TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 00, Number 0, Xxxx XXXX, Pages 000-000
S 0002-9947(XX)0000-0
SHEARED ALGEBRA MAPS AND OPERATION
BIALGEBRAS FOR MOD 2 HOMOLOGY AND
COHOMOLOGY
DAVID J. PENGELLEY AND FRANK WILLIAMS
Abstract. The mod 2 Steenrod algebra A and Dyer-Lashof al-
gebra R have both striking similarities and differences, arising
from their common origins in "lower-indexed" algebraic operations.
These algebraic operations and their relations generate a bigraded
bialgebra K, whose module actions are equivalent to, but quite dif-
ferent from, those of A and R. The exact relationships emerge as
"sheared algebra bijections", which also illuminate the role of the
cohomology of K. As a bialgebra, K* has a particularly attractive
and potentially useful structure, providing a bridge between those
of A* and R*, and suggesting possible applications to the Miller
spectral sequence and the A structure of Dickson algebras.
1. Introduction
Mod 2 "lower indexed" algebraic operations Di (i 0) arising via
F2-equivariance and the quadratic construction were used by Steenrod
[Stee1, Stee2, Stee3] to create "upper operations" Sqi (i 0) in the
cohomology of spaces, and his approach was extended by Araki and
Kudo [AK ] to create "upper operations" Qi (i 0) in the homology
of iterated loop spaces. Adem [Ade ] deduced relations amongst the
Sqi, giving us the Steenrod algebra A, and Dyer and Lashof [DL ] in-
dicated how this could be done for the Qi, giving us the Dyer-Lashof
algebra R, although these relations were first explicitly given by May
[May2 ]. The relations amongst the underlying algebraic operations Di
were first given, implicitly, by May in [May1 ], as part of his unified al-
gebraic approach to A and R, but neither the explicit relations nor the
(bi)algebra generated by these operations was studied until relatively
recently. The first mention of this bigraded bialgebra, which we shall
___________
1991 Mathematics Subject Classification. Primary 55S99; Secondary 16W30,
16W50, 55S10, 55S12, 57T05.
Key words and phrases. Steenrod algebra, Dyer-Lashof algebra, bialgebras,
sheared algebra map, Kudo-Araki-May algebra, Nishida relations.
cO1997 American Mathematical Society
1
2 DAVID J. PENGELLEY AND FRANK WILLIAMS
call the Kudo-Araki-May algebra K in recognition of their contributions
just cited, occurs in a note by Smirnov [Sm ] in 1987, but more extensive
study of it has not occurred until the mid-1990's, in independent work
by Postnikov [Po ], Bisson-Joyal [BJ ], and the present authors. Perhaps
one reason for this delay is that in K the gradings on products are not
additive in the now familiar sense in topology, but rather skew-additive
on the "topological degree" via a linear combination involving also the
"length degree". We will use the term bigraded algebra in this more
general sense. Note also that we call K a bialgebra, and not a Hopf
algebra, since, like the Dyer-Lashof algebra, it is not connected and
thus has no conjugation.
Our point of view here, rather distinct from the historical develop-
ment, will be to obtain the Steenrod and Dyer-Lashof algebras directly
from the bialgebra K via the basic conversion formulas
Sqixk = Dk-ixk
Qiyk = Di-kyk
on classes xk of degree -k in cohomology (negatively graded) and yk
of degree k in homology. This process creates two new bialgebras com-
pletely different from K , but with topological degree more simply
linked to their actions on spaces. Each conversion involves a minor
miracle from this point of view, namely a lack of entanglement between
k and the relations between the new operations under composition.
We intend to show that K and its dual K* are tractable, and have
some clear advantages over A, R, and their duals, while providing
a unifying view via the notion of "sheared algebra homomorphism".
Moreover, the duality between K and K* has a number of very nice
features. We believe that in certain circumstances, such as for the
Miller spectral sequence [Mill] and the A structure of Dickson algebras,
it may be more productive to consider the module actions available
over K (or comodules over K*), rather than over A or R. For instance,
our description can be used to provide a convenient way of explicitly
identifying the indecomposables in the cohomology of iterated loop
spaces on spheres in terms of the algebras of Dickson invariants, and
this identification is used in [GP ] to study their A-module structures.
We will also emphasize how the use of formal power series can motivate,
illuminate, and help prove many of our results.
In section 2 below, we shall describe the generators and relations of
K and list the principal results of this paper. In section 3 we shall see
how K acts on graded modules and graded algebras in topology, in-
cluding its relation to the suspension isomorphism in cohomology, and
the homology suspension on iterated loop spaces. We also examine the
SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 3
bialgebra structures of K and K* and how they interact. In section 4
we study actions of K on itself and on K*, arising from the Nishida re-
lations, including a description which may shed light on the A-algebra
structure of the Dickson algebras, and we present a formula analogous
to the Thom isomorphism theorem. In section 5 we formalize what
we mean by a sheared algebra map, prove a general theorem for pro-
ducing sheared algebra maps on Poincare-Birkhoff-Witt algebras, and
apply it to show the exact relationship between K and the Dyer-Lashof
algebra, and between their duals. In section 6 we provide the precise
relationship between K and the Steenrod algebra via another sheared
algebra bijection with domain an algebra K(1) constructed from K.
In section 7 we use this to shed light on the tantalizing similarities and
puzzling differences between the coproduct formulas in K* and A*. In
section 8 we compute the cohomology of the algebra K and show how it
is closely related to K(1) and A, again via sheared algebra bijections.
We also examine the effect of sheared algebra maps in homology and
cohomology, summarize how all our sheared algebra bijections are con-
nected to each other, and mention possible applications via the Miller
spectral sequence. Section 9 discusses future directions for research
and applications. As an illustration, we show that a theorem of J. Lin
about finite H-spaces can be reformulated elegantly in terms of K .
We note that some previous work involves certain aspects of K, for in-
stance in providing useful descriptions of the homology of iterated loop
spaces [CCPS , CPS , CLM , MM ], and calculating the mod p homology
of topological Hochschild homology [H ]. In [Sm ], K is characterized as
the free radical Hopf algebra on a generator of dimension one. In [Po ],
Theorem 2 gives the product and coproduct in K* , cf. Theorems B
and C below. And [BJ ] includes the product, coproduct, and Nishida
action in K as a main (and motivating) example in their study of Q-
rings. Perhaps both K and the concomitant notion of sheared algebra
homomorphism will find more widespread use.
We thank Dennis Sjerve for conversations with the second author
which were the genesis of this project. We also thank Bob Bruner,
Vince Giambalvo, Chuck McGibbon, Haynes Miller, Jack Morava, Pe-
ter May, Bill Singer, Jim Stasheff, and the referee for useful comments
on earlier versions of this paper.
2. Statements of principal results
For each principal result we indicate in parentheses which section
contains its proof.
4 DAVID J. PENGELLEY AND FRANK WILLIAMS
Definition 2.1. We define our bigraded bialgebra K = K*;*to be the
F2-algebra with identity in bidegree (0; 0) and generated by elements
Di2 K1;i(i = 0; 1; : :):, subject to the (Adem) relations
X k - 1 - j
DiDj = Di+2j-2kDk; (i > j):
k 2k - i - j
(Observe here that the binomial coefficient is zero unless i+j_2 k < i.)
The bidegree of elements in K is defined inductively by the require-
ment that the multiplication be a map
Km;i Kn;j! Km+n;i+2m j.
This definition is viable since the inductive formula is clearly consistent
with the Adem relations and with associativity. We will call the first
of the bidegrees the length, and the second the topological, degree. We
shall frequently use the notation | | to denote topological degree.
Finally, the coproduct of K is defined on generators by the usual
Cartan formula
Xi
OE(Di) = Dt Di-t,
t=0
and we shall see that the product and coproduct maps of K interact
as in a bialgebra. We note that K is a cocommutative component
coalgebra [CLM , p. 18] with counit Dn0in the n-th component Kn =
Kn;*. The behavior of the coproduct on bidegrees is
M
OE : Kn;i! (Kn;t Kn;i-t) ,
t
which one sees by induction on n.
The `skew' additivity of multiplication on the topological degree is
a very interesting feature, which we hope the reader will become con-
vinced is well worth becoming comfortable with (see the relation to
topology after Definition 3.4, and the categorical consequences of [Sm ]).
In fact, Haynes Miller pointed out to us that the skew additivity can be
interpreted as a grading of K over the nonabelian monoid constructed
as the semidirect product of Z+ with Z+ using the homomorphism
' : Z+ ! End(Z+ ) given by '(t)(x) = 2tx.
As with A and R, K also possesses a vector space basis of "admissible
monomials" Di1. .D.inwhere i1 . . .in. We shall find it convenient
to adopt the convention that Di = 0 for any i that is not a non-
negative integer. As mentioned in the introduction, K, like R, has no
conjugation since it is not connected.
SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 5
We next move to the structure of K*. The first step is
Lemma A (x3). The elements Di0Dj1(i + j = n) form a basis for the
coalgebra primitives of Kn.
From the lemma we see that K* is generated as an algebra by el-
ements xi;jdual to the elements Di0Dj1with respect to the basis of
admissible monomials. In fact, we prove
Theorem B (x3). The algebra summand K*nis isomorphic to the poly-
nomial algebra
F2[xi;j| i + j = n; 0 i n - 1]:
The identity element is xn;0. (Thus K*nis isomorphic to the Dickson
algebra on n generators [W ].)
We also determine the coalgebra structure of K*:
Theorem C (x3). The coproduct in K* is given by the formula
X p
(xi;j) = x2i-p;j-q xp;q.
p;q
The reader may wonder at the similarities and differences between this
formula and the coproduct formula on Milnor's polynomial generators
i for A* [Miln][Stee3, p. 133] or the formula of Madsen [Mad ] for the
coproduct in R*. These questions spurred much of our work, and are
resolved in sections 5 and 7.
There are Nishida relations in K, which give inductive formulas for
a left (downward) action of the opposite algebra Kop on K,
Kopm;i Kn;j! Kn;i+j_2m.
We check that this Nishida action is a map of coalgebras, and thus
the contragredient action
K K*n! K*n
makes K*ninto an unstable algebra over the bialgebra K.
We shall see that this action yields a formula analogous to the ac-
tion of the Steenrod squares on the Thom class of vector bundles.
Specifically,Precall that if Sq, the total Steenrod square, is defined by
Sq = i0 Sqi, if w is the total Stiefel-Whitney class of a vector bun-
dle, and if U is its Thom class, then there is the formula
Sq(U) = w . U:
P P
ToPstate our analogy, let x = i;j0xi;j, O = j0 x0;j, and D =
i0 Di. Then we have
6 DAVID J. PENGELLEY AND FRANK WILLIAMS
Theorem D. In the action of K on K*, there is the formula
D(O) = x . O:
This is a special case of the following theorem, which completely
determines the action of K on K*, and thus encodes the action on
the Dickson algebras (cf. [HP , Mad , W ]). Since this description of
the action is rather different from that of the Steenrod algebra on the
dual Dyer-Lashof algebra, although equivalent to it, we hope it may be
useful.
P
Theorem E (x4).P For fixed n, and 0 m n, let f+m;n= km xn-k;k
and f-m;n= j 0, and 0 k 1,
a vector space basis for the coalgebra primitives in H*kSm+k can
be identified with the admissible monomials in K involving only Dl
for 0 l < k, via the action on the fundamental class. When k is
8 DAVID J. PENGELLEY AND FRANK WILLIAMS
finite, we will denote by K(k) the primitives in H*kSm+k under this
identification:
Definition 2.6. Define K(k) to be the vector subspace of K spanned
by the admissible monomials of the form Di00Di11. .D.ik-1k-1.
While it appears at first that K(k) is merely a vector subspace, in
fact K(k) is a sub-bialgebra of K.
We may now define a new algebra K(1) by
K(1) = lim-K(k)
k
under algebra homomorphisms ff : K(k) ! K(k - 1) given on gener-
ators by ff(Di) = Di-1: (Recall that since Dt = 0 whenever t is not a
nonnegative integer, this formula gives ff(D0) = D-1 = 0:)
Theorem H (from Theorem 6.11). There is a sheared algebra bijec-
tion
1 : K(1) ! A:
We move on to examine the dual map *1: A* ! K(1)*.
Proposition I (from Proposition 7.3).A basis for K(1)* is given by
"monomials"
n-1X
x-l0;nxl11;n-1.x.l.n-1n-1;12 K(1)*n forl > li.
i=1
(Here n is a grading induced from the original length grading in K.)
The notation x-l0;nxl11;n-1.x.l.n-1n-1;1is intended to suggest an algebra
in which x0;nhas been inverted. In fact, in [BPW ] it is shown that
K(1)* is a subspace of such an algebra.
Finding a complete formula for *1 appears to be quite complicated
and is related to the work of Campbell, Peterson, and Selick [CPS ].
However, we are able to compute the "leading terms," in the following
sense.
P n L
Theorem J (x7). Let l = i=1li: If ln 1; then modulo m>n K(1)*m
*1(l11. .l.nn) = x-l0;nxl11;n-1.x.l.n-1n-1;1:
We caution that while K(1)* is endowed with a coproduct dual to
the multiplication in K(1), we do not claim any algebra structure for
K(1)*. We also do not know how to compute the coproducts of all
elements of K(1)*; but we can compute certain ones which, combined
with our leading term theorem, will shed light on the similarities and
differences between the coproducts in A* and K*.
SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 9
H*(^0)
K(1) _________H*(K)-!______-H*(R(-1))
Q Q | |6
Q Q || ||
1 Q QQs ||? ||?
AL _________H*(opp)-oe
Figure 1
The last group of results deals with cohomology. We use methods
of [Pr] to compute the cohomology H*(K) = ExtK(F2; F2) of K, and
then prove
Theorem K (from Theorems 8.3 and 8.5). There exist sheared alge-
bra bijections
! : K(1) ! H*(K)
and
: H*(K) ! A:
Sheared algebra morphisms induce maps on homology and cohomol-
ogy:
Theorem L (x8). Let (K; ^K; F; d) be an algebra with shifting, M an
algebra,_and__ : K !_M_a sheared algebra homomorphism. Then the
map B ( ) : B (K) ! B (M) defined for the reduced bar constructions
by
__ d(z ...z ) d(z )
B ( )([z1| . .|.zn-1|zn]) = [ (F 2 (zn1)| . .|. (F n(zn-1))| (zn)]
is a chain map, which then produces induced maps on homology and
cohomology.
Finally we summarize many of our results in a single commutative
diagram. Before presenting it, recall from [KL1 , Pr, Mill, Lo] that there
is an algebraic version R(-1) of the Dyer-Lashof algebra which has
R as a quotient, is isomorphic to the opposite of the -algebra, and
whose cohomology algebra H*(R(-1)) is isomorphic to AL; the Lie
Steenrod algebra, which is obtained by replacing Sq0 = 1 by Sq0 = 0
in the defining relations for A. See Figure 1.
All the maps in this diagram are degree-preserving isomorphisms
of vector spaces, the labeled maps are sheared algebra bijections, and
the unlabeled double-ended arrows are the algebra isomorphisms just
mentioned. While ! and are both sheared algebra maps and 1 =
10 DAVID J. PENGELLEY AND FRANK WILLIAMS
O !; it does not immediately follow that 1 will be a sheared algebra
map. In fact 1 and ! do not even use the same degree function
on their common domain K(1): This phenomenon can be explained
by a more detailed examination of the categorical structure of sheared
algebra morphisms which we shall treat in a subsequent paper.
3.Basic properties of K and K*
We begin this section with some instances of the relations in Defini-
tion 2.1.
Example 3.1. We have
D2iD0 = D0Di; D2i+1D0 = 0; DiDi-1= 0:
We also can see, by checking binomial coefficients, that if the term
D0D2a appears on the right hand side (RHS) of an Adem relation,
then the left side must be D2a+1D0, and that DlDl does not appear
on the RHS of any Adem relation. Finally, note that the RHS of the
Adem relation for DiDj involves only those Dl with l < i.
We believe that the nice features of these sample relations already
promise that the K-Adem relations will sometimes be simpler to work
with than those of A and R.
To be certain that in our definition the product and coproduct in-
teract as in a bialgebra, and that this is indeed the algebra of opera-
tions generated by the "algebraic Steenrod operations" Di, we refer to
[May1 ] where these operations are defined in a setting which applies to
both cohomology of spaces and homology of iterated loop spaces (see
[CLM ] for the latter). The symmetry that produces relations between
iterated operations is expressed there by equation (e) in the proof of
Theorem 4.7. The reader may find it satisfying to check that the some-
what elaborate equation there is merely a detailed expansion of the
attractive formal power series symmetry identity
D(u)D((u + v)v) = D(v)D((u + v)u),
P
where D(x) denotes the formal sum i0 Dixi. We will call this iden-
tity the symmetric K-Adem relations. Note that if we let u; v have
formal topological degree -1, the skew-additivity of topological degree
in K ensures that all the terms in the above relations are homogeneous
of degree zero. The coproduct in K can be expressed in this context as
D(x) = D(x) D(x):
Clearly this coproduct respects the symmetric K-Adem relations, and
thus does produce a bialgebra.
SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 11
To extract the Adem relations of our Definition 2.1 from the sym-
metric relations, we can use the clever residue method of [BM , Stei].
Letting w = (u + v)v and noting that for constant u, dw = udv (mod
2 of course), we use the symmetry identity in the second equality of
what follows.
D(u)D(w) du
DiDj = res res __________dw ____
u=0 w=0 wj+1 ui+1
D(v)D((u + v)u) du
= res res________________udv ___
u=0 v=0 ui((u + v)v)j+1 u
" #
X dv du
= res res DlDkvl-juk-i+1(u + v)k-j-1______
u=0 v=0 v u
k;l
X k - j - 1 du
= res DlDku(k-i+1)+(k-j-1)-(j-l)_
u=0 j - l u
k;l
X k - j - 1
= Di+2j-2kDk.
k 2k - i - j
Thus the relations in Definition 2.1 follow.
We must note, however (as did [May1 , BM , Stei] in analogous situa-
tions for the Steenrod and Dyer-Lashof operations), that these relations
are valid not only for i > j, but for all i; j 0 (with cd interpreted
as (-1)d -c+d-1d when c < 0). Is it not possible then, that the un-
stated formulas for i j could impose additional relations beyond the
"standard" ones in our definition?
Neither Adem nor Steenrod in the case of the Steenrod algebra
[Ade , Stee3], nor May in the case of the Dyer-Lashof algebra [May2 ],
express concern about this issue when they deduce explicit relations. In
each case this concern was presumably obviated by topological knowl-
edge of the time; Adem knew [Ade , p. 231] from Serre's calculations
of the cohomology of Eilenberg-MacLane spaces that the admissible
monomials in Steenrod squares formed a basis for all operations, and
May knew [May2 ] similarly from calculation of the homology of QS0
that the admissible monomials in Dyer-Lashof operations applied to
the fundamental class are independent. Thus in each case no further
relations were possible.
In our case we may also appeal to the homology of QS0, since the
isomorphism 0 of Theorem G (with K subject only to the "standard"
relations of Definition 2.1) commutes with the actions of R and K on
a zero-dimensional homology class. Thus the algebra of operations has
no more relations than those in our definition. Having gone to all
12 DAVID J. PENGELLEY AND FRANK WILLIAMS
this trouble, it would nonetheless be very nice to have purely algebraic
reasoning for why the relations for i j must be totally redundant,
but we know of none such at present.
It is also very useful to have these asymmetric K-Adem relations
expressed using formal power series, as follows. In the symmetric K-
Adem relations identity, let w = (u + v)v, and seek to eliminate v from
the identity. First write the quadratic v2 + uv + w = 0 (mod 2) as
v_2 v_ w_ w_
u + u + u2= 0: Then v = uf u2 where f(x) is any mod 2 formal
power series solution to f2 + f + x = 0: This leads to the asymmetric
K-Adem relations identity
i i w jj i i i w jjj
D(u)D(w) = D uf ___ D u2 1 + f ___
u2 ! u2
i i w jj w
= D uf ___ D ______ :
u2 f w_u2
Working with this identity involves choosing a solution f(x) for the
quadratic, and being able to identify the coefficients in its integer pow-
ers as well. Of course there are many integral combinatorial choices
for the mod 2 solutions. This important quadratic is well known and
studied, and Lemma 4.7 of [GPR ] points to much of what one needs to
know. For instance, one can obtain the K-Adem relations of our defi-
nition by this approach, rather than by the Bullett-Macdonald residue
method we used above.
The behavior of the cohomology of a space or the homology of an
infinite loop space in relation to K motivates the following definitions.
Definition 3.2. A graded module M* over K is one satisfying the de-
gree requirement
Km;i Mj ! Mi+2m j .
Note that K with its topological degree is thus a graded module over
K.
Definition 3.3. A graded module M over K, as above, is called un-
stable provided that
(1) Di: M-i ! M-i
is the identity for i 1, and that
(2) Di(Mk) = 0 (i > -k):
Definition 3.4. If in addition to being an unstable graded module
over K, as per Definition 3.3, M is also a graded algebra (with the usual
SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 13
additive grading convention) over K (with respect to the coproduct OE),
then M is called an unstable algebra over K if it satisfies the condition
D0(y) = y2
for all y 2 M.
Remark 3.5. In [May1 ] and [CLM ] it is implicit that the cohomology
of a space (graded negatively) and the homology of an infinite loop
space are functors to the category of unstable algebras over K.
At this point, traditional treatments of homology and cohomology
operations define "upper-indexed" Steenrod or Dyer-Lashof operations
in terms of the generators of K, then develop the algebraic properties
of the respective algebras of operations. We, however, shall continue
to focus on the bialgebra K and its dual.
Lemma 3.6. The elements Di1. .D.in, i1 . . .in, form a vector
space basis for K (which we will call the basis of admissible monomials.)
Proof.This follows in a standard way, cf. [Pr, p. 51], by using an
appropriate ordering of the set of all multi-indices (i1; : :;:in), since_K
is bigraded of finite type. |__|
We next prove Lemma A from section 2.
Proof of Lemma A. Clearly the elements Di0Dj1are primitive, since Di0
is grouplike (i.e. OE(Di0) = Di0 Di0) and D1D0 = 0. Since
OE(D0DI) = (D0 D0)OE(DI);
we need only examine admissible monomials DI in which the first sub-
script is at least 1. So let DI = Di1. .D.inbe admissible, with i1 1
and in 2. Now
OEDI = Dn1 (Di1-1. .D.in-1) +
terms involving other basis elements.
__
The lemma follows. |__|
Note that Di0Dj1has bidegree (i + j; 2i(2j - 1)).
Observe that K*;*is not of finite type in the second bidegree (the ele-
ments Di0all have zero topological degree). With this in mind, we define
K* to be the bigraded dual to K, i.e. K*n;-q= (Kn;q)*. We will con-
tinue to refer to the dual bidegrees of K* as "length" and "topological"
degrees, as in K. Since K is the direct sum of the cocommutative coal-
gebras Kn, K* is the direct sum of the commutative algebras K*n. From
Lemma A we saw that K* was generated as an algebra by elements
14 DAVID J. PENGELLEY AND FRANK WILLIAMS
xi;jdual to the elements Di0Dj1with respect to the basis of admissible
monomials. Of course the bidegree of xi;jis (i + j; -2i(2j - 1)).
We now move to the proof of Theorem B.
Proof of Theorem B. We note that the multiplication in K*nobeys the
usual (non-skewed) degree convention (i.e. is additive in the second
subscript; see the degree behavior of OE in Definition 2.1). Furthermore,
note that on admissibles the correspondence
Di1. .D.in! xi10;nxi2-i11;n-1.x.i.n-in-1n-1;1
provides a bijection of graded vector spaces between Kn and
F2[xi;j| i + j = n; 0 i n - 1].
Since K*nis generated by {xi;j| i + j = n; 0 i n - 1}, the theorem_
follows. |__|
We observe that since xi;jhas topological degree -2i(2j - 1), already
the K*nrealize the graded Dickson algebras [W ], even before we consider
their natural module structure over A or K arising from the Nishida
relations.
We now aim for a coproduct formula on the algebra generators xi;j2
K*. To that end, the next lemma gives another very useful property of
the primitive elements Di0Dj1.
Lemma 3.7. If DI and DJ are admissible monomials such that DIDJ =
Di0Dj1, then there exist non-negative integers a and b such that
DI = Di-a0Dj-b2a; DJ = Da0Db1:
__
Proof.This follows from Example 3.1. |__|
The algebra K has several interesting and useful self-maps. The next
definition describes one of these.
Definition 3.8. Let V : K ! K be the Verschiebung, the bialgebra
map dual to the Frobenius (squaring) map on K*.
Remark 3.9. Note that V is an epimorphism since the squaring map is
one-to-one.
Proposition 3.10. On algebra generators V is given by V (Di) = Di=2.
(Recall our convention that Di= 0 for any i that is not a non-negative
integer.)
Proof.Since the Frobenius map on K**;*preserves the length degree
and doubles topological degree, we must have V (Di) = Di=2since V_is
onto. |__|
SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 15
Remark 3.11. This shows that "halving", Di ! Di=2, extends to a
valid algebra endomorphism on K, which is not immediately obvious
from the explicit Adem relations, although it follows easily from the
symmetric power series form of the relations since V (D(u)) = D(u2).
We shall henceforth, for admissible elements y 2 K; use the notation
y* for the dual element to y with respect to the basis of admissible
monomials in K:
Corollary 3.12. Let DJ be admissible. We have the formula
a *
(D*J)2 = D2aJ .
Proof.By Proposition 3.10 we may compute, for DI admissible,
a * a * * a *
<(D*J)2 ; DI> = <(V ) (DJ); DI> = = ;
which is nonzero if and only if DI=2a = DJ, or equivalently, DI =_
D2aJ. |__|
We are now in a position to prove Theorem C.
Proof of Theorem C. Let DI and DJ be admissibles. We compute
<(xi;j); DI DJ> = ,
which is nonzero if and only if DIDJ = Di0Dj1.
By Lemma 3.7 this means that DI = Di-p0Dj-q2p, and DJ = Dp0Dq1. So __
by Corollary 3.12 we see that the terms of (xi;j) are as claimed. |__|
It is now interesting to observe that since clearly each Di is inde-
composable in K, each xi0;1is a coalgebra primitive, which at first sight
seems perplexing in an F2-bialgebra, but is in fact consonant with the
world of multiple components.
The K* coproduct has a very nice representation if we first define
the formal power series
X p p+q i i+j-q p
(x)(u; v; w) = x2i-p;;j-q xp;qu2 -1v2 w2 -2 ,
0pi
0qj
and note that it determines the coproduct since the exponents of u; v; w
determine i; j; p; q uniquely. Letting
X n-l n
x(v; w) = xn-l;lv2 w2 -1,
0ln
we have
16 DAVID J. PENGELLEY AND FRANK WILLIAMS
Corollary 3.13. The coproduct in K* is completely encoded by the
identity
(x)(u; v; w) = (1 x)((x 1)(v; w); u).
Note here also that if we assign formal topological degrees |u|= |w|= 1
and |v|= -1, then both x(v; w) and (x)(u; v; w) are homogeneous of
degree -1.
We end this section with some further particularly interesting fea-
tures of K and K*, some of which we shall use below.
Theorem 3.14. For 0 < j1 < . .<.jr n, we have the formula
xn-j1;j1. .x.n-jr;jr= (Dn-jr0Djr-jr-11.D.j.2-j1r-1Dj1r)*.
We preface the proof of this theorem with two lemmas.
Lemma 3.15. r
X
i2i-1= (r - 1)2r + 1:
i=1
__
Proof.Easy induction. |__|
Lemma 3.16. Let 0 < j1 < . . .< jr n. If DI 2 Kn;*; where
I = (i1; : :;:in); and if |DI| = |Dn-jr0Djr-jr-11.D.j.2-j1r-1Dj1r|; and if
in r - 1; then DI is not admissible.
Proof.We note
Xr
|DI| |Dn-r0D1D2. .D.r| = 2n-r i2i-1= (r - 1)2n + 2n-r;
i=1
by Lemma 3.15. But clearly, if DI were to be admissible, then |DI|__
in(2n - 1) (r - 1)(2n - 1): |__|
Proof of Theorem 3.14.Let DI be admissible. Write x = xn-j1;j1. .x.n-jr;jr;
x0= xn-j1;j1-1. .x.n-jr;jr-1; and I = (I0; in):
From the coproduct formula for K* it is immediate that is
zero if in > r: Furthermore, if in = r; we have
= <(x); DI0 Dr> = = :
If j1 - 1 > 0; then we may conclude inductively (on n) that this
is nonzero if and only if DI0= D(n-1)-(jr-1)0.D.j.1-1r; whence DI =
Dn-jr0. .D.j1r:
If j1-1 = 0; i.e. j1 = 1; then = ;
which, again inductively, is nonzero if and only if DI0= Dn-jr0. .D.j2r-1;
whence again DI = Dn-jr0. .D.j1r: So if in = r; the only possible DI is
the desired one.
SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 17
Finally, Lemma 3.16 applies to show that there are no admissibles
DI in this bidegree with in r - 1: Hence if DI is admissible, then
is nonzero if and only if DI is as desired. __
|__|
For more complex monomials in K*, the duality with admissibles in
K becomes more elaborate. An algorithm for dealing with this question
is given in [CPS ].
Let A be a bialgebra. An element g 2 A is called grouplike (cf. proof
of Lemma A) if
(g) = g g,
where denotes the coproduct in A:
Remark 3.17. If g 2 A is a grouplike element and if f : A ! A is
given by f(a) = ga, then f is a morphism of coalgebras and (hence)
f* : A* ! A* is a morphism of algebras.
Some useful self-maps of K and K* are as follows:
Example 3.18. In K, the coproductPformula shows that the element
D0 and the formal sum D = i0 Diare grouplike. We will denote the
algebra maps dual to left multiplication by D0 and D by * and *,
respectively. From Lemma 3.7 their values on the algebra generators
of K* are
*(xi;j) = xi-1;j
and
*(xi;j) = xi-1;j+ xi;j-1 .
P
In K*, the sum O = j0 x0;jsatisfies (O) = O O, so ff : K ! K
dual to multiplication by O is an algebra map. As with V *earlier,
multiplication by O preserves the length degree. Then the epimorphism
ff : K1;i! K1;i-1must be given by
ff(Di) = Di-1:
It is interesting to use the Adem relations to provide an alternative
proof that this formula for ff yields an algebra endomorphism. In fact,
this follows trivially from their symmetric power series form; since
ff(D(x)) = xD(x);
applying ff to the symmetric identity simply multiplies both sides by
u(u + v)v.
18 DAVID J. PENGELLEY AND FRANK WILLIAMS
Remark 3.19. The map ff has topological interpretations. If we regard
K as operating on the cohomology of spaces and let : H*(X) !
H*(X) be the suspension isomorphism, x 2 H*(X); and DI be a
monomial in K, then since is essentially a cup product with a one-
dimensional class, one checks as in [Stee3, May1 ] that
DI((x)) = (ff(DI)(x)):
Also, if X is an infinite loopspace with K acting on it, z 2 H*(X), oe*
is the homology suspension, and DI is a monomial in K, then it follows
from [AK , DL , CLM ] that
ff(DI)(oe*(z)) = oe*(DI(z)):
The self-maps we have described will be quite useful in describing
the relation between K and the Dyer-Lashof and Steenrod algebras, as
will the following proposition, which illustrates their use.
Proposition 3.20. If DI 2 Kn;*is admissible, then
xa0;nD*I= D*I+(a;:::;a):
Proof.We compute, for admissible DJ,
= = ;
which is non-zero if and only if ffa(DJ) = DI; i.e. when DJ = DI+(a;:::;a):_
|__|
4. Nishida actions
In K, the Nishida relations [N , May2 ] [May1 , p. 209] [Mad , pp. 244-5]
will give inductive formulas for a left (downward) action of the opposite
algebra Kop on K,
Kopm;i Kn;j! Kn;i+j_2m,
which we shall denote by DI* DJ, to distinguish from multiplication in
K. Traditionally, the Nishida relations mediate the interaction of the
Aop and R actions on the homology of an infinite loop space. If X is
an infinite loopspace and x 2 H*(X), then in terms of K they take the
form
X t+s_
Ds * (Dtx) = t-s2_ Dt+s_2-k(Dk * x):
k 2 + k
Here Ds*y is the left downward action of Kop on H*(X) that is equiva-
lent to the standard left downward action of Aop on H*(X). To obtain
SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 19
an action of Kop on K, we can use the action on a zero-dimensional
homology class and define inductive formulas
(
1; if s = 0;
Ds * 1 =
0; otherwise
and
X t+s_
Ds * (DtDJ) = t-s2_ Dt+s_2-k(Dk * DJ):
k 2 + k
The reader may again be pleased to check that this formula is merely
the formal power series identity
D(u) * (D(v) . ___) = D((u + v)v) . (D((u + v)u) * ___)
P
in different guise, where D(x)* denotes Dixi*. A straightforward
computation, which we leave to the reader, shows that this formal
power series formula for the action is compatible with (the opposite
of) the formal power series symmetric K-Adem relations in section 3.
This provides satisfying confirmation that we have really described a
bona fide algebra action of Kopon K, underlying the traditional Nishida
relations for Aop and R. It is also interesting to compare our formulas
with (3a) of [Stei].
As in the formal power series identity for the coproduct in K*, where
both sides were homogeneous of degree -1, in this case we find that
both sides of our action formula have homogeneous degree zero if we
let |u|= |v|= -1=2, where the degree of Ds * (Dt. ___) is (t + s)=2,
and that of Dl. (Dk* ___) is l + k, since these are the amounts by which
they increase the degrees they are applied to.
We also wish to note at this point that by using the action based
on fundamental classes in positive degrees, we obtain different actions
of Kop on K. We shall consider these actions and their topological
significance in a subsequent paper.
Example 4.1. Taking t = 0, we obtain
(
D0(Ds=2* DJ) if s is even;
Ds * (D0DJ) =
0 if s is odd.
Also,
8
>: 2 2 2
0 if s is even.
20 DAVID J. PENGELLEY AND FRANK WILLIAMS
Further,
Ds * (DtDJ) = 0 ifs; t have different parity.
And finally,
D0 * DJ = DJ=2;
so we see that the action of D0 produces the Verschiebung map V of
Definition 3.8.
We next wish to observe, as pointed out in section 2, that dualizing
from the Nishida action turns each K*ninto an unstable algebra over
Ka la Definition 3.4. First note that the Nishida action is a map of
coalgebras, as may be verified straightforwardly by induction on length
in K using the formal power series formulation above of the Nishida
action. Thus the contragredient action
K K*n! K*n
makes K*ninto an algebra over the bialgebra K. Moreover, one checks
by induction on length from the Nishida relations that * satisfies (the
dual of) the conditions in Definition 3.3. Finally, using Proposition
3.10 to dualize the last equation of Example 4.1, we obtain the squaring
condition of Definition 3.4.
We shall prepare for the proof of Theorem E with a lemma illustrat-
ing computations using the Nishida relations of Example 4.1.
Lemma 4.2. Let 0 a < d a + b and c 0. Then
D2a+b+2a-2d* (Da0Db1Dc2) = Dd-10Da+b+c-d+11:
Proof.By repeated applications of the formulas in Example 4.1, we
compute
D2a+b+2a-2d* (Da0Db1Dc2)
= Da0(D2b+1-2d-a* Db1Dc2)
= Da0(D2d-a(2b-d+a-1)+1* Db1Dc2)
= Dd-10(D2(2b-d+a-1)+1* Db-d+a+11Dc2)
= Dd-10D1(D2b-d+a-1* Db-d+a1Dc2)
= Dd-10D1+b-d+a1(D0 * Dc2)
= Dd-10D1+b-d+a+c1:
__
|__|
SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 21
Proof of Theorem E. Since K acts unstably on K*n, and xn-m;m lies in
topological degree -2n-m (2m - 1) in K*n, the element Dixn-m;m can
be nonzero only if it lies in a degree from -2n-m+1 (2m - 1) through
-2n-m (2m -1). In degrees from -(2n+1-2) to 0, a basis for K*nconsists
of the elements xn-j;jxn-k;kfor 0 j k n, lying in the distinct
degrees 2n-j+ 2n-k - 2n+1, together with the elements x2n-1;1xn-k;kfor
1 k n - 1, the latter lying respectively in the same degrees as
x2n-k-1;k+1in the first list.
Thus Dixn-m;m lands in a degree of zero rank unless i = 2n-j +
2n-k - 2n-m+1 for 0 j k n, in which degree the dual is spanned
by Dn-k0Dk-j1Dj2, together with Dn-j+10Dj-21D3 if also j = k 2. We
complete the proof by considering four cases discriminating primarily
where m lies in relation to j and k.
Case 1. If j < m k, then in the degree of Dixn-m;m , K*nhas rank
one, and we compute
=
= = 1;
taking a = n - k, b = k - j, c = j, and d = n - m + 1 in Lemma 4.2.
Thus Dixn-m;m = xn-j;jxn-k;kas claimed.
Case 2. If j = m = k, then D0xn-m;m = x2n-m;m, since K acts unstably
on K*, producing the leading term in the formula claimed.
Case 3. If j k < m, then for j 1, Dixn-m;m = 0 as claimed,
since computing as in Case 1 using the formulas from Example 4.1, we
obtain
=
=
= 0,
and, if also j = k 2, then similarly
= 0:
For j = 0 one checks that the result follows from instability.
Case 4. If m j < k or m < j k, then i < 0, which cannot occur._
|__|
22 DAVID J. PENGELLEY AND FRANK WILLIAMS
The action of K on K*; as described by Theorem E, lends itself to
formulation with formal power series. For m 0, let
X n-k n-m n-m n
f+m(u; v; w)= xn-k;ku2 -2 v2 w2 -1 ,
mkn
X n-j n-m n-m n
f-m(u; v; w)= xn-j;ju2 -2 v2 w2 -1 ,
0j 2j;
i_2ti-j-1 2t - i
P
and with coproduct given by Qi= Qt Qi-t[May2 ].
We call a nonnegative sequence I = (i1; i2; : :;:in); and the corre-
sponding monomial QI = Qi1. .Q.in, admissible provided it 2it+1
for each t, and recall [CLM , Mill] that the admissible monomials form
a basis for R(-1).
The excess of an admissible I and of QI is defined to be i1 - (i2 +
. .+.in), and we recall that an admissible vanishes on all k-dimensional
homology classes precisely if it has excess less than k [CLM , p. 17].
Following [CLM , Mill], we consider the ideal B(k) in R(-1) spanned
by all admissibles of excess less than k, and form the bigraded algebras
R(k) = R(-1)=B(k), with basis the admissibles of excess at least k.
Then R, the "geometric" Dyer-Lashof algebra, is defined to be R(0),
a quotient bialgebra of R(-1) [CLM , p. 17].
Before proceeding, we recall from Section 2 the shift map ffi : R(-1)n;i!
R(-1)n;i-(2n-1)defined by
n-1 i -2in-1
ffi(Qi1. .Q.in-1Qin) = Qi1-2 . .Q.n-1 Q
on admissibles. Note that ffi reduces excess by one.
For each fixed k 0, we inductively extend the conversion formula
above for the action on k-dimensional classes, and abstract from this
conversion on admissibles to define a map
^k: R(-1)n;i! Kn;i-(2n-1)k
of vector spaces, commuting with the respective actions on Hk(Y ),
given by the formula
^k(Qi1. .Q.in-1Qin) = Di1-i2-...-in-k.D.i.n-1-in-kDin-k:
24 DAVID J. PENGELLEY AND FRANK WILLIAMS
Note that at this time we only know that this formula for ^k holds on
admissibles.
From our remarks above we know that ^k factors through R(k),
creating
k : R(k)n;i! Kn;i-(2n-1)k
satisfying the same formula on admissibles, from which it is easily ver-
ified that
Proposition 5.1. For each k 0,
(i) k+1 = ff O k and k+1 = k O ffi;
(ii) k is an isomorphism of vector spaces; and
(iii) 0 is an isomorphism of coalgebras.
It follows that the dual of 0 is an algebra isomorphism. Combining
this fact with Theorem B yields us Madsen's theorem [Mad ] on the
algebra structure of R* . Further combining with Theorem E gives us
l
Madsen's computation of the A-action (of the Sq2 ) on generators of
the Dickson algebras R*n.
The bijections k will be some of our first examples of "sheared" al-
gebra maps. While not algebra maps in the traditional sense, we see
they respect products in a sheared sense (at least on admissible mono-
mials), using the shift map ffi. For example, while k maps Qi to Di-k,
it maps an admissible Qi1Qi2to Di1-i2-kDi1-k= k(ffii2(Qi1))k(Qi2).
Thus as k passes through the product it is sheared by a shifting of the
left factor. The shifting is by i2 applications of the shift map ffi, where
i2 is the topological degree of the right factor. This is reminiscent in
some respects of the notion of a semidirect product of groups.
We will show that this phenomenon has both general structure and
application. A sheared algebra map should behave as above on any
product, not just on monomials of special form like admissibles. This is
a pleasant circumstance, since it means the entire feature is compatible
with (Adem) relations, and thus it is a general property of the map,
unaffected by the representation of elements. In particular, the formula
we gave above for ^k will hold even on inadmissibles!
With these motivating examples in hand, we now proceed to describe
the general setting for sheared algebra maps, and prove a theorem
which can be applied to show that several of our constructions produce
sheared algebra bijections. In addition to the maps k :R(k) ! K,
these applications will include a map relating the cohomology of K and
the Steenrod algebra, i.e. a sheared algebra bijection H*(K) ! A. This
SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 25
will arise from sheared algebra maps involving a new graded algebra
K(1) which we will construct from K.
First we recall from Definition 2.4 what is meant by a sheared algebra
homomorphism.
Our general method for producing sheared algebra homomorphisms
requires K to be a Poincare-Birkhoff-Witt algebra (P-B-W algebra for
short), about which we briefly remind the reader, and refer to [Pr] or
[Lo ]. The Lie Steenrod algebra and algebraic Dyer-Lashof algebra are
examples [Lo , Mill, Pr], as well as our K and some other algebras we
shall encounter.
Begin with an augmented algebra with generating set K^ = {bi}i2I,
I a subset of Z. Suppose further that K has only homogeneous 2-
fold relations in these generators. Then K is called a homogeneous
pre-Koszul algebra, and has a multiplicative length grading, which we
denote by K*. (We are being somewhat more general than [Lo , Pr] by
not yet considering or requiring a second, topological, grading.)
Let I = (i1; : :;:in) denote a multi-index and call it the label of
bI = bi1. .b.in, the corresponding monomial in generators. Suppose S
is a set of multi-indices chosen so that B = {bI}I2S is a vector space
basis for K. Then (B; S) is called a labeled basis for K. The monomials
in B are called admissibles, the expression of any element in terms of
them is called its admissible expression, and the admissible expression
of any inadmissible 2-fold product of generators is called an admissible
relation. Now the set of all possible multi-indices is ordered, first by
length and then lexicographically. The labeled basis (B; S) is called a
P-B-W basis (and then (K; ^K; B; S) a P-B-W algebra) provided:
1. If I; J 2 S then either the juxtaposition (I; J) is also in S or else
the label of every monomial in the admissible expression for bIbJ
is strictly greater than (I; J), and
2. For n > 2, (i1; : :;:in) 2 S if and only if for each j < n we have
(i1; : :;:ij) 2 S and (ij+1; : :;:in) 2 S.
Now we will describe a process which begins with a P-B-W alge-
bra K, and constructs under three assumptions a particular type of
degree function d and shift map F which interact nicely. The result-
ing (K; ^K; F; d) will be called an "algebra with shifting", and we then
prove a theorem enabling the creation of various sheared algebra ho-
momorphisms with K as domain. We will apply this theorem first to
^k : R(-1)! K , and therefrom show that each k : R(k)! K is also
a sheared algebra bijection. We will later apply the theorem in several
other situations.
26 DAVID J. PENGELLEY AND FRANK WILLIAMS
So we start with a P-B-W algebra (K; ^K; B; S) and describe first the
construction of a degree function d under
Assumption 1. Suppose b is a positive integer and suppose dP: ^K!
Z+ is given satisfying the following condition: If k0k00= l0ml00mis an
admissible relation, then for each m
d(k0) + bd(k00) = d(l0m) + bd(l00m):
Examples. For b = 1, any additively graded algebra, e.g. topolog-
ical degree in R(-1), the negative of topological degree in A (i.e.,
positively graded), or length degree in R(-1) or K or AL (the Lie
Steenrod algebra [Lo , Pr]). For b = 2, topological degree in K:
To extend d to all monomials in K, consider first T (K^), the tensor
algebra on K^:
Definition 5.2. On the usual basis for T (K^) define
d^(k1 . . .kn) = d(k1) + bd(k2) + b2d(k3) + . .+.bn-1d(kn):
P
Lemma 5.3. If kiki+1 = l0ml00mis an admissible relation, then for
each m
^d(k1 . . .ki ki+1 . . .kn) = ^d(k1 . . .l0m l00m . . .kn):
Proof.We need only check that
bi-1d(ki) + bid(ki+1) = bi-1d(l0m) + bid(l00m),
__
which follows from d(ki) + bd(ki+1) = d(l0m) + bd(l00m). |__|
Corollary 5.4. If we define a map d on a typical monomial (of the
form k1. .k.nwhere each ki2 ^K) by the formula d(k1. .k.n) = ^d(k1
. . .kn); then this yields a well-defined d on each element of KPwhich
can be expressed as a monomial, in the sense that if k1. .k.n= l1m . .l.nm
expresses k1. .k.nas a sum of admissibles, then for each m, d(k1. .k.n) =
d(l1m . .l.nm):
Remark 5.5. Thus d of a monomial is independent of the various pos-
sible monomial representations of the element, even though d is defined
and calculated from any monomial representation.
We now state the main fact concerning d, whose easy proof we leave
to the reader:
Theorem 5.6. If z and z0 are monomials in K with z 2 Km , then
d(zz0) = d(z) + bm d(z0):
SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 27
Thus the degree d is in general "skew-additive" unless b = 1, which
is the traditional additivity on products.
Note that the algebra K*;*is now bigraded via length and degree.
We next describe the construction of a shift map F , a vector space
endomorphism of K, under
Assumption 2. Suppose a is a positive integer, and suppose a mapP
F : K^ ! K^[ {0} is given satisfying the condition: If k0k00= l0ml00m
is an admissible relation, then
X
F a(k0)F (k00) = F a(l0m)F (l00m)
is also an admissible relation.
Examples. For a = 1, K any P-B-W algebra with F the restriction
to ^Kof any algebra endomorphism that takes generators to generators
or to 0 (e.g., K with ff). For a = 2, R(-1) with F = ffi.
Definition 5.7. Define a linear transformation F^ : T (K^) ! K by
setting
F^(k1 . . .kn) = F an-1(k1) . .F.a(kn-1)F (kn)
and
^F(1) = 1
on the usual basis elements.
P
Lemma 5.8. If kiki+1= l0ml00mis an admissible relation, then
X
^F(k1 . . .ki ki+1 . . .kn) = F^(k1 . . .l0m l00m . . .kn):
Proof.We begin by noting that for any positive integer e,
X
F ea(ki)F e(ki+1) = F ea(l0m)F e(l00m):
We compute
F^(k1 . . .ki ki+1 . . .kn) =
n-1 an-i+1 an-i a
F a (k1) . .F. (ki-1)F (ki) . .F.(kn-1)F (kn):
On the other hand,
X
^F(k1 . . .l0m l00m . . .kn) =
X n-1 n-i+1 n-i
F a (k1) . .F.a (l0m)F a (l00m) . .F.a(kn-1)F (kn):
28 DAVID J. PENGELLEY AND FRANK WILLIAMS
Hence the proof is completed by checking:
n-i+1 an-i X an-i+10 an-i 00
F a (ki-1)F (ki) = F (lm )F (lm );
which follows from the remark at the beginning of the proof, taking_
e = an-i: |__|
Corollary 5.9. We can extend F to an endomorphism of K by the
formula F (k1. .k.n) = ^F(k1. .k.n). In particular, F is well-defined
on the set of monomials in KPusing the formula from Definition 5.7, in
the sense that if k1. .k.n= l1m .P.l.nmexpresses k1. .k.nas a sum
of admissibles, then F (k1. .k.n) = F (l1m . .l.nm):
Corollary 5.10. For each positive integer e,
n-1e e
F e(k1. .k.n) = F a (k1) . .F.(kn):
From the above, we obtain the main theorem concerning F .
Theorem 5.11. If e is a positive integer and z1 and z2 are monomials
with z2 2 Kn;*; then
ne e
F e(z1z2) = F a (z1)F (z2):
Assume now that d and F have been defined as above using Assump-
tions 1 and 2. Suppose in addition that they interact as in
Assumption 3. For k 2 ^K;
d(F (k)) = d(k) + b - a:
Then we are prepared for
Definition 5.12. Let K be a P-B-W algebra satisfying Assumptions
1,2,3, with shift map F and (skew-additive) degree d as above. We call
the quadruple (K; ^K; F; d) an algebra with shifting.
Examples. Any additively graded algebra with a = b = 1; F the
identity, and d the grading. Also R(-1) with a = 2, b = 1, F = ffi, and
d the topological degree. We will have two other important examples
soon.
Remark 5.13. While K has both degree and shifting maps, they do not
interact as in Assumption 3.
Theorem 5.14. Let (K; ^K; F; d) be an algebra with shifting. If z is a
monomial in Kn;*; then
d(F (z)) = d(z) + bn - an:
SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 29
Proof.Write z in monomial form k1. .k.n: We compute
n-1 a
d(F (k1. .k.n)) = d(F a (k1) . .F.(kn-1)F (kn))
n-1 an-2 n-2 a n-1
= d(F a (k1)) + bd(F (k2)) + . .+.b d(F (kn-1)) + b d(F (kn))
= (d(k1) + an-1(b - a)) + b(d(k2) + an-2(b - a)) + . . .
+bn-2(d(kn-1) + a(b - a)) + bn-1(d(kn) + b - a)
= (d(k1) + bd(k2) + . .+.bn-1d(kn)) +
(an-1 + an-2b + . .+.abn-2 + bn-1)(b - a)
= d(k1. .k.n) + bn - an:
__
|__|
We are now ready to prove the main theorem on algebras with shift-
ing, Theorem F, which will enable us to produce various sheared alge-
bra homomorphisms. We shall begin the proof with two preparatory
lemmas.
P
Lemma 5.15. Let e be a positive integer and let k0k00= l0ml00mbe an
admissible relation. Then under the hypotheses of the theorem,
00)+be0 e 00 X d(l00)+be0 e 00
(F d(k (k )) (F (k )) = (F m (lm )) (F (lm )):
Proof.We compute, using Assumption 3,
00)+be0 e 00 d(k00)+be-aeae0 e 00
(F d(k (k )) (F (k )) = (F (F (k ))) (F (k ))
e(k00))ae0 e 00
= (F d(F (F (k ))) (F (k )):
By Assumption 2,
X
F ae(k0)F e(k00) = F ae(l0m)F e(l00m)
is an admissible relation, so applying the hypothesis of the theorem,
the last term above is
X e 00
(F d(F (lmF))ae(l0m)) (F e(l00m));
which by reversing the above steps is
X 00
(F d(lm )+be(l0m)) (F e(l00m));
__
as desired. |__|
Lemma 5.16. For any monomial k1. .k.n;
(k1. .k.n) = (F d(k2...kn)(k1)) . . .(F d(kn)(kn-1)) (kn):
30 DAVID J. PENGELLEY AND FRANK WILLIAMS
Proof.Define ^ : T (K^) ! M by
^(k1 . . .kn) = (F d(k2...kn)(k1)) . . .(F d(kn)(kn-1)) (kn):
From Lemma 5.15, Corollary 5.4, and TheoremP5.6, we see by taking
e = d(ki+2. .k.n), that if kiki+1 = l0ml00mis an admissible relation,
then
^(k1 . . .ki ki+1 . . .kn) =
X
^(k1 . . .l0m l00m . . .kn):
__
|__|
Proof of Theorem F. Let z = k1. .k.mand z0= k01. .k.0nbe monomials.
Then
(zz0) = (k1. .k.mz0)
0) d(k ...km z0) d(z0) 0
= (F d(k2...km(zk1)) . . .(F i+1 (ki)) . . .(F (km )) (z ) (Lemma 5.1*
*6) :
m-id(z0)
For each i, write fi= F a (ki): Now, we also compute
0) d(z0)
(F d(z(z)) = (F (k1. .k.m))
m-1d(z0) am-id(z0) d(z0) 0
= (F a (k1) . .F. (ki) . .F. (km )) (Cor. 5.10 withe = d(z ))
= (f1. .f.i. .f.m)
= (F d(f2...fm()f1)) . . .(F d(fi+1...fm()fi)) . . .(fm ) (Lemma 5.16) :
So to finish the proof, it will suffice to show that for each i
0)
F d(fi+1...fm()fi) = F d(ki+1...km(zki);
i.e. that
m-id(z0) d(k ...km z0)
F d(fi+1...fm()F a (ki)) = F i+1 (ki);
which amounts to showing that
d(fi+1. .f.m) + am-id(z0) = d(ki+1. .k.mz0):
We have
d(fi+1. .f.m) + am-id(z0)
m-(i+1)d(z0) d(z0) m-i 0
= d(F a (ki+1) . .F. (km )) + a d(z )
SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 31
0) m-i 0
= d(F d(z(ki+1. .k.m)) + a d(z ) (Cor. 5.10)
= d(ki+1. .k.m) + (bm-i - am-i)d(z0) + am-id(z0) (Theorem 5.14)
= d(ki+1. .k.m) + bm-id(z0)
= d(ki+1. .k.mz0) (Theorem 5.6):
__
|__|
As promised, we can now immediately apply Theorem F to prove
the following theorem, which includes Theorem G as a special case.
Theorem 5.17. For k 0, ^k : R(-1) ! K is a sheared algebra
homomorphism, and k : R(k) ! K is a sheared algebra bijection. In
particular, the defining formula for ^k holds on arbitrary products, not
just admissibles.
Proof.R(-1) is a P-B-W algebra, and an algebra with shifting using
d = topological degree with b = 1 and F = ffi with a = 2, since
d(ffi(Qi)) = d(Qi-1) = i - 1 = d(Qi) + b - a;
verifying Assumption 3. Thus we can apply Theorem F by first defining
^k(Qi) = Di-kand then verifying that Adem relations are respected, as
required by the hypothesis of the Theorem, which is a straightforward
calculation. The earlier factorization of ^k through R(k) then yields
the claim that k is a sheared algebra map (since d and ffi both pass
to the quotient R(k); even without needing to know whether R(k) is
a P-B-W algebra or algebra with shifting). We have seen earlier in __
Proposition 5.1 that each k is a bijection. |__|
Remark 5.18. Dualizing Theorem 5.17 for k = 0 will shed light on the
similarities and differences between the coproduct formulae in K* and
R*, since *0: K* ! R* is an algebra isomorphism (Proposition 5.1)
and a co-sheared coalgebra bijection (Theorem 5.17). We will label
elements of R* using our description of K* (Theorem B). Our defining
formulas show that ffi and ff correspond under k, i.e. k O ffi = ff O k;
so we have the shearing formula
0 O R = K O (0 0) O fl = K O fi O (0 0);
0) 0 0 d(y0) 0
where fl(z z0) = ffid(z (z) z and fi(y y ) = ff (y) y . Since
ff is dual to multiplication by O in K* (Example 3.18) we can dualize,
32 DAVID J. PENGELLEY AND FRANK WILLIAMS
beginning with our formula (Theorem C)
X p
K*(xi;j) = x2i-p;j-q xp;q;
p;q
obtaining
X p q p
R*(xi;j) = O2 (2 -1)x2i-p;j-q xp;q
p;q
X 2p(2q-1) p
= x0;i+j-p-qx2i-p;j-q xp;q:
p;q
This recovers Madsen's formula [Mad ] for the coproduct in R*. We see
p(2q-1)
that the extra complicating factors x20;i+j-p-qin the coproduct in R*
are precisely those induced by the coshearing. The simpler coproduct
in K* is one of the reasons why K may prove more useful than R in
some applications.
6. Relationship between K and the Steenrod Algebra
As mentioned in Section 2, while relating K to R was quite easy to
accomplish, it is more challenging to relate K to the Steenrod algebra
A via a sheared algebra bijection, but this is our aim.. Let X be a
space, and grade cohomology negatively, as before (Remark 3.5). Let
x 2 H-k(X), k 0. When one defines the action of A on H-k(X) by
the conversion formula
Sqj(x) = Dk-j(x);
another minor miracle occurs and this produces a well-defined action of
a Hopf algebra A on H*(X) [May1 , Stee3]. As in the previous section,
any unstable algebra over K produces an unstable A-algebra and vice
versa. We will also grade the Steenrod algebra A negatively. In formal
power series we have
Sq(r)x(t) = D(r-1)x(rt)
with |r|= |t|= 1 (and the contragredient homology action
1_ 1_
Sq*(r)y(t) = D*(r-2 )y(r 2t)
with |r|= 1, |t|= -1).
As we did before with R(-1), for each k 0 we inductively extend
the conversion formula and abstract from it on admissibles to produce
a map, commuting with the actions on H*(X),
OEk : A ! K,
SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 33
given on an admissible monomial (al-1 2al> 0) by the formula
OEk(Sqa1. .S.qan) = Db1. .D.bn2 Kn;(2n-1)k-j
where Sqa1. .S.qan 2 A-j and
bl= (k + an + . .+.al+1) - al:
Remark 6.1. Some words of warning: It is curious that while this gen-
eral conversion formula is correct even on inadmissibles when used to
compare the two actions on HkX, it is nonetheless false for OEk it-
self. For instance, while OE4(Sq1Sq2) = OE4(Sq3) = D1, Sq1Sq2x4 =
D5D2x4, so D1 and D5D2 agree on every 4-dimensional class even
though D1 6= D5D2, and the formula for OE4 defined above does not
hold on Sq1Sq2, even though it does hold when followed by application
to a cohomology class. This is due to the identification of Sq0 with
1 in the Steenrod algebra for topological reasons, which in particular
prevents the Adem relations from being homogeneous, so A possesses
no length grading. As a slightly different and more perplexing warn-
ing example, Sq1Sq3 = 0, but Sq1Sq3x6 converts to D8D3x6. Now
D8D3 = D0D7, which is nonzero in K, but of course is zero on every
6-dimensional class. This more subtle type of conversion failure can be
avoided if one can arrange to encounter only "stable" Adem relations
SqaSqb = . . ., those for which b a < 2b; they exhibit several very
useful patterns and features. We could develop our analysis in this
direction, but will take a different tack since in the end we will need
to reverse the correspondence, ultimately mapping to, instead of from,
the Steenrod algebra, with a sheared algebra bijection. This illustrates
the noninvertibility of sheared algebra bijections.
Despite the caution and pessimism advised by this remark, we intend
to fit the OEk together to obtain a close relationship between A and K,
~=
noting first that it is for k large (quite contrary to 0 : R ! K) that we
should expect OEk to match A well with K. Given the warnings above,
one might expect there will be no general map relating the multipli-
cations in A and K in sheared fashion, but we will find, surprisingly,
that there is.
Recall from Definition 2.6 that K(k) is defined to be the vector sub-
space of K spanned by the admissible monomials of the form Di00Di11. .D.ik-1k-1.
We have
Proposition 6.2.
(i) K(k) is a sub-bialgebra of K; and
34 DAVID J. PENGELLEY AND FRANK WILLIAMS
(ii) there is an exact sequence of vector spaces
0 ! K(k) ! K(k) ff!K(k - 1) ! 0..
Proof.Immediate from the definitions. (Recall the maps ff and from_
Section 3.) |__|
The following proposition lists some information about OEk.
Proposition 6.3.
(i) The image of OEk is K(k).
(ii) The kernel of OEk is the span of the admissible monomials in A
with excess greater than k.
(iii) OEk-1 = ff O OEk.
Proof.Let SqA = Sqa1. .S.qan be an admissible monomial in A, i.e.
al-1 2al> 0. Since an > 0, bn = k - an < k. Furthermore,
bl- bl-1
= [(k + an + . .+.al+1) - al] - [(k + an + . .+.al) - al-1]
= al-1- 2al 0:
So OEk takes admissibles to admissibles, and Im (OEk) K(k). Now
suppose that the excess of SqA is less than k + 1. Then
b1 = k + (an + . .+.a2) - a1
= k - [a1 - (an + . .+.a2)]
= k - excess(SqA ) 0:
Since the n-tuple (b1; : :;:bn) completely determines (a1; : :;:an), the
restriction of OEk to the space of admissibles of excess k is an iso-
morphism. Finally, if the excess of SqA is greater than k, then the
preceding calculation shows that b1 < 0, i.e. that Db1 = 0 . This __
proves part (ii). Part (iii) is immediate from the definitions. |__|
Definition 6.4. We define K(1)*;*by
K(1)n;-i= lim-K(k)n;(2n-1)k-i forn; i 0
k
under the maps ff (recalling that ff : K(k)n;(2n-1)k-i! K(k-1)n;(2n-1)(k-1)-i).
Note K(1)*;*has negative topological grading. The products in the
K(k) combine under the bonding algebra maps ff to make K(1)*;*an
algebra with length and topological degrees behaving as in K:
1 : K(1)m;i K(1)n;j! K(1)m+n;i+2m j :
SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 35
Theorem 6.5. The algebra K(1) is described by generators D1-i 2
K(1)1;-i, for i > 0, with relations for i < j
X j - p - 1
D1-i D1-j = D1-(i+2j-2p)D1-p :
i + j - 2p
Furthermore, the maps OEk induce a map
OE1 : A ! K(1)
that is an isomorphism of vector spaces and preserves topological de-
grees (which are both negative).
Proof.Since we defined K(1) as the bigraded limit of the K(k), we see
that for every bidegree (n; i), K(1)n;iis isomorphic to K(k)n;(2n-1)k-i
for k sufficiently large.
Let D1-i be defined by the sequence {Dk-i | Dk-i 2 K(k)}. The
relations will follow from taking representatives Dk-iand Dk-j in K(k)
for k > max (i; j). Then if i < j, we have k - i > k - j, so
X q - 1 - (k - j)
Dk-iDk-j = D(k-i)+2(k-j)-2qDq:
2q - (k - i) - (k - j)
Setting p = k - q, we get
X j - p - 1
Dk-iDk-j = Dk-i-2j+2pDk-p;
i + j - 2p
T
as desired. Since ker(OEk) = {0}, it follows that OE1 is an isomorphism._
|__|
Remark 6.6. Given how we created K(1), it should be no surprise that
the relations of Theorem 6.5 have a formal power series representation
related to that for K. Indeed we have
D1 (u-1)D1 ((u + v)-1v-1) = D1 (v-1)D1 ((u + v)-1u-1),
P
where D1 (x) denotes D1-i xi, as may be verified by the residue
method of section 3.
Remark 6.7. OE1 is given on admissibles by the formula
OE1 (Sqi1. .S.qin) = D1-(i1-(i2+...+in)).D.1.-(in-1-in)D1-in:
As usual, K(1) has a vector space basis of admissibles, where a mono-
mial D1-i1 . .D.1-inis admissible provided that ik ik+1 for all k,
and we see that OE1 is a bijection between both sets of admissibles.
We shall relate the product in A to the product in K(1) by showing
that the inverse of OE1 is a sheared algebra bijection. The map OE1 itself
fails because of the presence of unstable Adem relations.
36 DAVID J. PENGELLEY AND FRANK WILLIAMS
Definition 6.8. Let the inverse of OE1 : A ! K(1) be denoted by
1 : K(1)*;i! Ai.
A short calculation based on Remark 6.7 yields the formula for 1
on admissibles:
Lemma 6.9.
1 (D1-i1 . .D.1-in) =
n-2in i +i +2ini +in in
Sqi1+i2+2i3+...+2 . .S.q n-2 n-1 Sq n-1 Sq :
Note already that 1 has the substantial virtue that the RHS of
the above formula at least makes sense for any input (not necessarily
admissible) on the LHS, while this failed for the OE1 formula.
Remark 6.10. While we could have defined 1 directly without first
producing OE1 via the OEk, we would not be able to have 1 arise directly
from maps k out of the inverse limit, since maps out of an inverse limit
are not generally forthcoming.
The map ff induces an algebra endomorphism
ff1 : K(1)n;-i! K(1)n;-i-(2n-1),
with ff1 (D1-i ) = D1-(i+1), which we will use in our analogue of
Theorem 5.17. Theorem H follows from
Theorem 6.11. The map 1 : K(1) ! A is a sheared algebra bijec-
tion, given on arbitrary products by the formula of Lemma 6.9.
Proof.To apply Theorem F, we first note that K(1) is clearly a P-
B-W algebra. It is also an algebra with shifting, taking d to be the
negative of topological degree and b = 2, and F = ff1 with a = 1,
since ff1 is an algebra map. Assumption 3 is verified by
d(ff1 (D1-i )) = d(D1-(i+1)) = i + 1 = d(D1-i ) + b - a:
We then apply Theorem F by first defining 1 (D1-i ) = Sqi, and
verifying the hypothesis that Adem relations are respected, which is_
again a straightforward calculation. |__|
7.The relationship between K* and A*
The relationship in Remark 5.18 between the coproduct formulas in
K* and R* arose since 0 : R ! K was a sheared algebra bijection. We
intend to shed an analogous light on why the coproduct formula in A*
resembles that of K* with xi;jreplaced by i, a most curious relationship
since this replacement puzzlingly converts the nonunits x0;ninto the
SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 37
unit 0 = 1, and converts each unit xn;0into a nonunit n. We shall
show how to make sense of this conundrum. It will be elucidated by
the connection between the coproduct formulas in A* and K* given
by the map *1 : A* ! K(1)* dual to the sheared algebra bijection
1 : K(1) ! A. As with K*; K(1)* is defined to be the bigraded
dual to K(1); and we shall call the dual bidegrees the "length" and
"topological" degrees. Thus K(1)*n;q= (K(1)n;-q)*: (We note that
K(1)* appears in positive topological degrees.) We shall first describe
a standard basis for K(1)*; then interpret *1 in terms of this basis
and the standard monomial basis of A*:
We begin by finding appropriate bases for the bialgebras K(k)*:
Proposition 7.1. A basis for K(k)*n;*is given by the images under
the projection K* ! K(k)* of all monomials in x0;n; : :;:xn-1;1of total
exponent k - 1:
Proof.We dualize the exact sequence of Proposition 6.2 and obtain
* * * *
0 ! K(k - 1)*n;*ff!K(k)n;*! K(k)n-1;*! 0:
The proposition follows from an induction on k + n based on the facts
that K(k)*0;*has the single non-zero element x0;0; and that K(0) = 0:
To see that the elementsPofPthe specified form are independent, we form
in K*n;*a sum x = jXj+ ix0;nYi of distinct monomials, where the
Xj are of exponent k - 1 and do not have x0;nas a factor, and the
Yiare of exponent k - 2:We ask whetherPor not x projects to zero in
K(k)*n;*: By Example 3.18, *(x) = jX0j;where X0jis Xj with the first
indices of all factors reduced by one, and we know inductively that these
projections are distinct basis elements of K(k)*n-1;*: So if x projects to
zero, all of its summands must havePx0;nas a factor. In this case,
ExampleP3.18 shows that x = ff*( iYi): Again, we know inductively
that iYi projects nonzero to K(k - 1)*n;*: Since ff* is monomorphic
on K(k - 1)*, we see that if x projects to zero, all the Yi must also
be zero and hence so is x: Thus the images of these monomials are
independent. Inductively the X0jand Yi form bases for the ends of the
exact sequence, and thus the Xj and x0;nYi are a basis for K(k)*n;*,_
hence the proposition. |__|
Remark 7.2. The algebra epimorphism K* ! K(k)* has been studied
by [CPS ], in which they prove Proposition 7.1. They also describe an
algorithm to resolve the rather complicated problem of determining
the images of monomials with higher exponent. While in general thesea
images can be complicated, it follows from Corollary 3.12 that x2i;j=
j * a
Di0D2a ; so x2i;j= 0 in K(k)* if 2a k:
38 DAVID J. PENGELLEY AND FRANK WILLIAMS
We again recall from Example 3.18 that ff* : K(k)*n;*! K(k +
1)*n;*is given by multiplication by x0;n; which is monomorphic. As a
consequence, K(1)* can be identified as the union of the K(k)* under
the inclusions ff*: Thus a monomial xl00;nxl11;n-1.x.l.n-1n-1;1in K(k)* is
identified with the monomial xl0+10;nxl11;n-1.x.l.n-1n-1;1in K(k + 1)*; and
hence we obtain the following result, which includes Proposition I.
Proposition 7.3. A basis for K(1)* is given by "monomials"
n-1X
x-l0;nxl11;n-1.x.l.n-1n-1;12 K(1)*n;t for l > li,
i=1
P
where t = l(2n-1)- li2i(2n-i-1): The "monomial" above represents
the elements xk-l0;nxl11;n-1.x.l.n-1n-1;1in K(k)* for k l:
As mentioned in Section 2, we do not know how to compute the
coproducts of all elements of K(1)*; due to the challenge mentioned
above involved in identifying the images of elements under the projec-
tions K* ! K(k)*. But we can compute certain ones. For instance, if
n m, then using the formula for the coproduct in K* and Remark
7.2,
X m-j
*K(p)(xm;n-m ) = x2j;k xm-j;(n-m)-k
0kn-m; max{-1;m-log2p} 0; then
X
*1(x-l0;n) = x-l0;k x-l0;n-k:
Before proving our "leading term" theorem, Theorem J, we shall first
see how it sheds light on the relationship between the coproducts in K*
and A*:
SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 39
We begin by noting, using Example 3.18, that the dual of ff1 :
K(1) ! K(1),
ff*1: K(1)*n;i! K(1)*n;i+(2n-1);
takes the form ff*1(x-l0;nx) = x-l+10;nx. From Remark 2.5, the shearing
property of 1 in Theorem 6.11 may be written as
1 O 1 = A O ( 1 1 ) O fl
where
0) 0 -|z0| 0
fl(z z0) = F d(z(z) z = (ff1 ) (z) z
and |z0|is the (negative) topological degree of z02 K(1)*;*. We dualize
to obtain
fl* O ( *1 *1) O *A= *1O *1;
where fl* is given by (ff*1)t 1 on K(1)* K(1)**;t: We first see what
is revealed by applying this equality to n: All calculations in K(1)*
will be modulo length degree > n:
We start with
*1(n) = x-10;n;
from Theorem J. Then by Lemma 7.4 we have
Xn
*1 *1(n) = x-10;j x-10;n-j
j=0
for the right side of the equality.
On the left side,
Xn
n-j
*A(n) = 2j n-j;
j=0
hence
Xn
n-j -1
( *1 *1)*A(n) = x-20;j x0;n-j:
j=0
Now x-10;n-j2 K(1)*n-j;2n-j-1; so
Xn
n-j-1 -2n-j -1
fl*( *1 *1)*A(n) = (ff*1)2 (x0;j ) x0;n-j
j=0
Xn
= x-10;j x-10;n-j;
j=0
40 DAVID J. PENGELLEY AND FRANK WILLIAMS
for the left side. So we see first that the coproduct formula on n is
essentially forced by the apparently simpler coproduct formula on x0;n;
together with the co-shearing feature of the map *1:
Next we shall turn the tables around to see how, in the other direc-
tion, the coproducts of the xi;jare essentially forced by the coproducts
of the n and the co-shearing of *1: Let E = 2m and suppose that
n m: We compute, using Theorem J and Lemma 7.4, that
*1 *1(Enm ) = *1(x-(E+1)0;nxm;n-m )
X -(E+1) m-j -(E+1)
= x0;j+k x2j;k x0;n-j-kxm-j;(n-m)-k:
0kn-m; 0jm
On the other hand,
X n-i m-j
*A(Enm ) = 2i E2j En-im-j :
0in; 0jm
The pairs of indices break into two cases: (1) i j and n - i m - j;
and (2) otherwise. In case (2), the image of this under *1 *1 lands
in total length degree > n: So modulo this filtration we get, again from
Theorem J,
( *1 *1)*A(Enm ) =
X -(2n-iE+2m-j) m-j -(E+1)
x0;i x2j;i-j x0;n-i xm-j;(n-i)-(m-j):
0jm; 0i-jn-m
The topological degree of the righthand factor, x-(E+1)0;n-ixm-j;(n-i)-(m-j),
is (E +1)(2n-i-1)-2m-j (2(n-i)-(m-j)-1), which simplifies to 2n-iE +
2m-j - (E + 1): So
fl*( *1 *1)*A(Enm ) =
X -(E+1) m-j -(E+1)
x0;i x2j;i-j x0;n-i xm-j;(n-i)-(m-j);
0jm; 0i-jn-m
which, letting k = i - j; matches our formula above for *1 *1(Enm ).
Thus we see the coproduct formula on xm;n-m emerge as essentially
forced by the coproduct formulae on n and m .
For us these sample calculations of mutual forcing via the cosheared
feature of *1 provide a satisfying answer to the earlier puzzle of how
the coproduct formulae in K* and A* seem so similar in some ways,
and yet in other ways so different.
We shall prove Theorem J using some related maps *k: A* ! K*
which we now describe.
SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 41
First fix a value for n. Let I = (i1; : :;:in) be an n-tuple of integers,
and DI = Di1. .D.in.
Definition 7.5. Let : Zn+1 ! Zn be given by
(k; I) = (2n-1k - 2n-2in - . .-.2i3 - i2 - i1; : :;:2k - in - in-1; k - in):
Note that is linear in the (n + 1)-tuple (k; i1; : :;:in), and each
(k; _) is an isomorphism. We define linear transformations k : K !
A for k 0 by
Definition 7.6. If DI is admissible, we set k(DI) equal to the mono-
mial string Sq (k;I), which represents zero in A if any indices are nega-
tive, i.e. if in > k. (We will often need to focus on the specific monomial
string Sq (k;I), not merely the element of A which it represents.)
Remark 7.7. The formula for shows that k(DI) represents an ad-
missible in A provided in < k, and that if in = k then it is an admissible
string with some trailing Sq0s appended.
Remark 7.8. We note that this definition gives the correspondence be-
tween K and A acting on a cohomology class of degree -k: As such,
if DI is an arbitrary (not necessarily admissible) monomial, then we
can see, from a universal example such as K(Z=2; k), that it follows
that k(DI) is given by the same recipe, provided it is then rewritten
in terms of admissibles and those terms with excess greater than k
dropped.
We now prepare to prove a leading term theorem for *k: A* ! K*,
from which the proof of Theorem J will follow.
Definition 7.9. If SqJ = Sqj1. .S.qjn is admissible except possibly
for trailing Sq0s, l 0; and 0 j n; set
X
V j(SqJ) = {SqJ-oe| oe is a(j; n - j)-shuffle of(2j-1; 2j-2; : :;:1; 0; : :;*
*:0)};
and let Vlj(SqJ) denote V j(SqJ); expressed in terms of admissibles,
deleting all summands of length less than n and all those of excess
greater than l - 1:
Definition 7.10. For DI admissible, define
X
W j(DI) = {DI- -11(oe)| oe is a(j; n - j)-shuffle of(2j-1; 2j-2; : :;:1; 0; *
*: :;:0)}:
42 DAVID J. PENGELLEY AND FRANK WILLIAMS
Lemma 7.11. If DI is admissible, then modulo the subspace of A with
basis the set of admissibles of length less than n;
Vkj( k(DI)) = k-1(W j(DI)):
Proof.Let
X = {oe | oe is a(j; n - j)-shuffle of(2j-1; 2j-2; : :;:1; 0; : :;:0)}:
Then modulo the stated subspace of A, and using Definitions 7.6 and
7.10, Remark 7.8, and the linearity of ,
X
k-1(W j(DI)) = { k-1(DI- -11(oe)) | oe 2 X}
X -1
= {Sq (k-1;I- 1 (oe))| oe 2 X}
(with summands of excess> k - 1 dropped after conversion to admissibles)
X
= {Sq (k;I)-oe| oe 2 X}
(with summands of excess> k - 1 dropped after conversion to admissibles).
This in turn is by Definition 7.9 and Remark 7.7 equal to Vkj(Sq (k;I)) if
(k; I) is nonnegative, and also otherwise since both will then represent
__
zero. Finally, Vkj(Sq (k;I)) = Vkj( k(DI)): |__|
Theorem J will follow from
P n
TheoremL7.12. Let l = i=1li: If ln 1 and k l; then modulo
*
m>n Km;*,
*k(l11: :l:nn) = xk-l0;nxl11;n-1.x.l.n-1n-1;1:
Our proof of this will rely first on a special case in slightly stronger
form.
ae
L xk-l ifk l
Lemma 7.13. Let l 1. Modulo m>n K*m;*, *k(ln) = 0;n
0 otherwise.
Proof.Let DI be admissible of length at most n. We evaluate
< *k(ln); DI> =
=
SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 43
8
< 1 ifl-1( k(DI)) contains the term
= (Sq2n-1. .S.q2Sq1)l when expressed using admissibles
: 0 otherwise.
But the only admissible SqJ of length n for which l-1(SqJ)
n-1l l
contains this term is Sq2 . .S.q , and since k(DI) contains this
term in its admissible representation if and only if DI = Dnk-l, we
* k-l
have *k(ln) = Dnk-l modulo length n. By Proposition 3.20, x0;n=
* * __
Dnk-l in K , whence the lemma. |__|
Proof of Theorem 7.12.We shall freely use the following facts: If w =
Sqi1. .S.qim is admissible with m < n; then w pairs to 0 with the ideal
generated by n; n+1; : :.:If the excess of an admissible SqJ is greater
than q; then SqJ pairs trivially with any monomial in the 's of total
exponent q or less.
Inductively, based on Lemma 7.13, which includes the case l =
1; assume the desired formula holds for total exponents less than l.
Let j < n be the smallest j for which lj > 0 (the case where this
smallest j equals n is already covered by Lemma 7.13). Write =
lj-1jlj+1j+1:l:n:-1n-1and let x 2 K be defined correspondingly. Let DI be
admissible of length n: If the length of DI is strictly less than n;
then
< *k(jlnn); DI> = 0 = ,
since k(DI) also has length < n.
So assume that DI is admissible of length n: We compute:
< *k(jlnn); DI> =
=
= by Remark 7.7 and Definition 7.9
= by Definition 7.9
= sincek l
= from Lemma 7.11
= < *k-1(lnn); W j(DI)>
= by induction, sincek - 1 l - 1, and
sinceW j(DI) has length n.
Finally, = from Definition 7.10, since
using Lemma 3.7 one can check that the elements -11(oe) are precisely
44 DAVID J. PENGELLEY AND FRANK WILLIAMS
Yk ______K(1)-fflk
H H
|| H H 1
ae|k H H
| H H
|? HHj
K(k) ______-Kj________-A
k k
Figure 2
the length n monomials which involve the admissible basis element __
Dj0Dn-j1(in fact they each equal it). |__|
Proof of Theorem J.Choose k to be at least the topological degree of
l11: :l:nn. Let Yk denote the subspace of K(1) with basis the set of
admissibles
{D1-I = D1-i1 . .D.1-iq| q 0 and i1 k}:
We note first that in topological degrees -k; Yk is isomorphic to
K(1): Second, let aek : K(1) ! K(k) denote the projection, and
jk : K(k) ! K and fflk : Yk ! K(1) the inclusions. By following the
bases we see that aek O fflk is an isomorphism, and that
1 O fflk = k O jk O aek O fflk
by direct calculation using our formulas on admissibles for k and 1
(even though we caution that 1 6= k O jk O aek). We summarize this
in Figure 2, in which the triangle is not commutative until composed
with fflk.
Therefore
ffl*kO *1(l11. .l.nn) = ffl*kO ae*kO j*kO *k(l11. .l.nn):
L
Now *k(l11. .l.nn) = xk-l0;nxl11;n-1.x.l.n-1n-1;1modulo m>n K*m;*from The-
orem 7.12 since k deg l11: :l:nn l. Thus, since fflk, aek, jk are each
length-preserving, we have
*
* M
ffl*kO ae*kO j*kO *k(l11. .l.nn) = ffl*kO ae*kO j*k(xk-l0;nxl11;n-1.x.l.n-1n-1*
*;1) modulo K(1)*m;*;
*
* m>n
which in turn equals
ffl*k(x-l0;nxl11;n-1.x.l.n-1n-1;1)
since the exponent of xk-l0;nxl11;n-1.x.l.n-1n-1;1is less than k: Now since_ffl*
**k
is an isomorphism in this degree, the result follows. |__|
SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 45
8. The cohomology of K
In this section we shall use methods of [Pr] to compute the cohomol-
ogy ExtK(F2; F2) of K. (We suggest [Lo ] as a concise reference for the
cohomology of P-B-W algebras.) We shall then construct a sheared
algebra bijection to it from the algebra K(1) of Section 6. Finally, we
shall construct a sheared algebra bijection from it to A, consider the
effect of sheared algebra maps on homology and cohomology, summa-
rize how our sheared bijections fit together, and suggest applications
via the Miller spectral sequence.
We begin by recalling the defining relations of K from Definition 2.1:
X k - 1 - j
DiDj = Di+2j-2kDk; (i > j):
k 2k - i - j
Following [Pr], we define
m = i + 2j - 2k:
We note that
k - 1 - j k - 1 - j
= :
2k - i - j j - m
It is clear that K is a P-B-W algebra in our setting, with skew-
additive topological degree with b = 2 in Assumption 1. The next
proposition then follows from [Pr]. Before stating it, we shall make a
brief remark on gradings:
Remark 8.1. The cohomology of K is H*(K) = ExtK(F2; F2). As is
pointed out in [Pr], since a P-B-W algebra is bi-graded, the cohomology
is tri-graded. The elements in Hi;n;k(K) are defined by Priddy to be
of cohomological degree i; length degree n; and internal degree k: For
consistency with our convention of grading cohomology negatively, we
define the topological degree of such elements to be -(i + k):
Proposition 8.2. The cohomology of K, H*(K); is the algebra with
generators oek 2 Ext1;1;k(k 0) and Adem relations, for m k,
X k - 1 - j
oem oek = oem+2k-2joej:
j - m
The internal degree of products behaves in analogous fashion to that of
K; i.e. skew-additively with b = 2 in Assumption 1.
As usual, H*(K) has a basis of admissibles, where a monomial oei1. .o.ein
is admissible provided that il> il+1for all l. The relations for H*(K)
46 DAVID J. PENGELLEY AND FRANK WILLIAMS
are encoded by
uoe(u-1)oe((u + v)-1v-1) = voe(v-1)oe((u + v)-1u-1),
which may again be verified by a residue calculation.
We next proceed to develop a sheared algebra bijection
! : K(1) ! H*(K):
We recall that K(1) is an algebra with shifting map ff1 : Previously
we used the absolute value of the topological degree as degree map,
but we now observe that the length degree n can also serve as a de-
gree map, with b = 1; and n interacts with ff1 (with a = 1) in the
desired way (Assumption 3) to again produce an algebra with shifting
(K(1); ff1 ; n): In the now-standard way, we define ! on generators by
the formula
!(D1-i ) = oei-1
Theorem 8.3. The map ! is a sheared algebra bijection which pre-
serves topological degree. Thus it is given by
!(. .D.1-cD1-b D1-a ) = . .o.ec+1oeboea-1
on any monomial, not necessarily admissible.
Proof.A direct calculation confirms that ! respects Adem relations as
required by Theorem F. Alternatively, one may do a pleasant calcu-
lation comparing the formal power series forms of the Adem relations.
This yields the sheared algebra map and formula. It is then immediate __
that it preserves topological degree and is a bijection. |__|
We now turn to the relationship between H*(K) and A:
As we observed in Proposition 8.2, the internal degree of products
in H*(K) behaves analogously to that in K and so internal degree will
serve as a degree function for H*(K); with constant b = 2: To obtain a
shifting map, we need only define
s(oei) = oei+1,
and verify Assumption 2. This is easily checked, with constant a = 1,
by comparison of binomial coefficients, or by formal power series via
the method mentioned for ff after Example 3.18, since here s(oe(x)) =
x-1oe(x). Finally, s interacts with internal degree i as required by
Assumption 3. We have
Proposition 8.4. The map s is an algebra endomorphism, and we
have an algebra with shifting (H*(K); s; i):
SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 47
We intend to apply Theorem F to obtain yet another sheared algebra
bijection, this time from H*(K) to A. But first, note that we could
simply compose the isomorphisms !-1 : H*(K) ! K(1) and 1 :
K(1) ! A to obtain an isomorphism of vector spaces,
: H*(K) ! A;
given on admissibles by
(oei1. .o.ein) =
n-2in+1 i +i +2in+1i +in+1 in+1
Sqi1+i2+2i3+...+2 . .S.q n-2 n-1 Sq n-1 Sq :
It is immediate that preserves topological degree. It would be nice to
claim now that this is a sheared algebra map, i.e. that this formula
holds on any monomial, simply by combining Theorems 6.11 and 8.3,
but unfortunately Theorem 8.3 does not yield adequate information
about !-1 (recall that sheared algebra bijections are only one-way af-
fairs). The following theorem, which emerges immediately in the now
familiar way using Theorem F, accomplishes this goal of relating prod-
ucts in H*(K) directly to those in A.
Theorem 8.5. The map : H*(K) ! A is a sheared algebra bijection.
We might equally well consider the Lie Steenrod algebra, AL [Lo ,
Pr],which is formed by replacing the relation Sq0 = 1 with the relation
Sq0 = 0: Since the image under of any monomial in H*(K) produces
a monomial in A for which the reduction to a sum of admissibles via
Adem relations does not produce any terms with factors of Sq0 (cf. Re-
mark 6.1), can equally well be regarded as a sheared algebra bijection
from H*(K) to AL:
Theorems 8.3 and 8.5 give Theorem K of Section 2. We now turn to
the effect of sheared algebra morphisms on homology and cohomology,
and note that the proof of Theorem L is straightforward using the
definition of a sheared algebra homomorphism and Theorems 5.6, 5.11,
and 5.14.
Using the formulas of [Pr], it is easy to see that if K; M are both P-
B-W algebras, and if carries generators to generators, one can write a
formula for the induced map on cohomology. In particular, one can im-
mediately see that the sheared algebra morphism ^0:iR(-1)j! K of
Section 5 induces a vector space isomorphism H* ^0 on cohomology
that is also a sheared algebra map.
We hope to apply the results of this section to the setting of the Miller
spectral sequence [KL1 , Mill]. The built-in "unstableness" of the action
of K on the homology of infinite loopspaces should enable the use of
48 DAVID J. PENGELLEY AND FRANK WILLIAMS
the standard Ext functors in place of the somewhat artificial Unext,
and thereby result in shortened calculations. Also we feel that, with
the simpler coproduct formula of K* as compared to R*, the range of
applicability of the Miller spectral sequence can be extended, leading
to more results in the direction of those of [KL2 ]. We are presently
working on such computations.
9.Future directions
As we have seen, the algebra K has a very rich internal structure.
It may be regarded as being closer to the actual topology of spaces
than either A or R, and we hope this, along with the "nice" coproduct
formula in K* , may facilitate applications. We have also seen how K,
and its cohomology, are related to A and R via various interrelated
sheared algebra bijections.
There are several avenues of further research and application that
we are currently pursuing. We briefly list a few.
(1) The work of Bisson-Joyal [BJ ] approaches K from a very different
perspective. Combining our point of view with theirs is providing new
insights as well as new results. In particular, in [BPW ] we use the
extended Milnor Hopf algebra [BJ ] to explain the shearing that occurs
in our map 1 : K ! A.
(2) Although the present paper provides a unifying treatment of
homology and cohomology operations through the use of K and its
related algebras, it would be nice to see a direct connection between the
Steenrod algebra and the Dyer-Lashof algebra. We treat this question
in [PW1 ] in which we develop an inverse limit process, based on the
way K(1) was formed from K, to produce the Steenrod algebra from
the Dyer-Lashof algebra and vice versa.
(3) As mentioned above, we plan to examine the categorical structure
of sheared algebra morphisms in a future paper. We hope to relate this
to the concept of gradings over various monoid structures on Z+ x Z+
that was discussed following Definition 2.1.
(4) The "Nishida" actions of Kop on K based on fundamental classes
in positive degrees, mentioned above in section 4, are closely related to
the actions of the Steenrod algebra on the homology and cohomology
of iterated loopspaces of spheres. We are interested in describing these
actions through use of the maps ; ff, and V of this paper.
(5) As mentioned in section 8, we also plan to use homological calcu-
lations based on K to improve and extend the use of the Miller spectral
spectral sequence [KL2 , Mill] in the computation of the homology of
infinite loopspaces.
SHEARED ALGEBRA MAPS AND OPERATION BIALGEBRAS 49
(6) In many cases the statements of theorems become more concise
through use of K. For instance, the description of the homology of
iterated loop spaces in terms of K is well-known [CLM , CPS , CCPS ,
May2 ]. Use of K in cohomology can also produce elegant results. In
[PW2 ] we use K to describe bases for certain cyclic unstable A-modules.
In particular, we show that the module M(n) which is the quotient of
the free unstable A-module on a generator x2nof degree 2n by the action
n-2
of A(Sq1; : :;:Sq2 ) can be expressed as a K-module with basis the
set {DIx2n} where DI is admissible and I consists only of entries of the
form 0 and powers of 2 that are less than 2n. We use this description to
give the following characterization of the Dickson algebras (our version
of a recent result of F. Peterson). The Dickson algebra K*n+1;*is the K-
algebra quotient of the free unstable K-algebra on M(n) by the single
additional relation D2n-1D2n-1x2n = x2nD2n-1x2n:
Finally, we close with the following particularly elegant formulation
of the theorem of J. Lin [Li] about the action of the Steenrod algebra
in the cohomology of finite H-spaces.
Let QH-iX denote the indecomposable quotient of H*X in de-
gree -i (i 0). Let Nj be the natural numbersLwith j ones in
their dyadic expansion, and let QH(j)X = i2NjQH-iX. Clearly
L
QHe*X = jQH(j)X. Then
Theorem 9.1. Let X be a simply connected finite H-space, and let
K = {Di11Di23: :D:ik2k-1.|.{.ik} is eventually zero}:
Then for each j,
j-1)
QH(j)X = K . QH-(2 X:
The theorem may be proved by a straightforward induction from
Lin's theorem, which says that under the hypotheses on X, any inde-
composablerin degree -(2r+1k + 2r - 1) for k > 0 is in the image of
Sq2 k(which turns out to be just D2r-1).
But the theorem is perhaps better understood as follows, and per-
haps even stated a little more strongly. Elements of K acting on
j-1)
x 2 QH-(2 X retain exactly j dyadic ones in the resulting degree
(indecomposability is not needed for this), and moreover every inde-
composable y 2 QH(j)X arises in this way. In actual fact, for y in
fixed degree, the element DI 2 K; for which y = DIx, is uniquely
determined (provided we do not redundantly insert D2j-1, for which
D2j-1x = x), and is easily calculated from the dyadic expansion.
For instance, any indecomposable y in dyadic degree -101111100111
is the image of precisely D223-1D28-1on an indecomposable x in dyadic
50 DAVID J. PENGELLEY AND FRANK WILLIAMS
degree -111111111. As the class y is traced back towards x by suc-
cessive individual operations and intermediate classes, the number l of
trailing ones in each intermediate dyadic degree dictates the subscript
2l- 1 of the next operation; and each operation always simply excises
the rightmost zero in the expansion. Thus the lengths of blocks of con-
secutive zeros in the expansion of the degree of y are the exponents, in
reverse order, of DI.
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New Mexico State University, Las Cruces, NM 88003
E-mail address: davidp@nmsu.edu
E-mail address: frank@nmsu.edu