one less than the
degrees of the elements
{DIt2p: I = (2a1, . .,.2al), where 0 a1 . . .al< p}
___________
2Kochman describes degrees of generators in terms of ff(m) + (m) ( is the
2-divisibility), but we equivalently use ff(m - 1) = ff(m) + (m) - 1.
14 DAVID J. PENGELLEY AND FRANK WILLIAMS
are exactly p n, so these are all in degrees where B*(n) has gener-
ators. Since we showed in the previous proof that these elements are
also in distinct degrees, this similarly completes the proof. Minimality_
follows as in the previous proof. |__|
Proof of Theorem 4.3.It is clear that the presentation of S collapses
in the manner stated. Minimality follows for most of the relations as
in the previous proofs. We comment only that to confirm that the
collapsed top relations
n 2i
0 = ` (n + 1, i) Sq2 Sq t2n+ decomposables fori n - 1
are also all nonredundant, one can observe that there is a natural map
of the new presentation without these final relations to the presentation
n 2i
for S, and compute that on indecomposables, each Sq2 Sq t2n maps
to w2n+1+2i. Now from the Wu formulas, QH*BO is filtered over A by
FpQH*BO = {wm : ff(m - 1) p}, and w2n+1+2iis in filtration exactly
i + 1. Thus {w2n+1+2i: i n - 1}must be a minimal generating set
for the A-submodule it generates in QH*BO. The same then must
be true of {` (n + 1, i): i n - 1}in the indecomposables of the new __
presentation without these final relations; so they too are minimal. |__|
Proof of Theorem 4.4.In [9] we proved that the (n + 1)-st Dickson al-
gebra Wn+1 is isomorphic to the quotient of the free unstable A-algebra
on the module M(n, 0) on generator x2n by the single A-algebra rela-
tion
n 2n-1 2n-1
Sq2 Sq x2n = x2nSq x2n,
and that M(n, 0) injects into Wn+1 ([9], proof of Theorem 2.11). In
other words, this is a minimal presentation in our sense.
Now let us turn to the quotient of the symmetric algebra that com-
bines the relations from the previous two theorems, i.e., the quotient by
the left A-ideal generated by {t2k, k 6= n}. Let us denote this quotient
by Jn. In Jn, the relations ` (k, i)are all trivial except when k is n + 1
i
or n. When k = n, they reduce to Sq2 t2n = 0, i n - 2, the defining
relations for M(n, 0). When k = n + 1, we have the relations
n 2i 2i
0 = ` (n + 1, i) Sq2 Sq t2n+ Sq t2n+1+
2n-i-2X
n
Sq2 (t2nt2i)+ t2n-2ilt2n+2i+2il
l=0
for i n - 1. These reduce to
n 2i
Sq2 Sq t2n = t2nt2n+2i.
STRUCTURE OF MOD TWO SYMMETRIC ALGEBRA 15
Now since
i i j i
t2nt2n+2i= t2n Sq2 t2n+ t2it2n= t2nSq2 t2n,
the relations can be rewritten as
n 2i 2i
Sq2 Sq t2n = t2nSq t2n.
i
Since Sq2 t2n = 0 for i < n - 1, these are trivial for i < n - 1, and yield
n 2n-1 2n-1
Sq2 Sq t2n = t2nSq t2n
for i = n - 1. This precisely matches the single relation (stated above)
characterizing the Dickson algebra, so we obtain an isomorphism of
A-algebras from Jn to Wn+1 by taking t2n 2 Jn to the generator x2n_2
Wn+1. |__|
__
6. Appendix I: The unstable modules F (2p - 1)=AA p-2 and
a minimal A-presentation for H* (RP 1)
For each p 0 , the module M(p, 1) is defined in [9] as the quotient
of the free unstable A-module on a class x2p-1in degree 2p- 1 modulo
i
the action of Sq2 for i p - 2; in other words, in usual notation,
__
M(p, 1) = F (2p - 1)=AA p-2.
These modules are tractable, important, and interesting, and we shall
show they are the filtered quotients of a simple minimal A-presentation
for H*RP 1.
In the proof of our primary Theorem 3.5 above, we appealed to our
development in [9, Theorem 2.11] of bases for these modules. The
proof used the bases to öc unt" that the direct sum of the modules
(we were actually dealing with their suspensions in that theorem) has
rank exactly one in each nonnegative degree. In fact we know the rank
separately for each module:
Theorem 6.1 (Rank of M(p, 1)). The module M(p, 1) has precisely
a single nonzero element in each degree with alpha number p, i.e., with
p ones in its binary expansion, and nothing else.
Proof.The basis for M(p, 1) provided in [9, Theorem 2.11] is
{DIx2p-1: the multi-indexI consists of nonnegative,
nondecreasing entries of form2k - 1, k < p}.
The reader may check that the degrees of these elements are precisely
those with alpha number p (see Appendix II for a recollection of essen-_
tials regarding the elements DI in the Kudo-Araki-May algebra K). |__|
16 DAVID J. PENGELLEY AND FRANK WILLIAMS
This suggests a connection to the cohomology of RP 1. Recall that
` '
l l+j
(6.1) H*RP 1 ~=F2[y] with Sqjyl= y ,
j
from which one sees that H*RP 1 is A-filtered by the number of ones
in the binary expansion of degrees. Indeed it is now not hard to prove
Theorem 6.2 (M(p, 1) and H*RP 1). The A-module M(p, 1) is iso-
morphic to the p-th filtered quotient of H*RP 1.
Proof.The module M(p, 1) clearly maps nontrivially to the p-thpfil-
tered quotient of H*RP 1, since the quotient begins with y2 -1, and
i 2p-1
Sq2 y lies in lower filtration for i p - 2. The map is onto because
one sees from (6.1) that the p-th filtered quotient of H*RP 1 is gener-
ated over A from degree 2p- 1. Now the previous theorem shows that __
the ranks agree, so the two are isomorphic. |__|
Remark 6.3. This result also follows from [2], where it essentially
appears in a stabilized form. Indeed, in [2] the A-modules
p-1 n 2j o
2 A=A Sq : j 6= p - 1
are studied with stable purposes in mind. Each of these modules ob-
viously maps onto the corresponding M(p, 1), and thus the two would
clearly be isomorphic if it were known that the domain module is un-
stable, which does not seem obvious. In fact, though, it is proven in
[2] that these modules are isomorphic to the same filtered quotients of
H*RP 1. Thus they are indeed unstable and isomorphic to the modules
M(p, 1). The theorem follows.
Remark 6.4. The modules M(p, 1) are also used in [5], where Remark
2.6 claims that in an unpublished manuscript [7], William Massey cal-
culated that M(p, 1) is A-isomorphic to the p-th filtered quotient of
H*RP 1, i.e., the theorem above. However, this does not actually seem
to appear explicitly in [7]. Finally, we note that the filtered quotients
of H*RP 1 arise again in [1, after Prop. 3.1] in a fashion closely related
both to [5] and [7].
We are now equipped to show
Theorem 6.5 (Minimal A-presentation of H* (RP 1)). There is a min-
imal unstable A-module presentation of H* (RP 1; F2), as the quotient
of the free unstable module on abstract classes s2k-1 in degrees 2k - 1
by the relations
i 2k-1 2i
Sq2 s2k-1= Sq Sq s2k-1-1, i k - 2.
STRUCTURE OF MOD TWO SYMMETRIC ALGEBRA 17
Proof.There is an A-module map from the abstract quotient to H*RP 1,
carrying each A-generator nontrivially, since the given relations are
easily calculated also to hold amongst the nonzero classes in H*RP 1.
Moreover this is epic, since H*RP 1 is generated over A from degrees
one less than a two-power. To see that the two are isomorphic, we need
merely show that these relations are enough, i.e., that the abstract quo-
tient has only rank one in each degree. This we do by considering the
A-filtration of the abstract quotient in which the p-th filtration is the
A-submodule generated by {s1, . .,.s2p-1}. The p-th filtered quotient
is clearly M(p, 1). That the union of these has rank one in each non-
negative degree follows from either of the two previous theorems.
Minimality of the presentation is clear. The nonzero classes in H*RP 1
in degrees one less than a power of two cannot be reached from below,
so the generating set is minimal, and unique. The nonredundancy of
all the relations is clear from the filtered quotients and the fact that
two-power squares are minimal generators of A.
An alternative proof would be to obtain this presentation simply by
collapsing the relations (3.4) in the A-algebra presentation of H*BO
in Theorem 3.5 to the indecomposable quotient, since H*RP 1 ~=_
QH*BO as A-modules (Wu formulas). |__|
7. Appendix II: The Kudo-Araki-May algebra K
We recall here just the bare essentials about K needed to understand
the proofs in this paper. We refer the reader to [10] for much more
extensive information about K.
The mod two Kudo-Araki-May algebra K is the F2-bialgebra (with
identity) generated by elements {Di : i 0} subject to homogeneous
(Adem) relations [10, Def. 2.1], with coproduct OE determined by the
formula
Xi
OE(Di) = Dt Di-t.
t=0
It is bigraded by length and topological degrees (|Di|= i), which be-
have skew-additively under multiplication [10, Def. 2.1].
The F2-cohomology of any space is an unstable algebra over the
Steenrod algebra, and there is a correspondence between unstable A-
algebras and unstable K-algebras, completely determined by iterating
the conversion formulae: On any element xl of degree l, and for all
j 0, one has
Djxl= Sql-jxl, equivalently,Sqjxl= Dl-jxl.
18 DAVID J. PENGELLEY AND FRANK WILLIAMS
Since the degree of the element is involved in the conversion, and this
changes as operations are composed, the algebra structures of A and K
are very different, and the skew additivity of the bigrading in K reflects
this.
The requirements for an unstable K-algebra, corresponding to the
nature and requirements of an unstable A-algebra, are: On any element
xl of degree l,
Dlxl= xl,Djxl= 0 forj > l, and D0xl= x2l.
Finally, and used in our proofs, the K-algebra structure obeys the
(Cartan) formula according to the coproduct OE in K:
Xi
Di(xy) = Dt(x)Di-t(y).
t=0
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New Mexico State University, Las Cruces, NM 88003
E-mail address: davidp@nmsu.edu
New Mexico State University, Las Cruces, NM 88003
E-mail address: frank@nmsu.edu