UNSTABLE MODULE PRESENTATIONS FOR THE
COHOMOLOGY OF REAL PROJECTIVE SPACES
DAVID J. PENGELLEY AND FRANK WILLIAMS Abstract. There is much we still do not know about projective spaces. We describe here how the mod two cohomology of each real projective space is built as an unstable module over the Steenrod algebra A, or equivalently, over K, the algebra of inherently unstable mod two "lower operations" originally introduced by Steenrod. In particular, to produce the cohomology of projective space of each dimension we consider the well-known minimal set of unstable module generators and construct a minimal set of unstable relations. Three new perspectives we blend for this purpose are:*
to focus solely on the two-power Steenrod squares that generate A to understand the A-action in a process we call "shoveling ones";* to describe every element in a canonical way from a particular unstable generator by composing operations from the algebra K;* to shift attention when studying an unstable A-module to considering and analyzing it directly as an equivalent K-module.
1. Introduction Much of the structure of the real projective spaces RP (m) and their limitR P (1) is captured in their mod two cohomologies, which are unstablealgebras over the Steenrod algebra A. As algebras their structure could
hardly be simpler: H*RP (1) ,= F2 [t] with deg (t) = 1, and the inclusionsR
P (m) ae RP (1) induce H*RP (m) ,= F2 [t] / \Gamma tm+1\Delta . As unstable algebrasover A, their structure is uniquely determined by the basic properties of
unstable A-algebras, namely the unstable dimension condition, the Cartanformula, and the topological consequence that
Sq0 acts as the identity [8].From these it is obvious that the action is given by
(1) Sqk (tn) = `nk'tn+k, where A is generated as an algebra by the Steenrod squares \Phi Sqk | k >= 0\Psi ,which are subject to the Adem relations [8]. However, the structure of these cohomology algebras as unstable A-modules is nonetheless very elaborate.
1991 Mathematics Subject Classification. Primary 55R35; Secondary 55S05, 55S10. Key words and phrases. Steenrod algebra, unstable module, Kudo-Araki-May algebra, real projective space, presentation.
We thank the referee for very helpful suggestions on exposition.
1
2 DAVID J. PENGELLEY AND FRANK WILLIAMS For instance, we can ask what are minimal sets of A-generators and A-relations, an important question for applications.
We will describe such minimal unstable presentations for the cohomol-ogy of all these projective spaces in the category of unstable A-modules. First we describe minimal A-presentations for H*RP (1) and its naturalA-filtration pieces
FsH*RP (1). Our A-presentation for H*RP (1) is in [7,Theorem 6.5], but the proofs there are circuitous and rely on methods that
are ill-suited to our primary goal here of minimally presenting the coho-mologies
H*RP (m) of the finite projective spaces from the presentations of FsH*RP (1)). We thus propose a new perspective, taking three distinctiveand atypical points of view on unstable modules, and we provide proofs of
all our results from this perspective.As an initial indication of our results, we state now a form of our minimal presentation of H*RP (1), which will emerge from the detailed theoremsthat follow.
Recall first that there is a filtration Fs = FsH*RP (1), for s = 0, 1, . . . , 1,of A-submodules given by the F
2-subspace Fs = F2 {tr | ff(r) <= s}, where ff(r) is the number of ones in the binary expansion of the nonnegative integer
r. That these vector subspaces are A-closed is clear from the action formulaabove.
Minimal presentation for FsH*RP (1): see Theorem 3.1 and Re-mark. For 0 <=
s <= 1, a minimal presentation of FsH*RP (1) as anunstable module is given by generators
x2j-1, for j an integer with 0 <= j <= s (degree of x2j-1 is 2j - 1), and relations
Sq2
k x
2j-1 = Sq2
j-1 Sq2kx
2j-1-1, for 0 <= j <= s and 0 <= k <= j - 2.
To prove this and our minimal presentations for the finite projectivespaces, we develop the following three points of view.
1. [Shoveling ones] Our first point of view is to study the A-action entirely in terms of the set of two-power Steenrod squares nSq2
i | i >= 0o
that minimally generates A. We do this partly because the cohomology ofa projective space has at most one nonzero element in each degree, and the
two-power squares connect these in a manner particularly easily viewed interm of the binary expansions of their degrees; so all calculations will reduce to simple relationships between binary expansions. Additionally, we shallsee that the minimal set of A-relations forming a projective space is easily described using just these two-power squares, in a process we call `shovelingones', especially when combined with our second point of view.
2. [Represent with K] Our second point of view appears at first sightto contradict the first. It is to label and conceive of all elements in unstable modules primarily via iteration of Steenrod's original unstable "cup-i"squares on the A-generators of our module. We denote these operations
PRESENTATIONS FOR COHOMOLOGY OF REAL PROJECTIVE SPACES 3 by {Dj | j >= 0}. They generate the Kudo-Araki-May algebra K, with itsown set of K-Adem relations [5]. The
Dj are related to Steenrod squares via Steenrod's defining relationship Sqixm = Dm-ixm on a class xm of de-gree
m. The homology analogue to our point of view is already standardin studying the homology of loop spaces [1, 2, 3, 4]. While the
Sqi arestable operations in that they are preserved by suspension, the Dj are not.We find that the form of monomials using the unstable Dj is often much more useful than those in the stable Sqi for describing elements and bases inunstable modules. For projective spaces the
Dj are particularly efficacioussince we can relate their iterations well to the binary expansions of degrees
of elements. This allows us to choose a very useful canonical representationfor each element in terms of a particular monomial in the
Dj on a particularunstable module generator, which in turn enables us to analyze the global
module structure. To wit, we will prove (cf. [7, Theorems 6.1, 6.2]) A basis for H*RP (1) in terms of K: from Theorem 2.1. For s ? 0,a basis for
FsH*RP (1)/Fs-1H*RP (1) in terms of admissibles from K on t2
s-1 is given by
{De00 * * * Dei2i-1 * * * Des-12s-1-1t2
s-1 | each e
i ? 0},
with corresponding degrees Ps-1i=0 2e0+***+ei+i having ff-number s. (Noticethat the
ei are simply the lengths of the blocks of consecutive zeros in thebinary expansions of degrees.)
The first two viewpoints are not contradictory: we use the Dj to represent elements for a basis, and the Sq2
i to see how the module action connects
them, with both viewpoints closely tied together via binary expansions ofdegrees. Of course to meld the two points of view effectively we must be
comfortable working flexibly and simultaneously with both the Sq2
i and the
Dj in whatever combination is most judicious.3. [Study unstable A-modules as K-modules] This brings us to
our third viewpoint, which goes beyond the second, to view our unstablemodules interchangeably over the Steenrod algebra A or the Kudo-ArakiMay algebra K. In particular, the proof of the minimal presentation theoremabove for
FsH*RP (1) relies on shifting to module analyses entirely over K.We end the introduction with a statement of our main result on presentations for finite projective spaces. To state it, we first note how to recastthe A-relations of the presentation of
FsH*RP (1) in terms of the points ofview just espoused.
Remark. By induction on j one sees that the set of relations
Sq2
k x
2j-1 = Sq2
j-1 Sq2k x
2j-1-1, for 0 <= j <= s and 0 <= k <= j - 2,
is equivalent to the set
Sq2
kx
2j-1 = Sq2
j-1Sq2j-2 * * * Sq2kx
2k+1-1, for 0 <= j <= s and 0 <= k <= j - 2,
4 DAVID J. PENGELLEY AND FRANK WILLIAMS and the latter is equal to the set
Sq2
kx
2j-1 = Dj-k2k-1x2k+1-1, for 0 <= j <= s and 0 <= k <= j - 2.
Our main theorem of this paper is the following. Minimal presentation for finite projective spaces: Theorem 4.2.Let 2
s - 1 <= m <= 2s+1 - 2. Let ff be the number of ones in the binary
expansion of m + 1, and write the expansion as m + 1 = Pff-1i=0 2e0+***+ei+i(each
ei >= 0). Also let fi be the number of leading ones in this expansion;in other words, if there is a nonzero
ei, then ff - fi is the largest index i forwhich ei is nonzero.Then a minimal presentation for
H*RP (m) is given by generators
x2j-1 in degree 2j - 1, for 0 <= j <= s, and the three sets of relations (a) Sq2
k x
2j-1 = Dj-k2k-1x2k+1-1, for 0 <= j <= s and 0 <= k <= j - 2;
0 = Dei-l2i+r-1Dei+12i+1+r-1 * * * Deff-12ff-1+r-1x2ff+r-1,(b) for 0 <= i <= ff - 1 and 0 <= l < ei, where r = e0 + * * * + ei-1 + l; and
(c) 0 = Ds+1-k2k-1 x2k+1-1, for s + 1 - fi <= k <= s - 1. Remark. While the first set of relations are simply those we used to present FsH*RP (1) above, it is not easy to get a sense for the second and thirdset of relations from this formulaic description; they are best understood
in terms of visualizing binary expansions. In section 4 where we prove thetheorem, we first illustrate the theorem with the special family where
m is ofthe form 2 s+1 -2 (in which case the second set of relations is empty). Notice
that the third set of relations for H*RP (2s+1 - 2) consists of the additionalrelations that would appear if we instead presented
Fs+1H*RP (1) as above,and then set the unnecessary generator x2s+1-1 on the left side equal to zero.This is why we write relations (b) and (c) with zero on the left.
Example. For m = 6, of form 2s+1 - 2 with s = 2, we have that H*RP (6)is minimally presented as an unstable A-module by generators
x1 and x3with the relation Sq1x3 = D20x1 = Sq2Sq1x1 of the first kind, and relations0 = D30x1 = Sq4Sq2Sq1x1 and 0 = D21x3 = Sq4Sq2x3 of the third kind.
Remark. In section 4 we also discuss in the general case all the relationsfrom the point of view of binary expansions, the interplay between the second
and third sets of relations, and give another concrete example, for a moregeneral
m, in terms of binary expansions. When thought of in terms ofbinary expansions, it is not difficult to visualize how the main theorem
follows from the minimal presentation theorem above for FsH*RP (1).
PRESENTATIONS FOR COHOMOLOGY OF REAL PROJECTIVE SPACES 5
2. Two points of view 2.1. Two-power squares and shoveling ones. To flesh out our firstpoint of view, given
n >= 0, with binary expansion n = Pi=0 ai2i (i.e., ai = 0 or 1), we will often write (a0 * * * ai * * * ) for n, in other words, we maydenote
n simply by listing the digits of its binary expansion (enclosed inparentheses), but in the reverse order from what is ordinary when writing
numbers in decimal form. The reader will soon see why reverse listing ofthe binary digits is helpful to our work. Sometimes we need explicitly to indicate the exponent of the power of two corresponding to a digit (alsocalled its `place' or `position'), so we may write
(a0 * * * ai
i * * * )
to indicate this. Note that an ellipsis may stand for any combination ofdigits. We will write (* * *
al $ ak * * * ) to mean that al = aj = ak for all l <= j <= k. Although the nonzero elements of the cohomology of H*RP (1) ,=F
2 [t] are in one-to-one correspondence with the binary expansions of theirdegrees, we wish explicit corresponding labels for the elements themselves,
so in degree n = (a0 * * * ai * * * ) we write t (a0 * * * ai * * * ) for the element tn.Now we can observe how 2-power squares act in relation to binary expansions. From equation (1), the action of Sq2
i on an arbitrary element
x = t (a0 * * * ai * * * ) is clearly zero if ai = 0, while if ai = 1, the result isnonzero and can be written as follows, since it is the unique nonzero element in degree
(a0 * * * ai * * * ) + (0 $ 0 1i ).
Write
x = t(* * * 1i 1 $ 1 0j * * * ),
i.e., j is the first place above i with a zero digit. Then since
(* * * 1i 1 $ 1 0j * * * ) + (0 $ 0 1i ) = (* * * 0i 0 $ 0 1j * * * ),
we have
Sq2
i x = Sq2it(* * * 1
i 1 $ 1 0j * * * ) = t(* * * 0i 0 $ 0 1j * * * ).
Notice in sum then, that the effect of applying Sq2
i to the nonzero element
in a degree with digit 1 in place i is to replace that digit 1, along with anyconsecutively above it, with zeros, and to convert the first zero encountered
above place i into a 1. We refer to this procedure on binary expansions as`shoveling ones', since we imagine pushing the 1 in place
i rightwards intothe first open (i.e., value 0) place j, and sweeping away any consecutiveintervening ones. We will call a shovel `simple' if there are no intervening
ones (i.e., j = i + 1), and `longer' if j > i + 1, so that some ones are actuallyswept into oblivion by our shovel.
6 DAVID J. PENGELLEY AND FRANK WILLIAMS
From this shoveling perspective we immediately notice several things(some already well-known) intimately relating the A-action on projective spaces with ff-number of degrees. Indeed, since 2-power squares generateA, and since shoveling obviously never increases
ff-number, the A-action re-spects the filtration FsH*RP (1) by ff-number defined in the introduction.Notice too that a simple shovel leaves
ff-number unchanged, while a longershovel reduces it. Moreover, from this point of view it is obvious that
{t2
j -1 = t(1 $ 1
j-1) | j >= 0}
is the unique minimal set of A-generators of H*RP (1), since clearly none ofthese elements is in the image of a shovel (which always produces a 0 to the
left of a 1), and any binary expansion is the image of simple shovels from thegenerator with the same
ff-number, for instance by `simply' shoveling onesrightward as needed into their desired places, beginning at the right. And
it is equally clear that these same elements for j <= s are the minimal set ofgenerators of the filtration
Fs. We shall call the elements of this minimalset `minimal generators'.
2.2. Candidates for a minimal set of A-relations. Clearly in seekingnecessary A-relations in a minimal presentation for
FsH*RP (1), we should
first consider, for a fixed minimal generator t2
j-1 with j <= s, which generators of lower filtration it may be related to and how. Since 2-power squares generate A, the first possibilities to consider are Sq2
k (t2j-1) for 0 <= k <= j-1.
We have, from above, the shovel
Sq2
k (t2j-1) = Sq2k t(1 $ 1
j-1) = t(1 $ 1 0k $ 0 1j ) if k <= j - 1.
While for k = j - 1 this equation involves a simple shovel, and yields noconnection to lower filtration, for 0 <=
k <= j - 2 it involves longer shovels,and we have thus discovered essential and independent connections between
filtrations, so that relations corresponding to these, at a minimum, must beincluded in any presentation of
FsH*RP (1). We shall show that preciselythis family of relations minimally suffices to present
FsH*RP (1).First we need to decide how each element on the right side in the above
equalities should best be represented via A-action on a minimal generator,in order to write down a claimed abstract presentation. For this we need to flesh out our second point of view, since there may be a multitude of waysto represent such an element via module action on minimal generators.
2.3. Representing basis elements by iterating unstable operations Dj from K. We will do more than just represent the particular elements ofour candidate relations above. We will represent every element of
H*RP (1)as a preferred chosen monomial in the Dj's applied to a preferred minimalA-generator.
We remind the reader that the unstable operations Dj compose very dif-ferently from the Steenrod squares, and form their own algebra of operations
PRESENTATIONS FOR COHOMOLOGY OF REAL PROJECTIVE SPACES 7 K, the Kudo-Araki-May algebra. Despite this algebra being extremely dif-ferent from A, unstable modules over K are in perfect correspondence with those over A. Let us recall [5] just a few basic features of K, and how it willinform what we mean by an `unstable module'.
The bigraded algebra K is generated by elements Di 2 K1,i, i = 0, 1, . . .,subject to the (Adem) relations
DiDj = X
k `
k - 1 - j 2k - i - j'Di+2j-2kDk, (i > j).
The degree of elements in K satisfies the condition that multiplication bea map K
m,i \Omega Kn,j ! Km+n,i+2mj. The identity element is 1 2 K0,0.A monomial
DI = Di1 * * * Din is admissible provided the multi-index I =(
i1, . . . , in) is nondecreasing, and the admissibles form a basis for K. Agraded K-module
M* is one that satisfies the requirement Km,i \Omega Mj ! M2mj-i. It is unstable provided that each Di : Mi ! Mi is the identityand that
Di(Mk) = 0 for i > k. Unstable modules over K and A alwayscorrespond, via the relationship
Sqixm = Dm-ixm on a class of degree m, soin the sequel we shall refer simply to an unstable module without preference
for either A or K. Note too that D0xm = Sqmxm. In particular, this shouldpermanently dispel any misapprehension that might confuse
D0 2 K1,0 withthe identity element 1 2 K
0,0. For the cohomology of a space, the structureand correspondence extends further to that of an unstable K-algebra, with
its own Cartan formula, but we will not need that here.To advance our second point of view, we now assign our preferred representation to each element in H*RP (1) (and thence in its filtrations Fjand quotients
H*RP (m)) as a single admissible monomial in K applied to apreferred minimal module generator. It is clear from our shoveling discussion above that an arbitrary element may have a myriad of representationsvia iterated operations on various possible minimal module generators. We intend to single one out first by matching the ff-number j of the degree ofour arbitrarily chosen element
x with the minimal generator of the same ff-number, shunning all representations that reduce ff. Even with the same ff-number, there will generally be numerous ways to choose and to order iteration of operations that will carry our chosen generator t2
j-1 = t(1 $ 1)
to the given x. Since the binary expansions of these two degrees both have j ones, they differ by the placement of the j blocks of zeros (some possiblyempty) in the expansion of the degree of
x, one to the left of each of its jones.
We will use these blocks to dictate a specific admissible in the D's to produce x from t2
j-1. First note that in H*RP (1) the criterion for whether
Di(tn) is nonzero is given by \Gamma ni \Delta , since
Di(tn) = Sqn-i(tn) = ` nn - i't2n-i = `ni 't2n-i.
8 DAVID J. PENGELLEY AND FRANK WILLIAMS
Our plan is to express x from t2
j-1 not necessarily by iterating simple shovels (via 2-power squares), but more efficiently, by inserting a newneeded zero in the degree expansion with each operation, in a way that
pushes the entire part of the degree expansion to the right of the insertionpoint rightwards one position to accommodate the new zero. And we insert the new zeros working from right to left. For the first block of zeros thisamounts to simple shoveling via 2-power squares, since only a single one is being shoveled, but after this block non-shovels will be used, since multi-ple ones separated by zeros are being pushed rightwards. Nonetheless, we shall now see that when all is expressed in terms of D's, the process is quitetransparent.
The key is to observe how a single operation can insert a zero after (i.e.,to the right of in the reversed binary expansion) trailing ones (we call ones "trailing" if they are at the left of the reversed binary expansion, represent-ing the least significant digits), thus shifting an entire initial part of the expansion rightwards one position. We claim this will be accomplished by
D2k-1t(1 $ 1k-1 * * * ) = t(1 $ 1k-1 0 * * * ),
where the two ellipses represent the same string (shifted right by the oper-ation
D2k-1). This holds because the resulting degree is clearly
2 * (1 $ 1k-1 * * * ) - (2k - 1) = (1 $ 1k-1 0 * * * ),
and the result is nonzero because`
(1 $ 1k-1 * * * )
(1 $ 1k-1) ' = 1.
Thus it is clear that for an arbitrary degree
(2) (0 $ 0-- -z ""
e0
1 * * * 1 0 $ 0-- -z ""
ei
1 * * * 1 0 $ 0-- -z ""
ej-1
1) =
j-1X
i=0 2
e0+***+ei+i
with ff-number j (note some of the ei may be zero), iterations of the above nature on t2
j-1, working from right to left through the blocks, lead to a
particular admissible monomial involving only D's of the form D2k-1, for k < j, applied to t2
j-1, and that the degrees and their elements are in one
to one correspondence with such monomials in K. For example, for j = 8,
t (0011000110110011) = D20D33D15D263t (11111111) . (Notice here e0 = 2, e1 = 0, e2 = 3, etc., and that this correspondence isbest conceived using reverse binary digit representations.)
An analogous phenomenon in the context of finite H-spaces was alreadyobserved and illustrated in [5, p. 1490f]. In general we now clearly have
PRESENTATIONS FOR COHOMOLOGY OF REAL PROJECTIVE SPACES 9 Theorem 2.1 (A basis for H*RP (1) in terms of K). For j ? 0, a basis for FjH*RP (1)/Fj-1H*RP (1) in terms of admissibles from K on t2
j-1
is given by
{De00 * * * Dei2i-1 * * * Dej-12j-1-1t2
j-1 | each e
i ? 0},
with corresponding degrees Pj-1i=0 2e0+***+ei+i as in displayed equation (2)above.
This form of basis thus allows us to state specific module relations between the minimal generators {t2
j-1 | j >= 0} in H*RP (1), based on the
preliminary calculations above of candidates for a minimal set of modulerelations:
Sq2
k (t2j-1) = t(1 $ 1 0
k $ 0 1j ) for k <= j - 2. Thus we have
Sq2
kt2j-1 = Dj-k
2k-1t
2k+1-1 for k <= j - 2.
3. Minimal module presentations for the filtrations of
H*RP (1)
From directly above, our theorem about FsH*RP (1) will be Theorem 3.1 (Minimal presentation for FsH*RP (1)). For 0 <= s <=1, a minimal presentation of
FsH*RP (1) as an unstable module is givenby generators
x2j-1, for j an integer with 0 <= j <= s (degree of x2j-1 is 2j - 1), and relations
Sq2
k x
2j-1 = Dj-k2k-1x2k+1-1, for 0 <= j <= s and 0 <= k <= j - 2.
Remark. Already in the introduction we noted that these relations can berecast solely in terms of the A-action. They are equal to the set
Sq2
k x
2j-1 = Sq2
j-1 Sq2j-2 * * * Sq2k x
2k+1-1 for 0 <= j <= s and 0 <= k <= j - 2,
and equivalent to the set
Sq2
kx
2j-1 = Sq2
j-1 Sq2kx
2j-1-1 for 0 <= j <= s and 0 <= k <= j - 2.
In preparation for the proof of Theorem 3.1, notice that from above wealready have a basis in terms of K for the filtration quotients
Fj/Fj-1 of H*RP (1). Therefore, since our goal is to present FsH*RP (1) via gen-erators and relations, we next endeavor to identify abstract minimal Kpresentations for these filtration quotients Fj/Fj-1. These abstract presen-tations, when recast via A-presentations, will perfectly match the obvious filtered quotients of the presentation in the theorem, and the rest will followeasily.
10 DAVID J. PENGELLEY AND FRANK WILLIAMS 3.1. A third point of view: analyze as a K-module. We wish to prove(cf. [7, Theorems 6.1, 6.2])
Theorem 3.2 (Filtration quotients presented as cyclic modules).For
j >= 0, the filtered quotient FjH*RP (1)/Fj-1H*RP (1) is A-isomorphic to the free unstable cyclic module on a generator x2j-1 in degree 2j -1, modulo the left ideal generated by nSq2
k | k <= j - 2o, which module we denote
by M(j, 1) for consistency with [7]. The abstract cyclic module M(j, 1) thus has basis as described in Theorem 2.1, with x2j-1 replacing t2
j-1. In particular, the abstract cyclic module is nonzero, and of rank one, precisely inthose degrees with
ff-number j.
Our method of proof will go beyond just describing elements in these A-modules via K-operations and a basis in terms of K. We will actually shift
to viewing these unstable A-modules equivalently as K-modules, since thereis a special feature of the product structure of K that we wish to exploit. The existence of this feature, inherent to the algebraic structure of K asopposed to that of A, demonstrates why shifting to the K-module view is important. Notation. Let Fx2j-1 denote the free unstable module on x2j-1 in degree 2j - 1.
To prove Theorem 3.2, indeed to obtain a basis for Fx2j-1 /A iSq2
k | k <= j - 2j x
2j-1written in terms of K, we will analyze it by shifting to view it as a K-module,
for which purpose we make two definitions. Definition. Let Ij = A iSq2
k | k <= j - 2j, a left ideal in A depending on
j. Definition. Let J = K \Gamma Di | i is not of the form 2l - 1 for any l\Delta , a leftideal in K not depending on
j.
Despite the fact that J does not depend on j, while Ij does, we can provethat
Theorem 3.3 (Shift presentation to K-module). For each j >= 0, inF
x2j-1 the submodules Ijx2j-1 and J x2j-1 are equal. Equivalently,
Fx2j-1 /A iSq2
k | k <= j - 2j x
2j-1 = Fx2j-1/K iDi | i 6= 2l - 1j x2j-1.
This converts the minimal A-presentation of this abstract module into aminimal K-presentation.
Remark. To provide some motivation for why these very different lookingquotients might be the same, consider which Steenrod squares we expect to survive on the left side. Certainly Sq2
j-1x
2j-1 survives, and we noteit equals D
2j-1-1x2j-1, which also survives on the right. Those familiar
PRESENTATIONS FOR COHOMOLOGY OF REAL PROJECTIVE SPACES 11 with the Adem relations in A will also expect that Sq2
j-1+2j-2 x
2j-1 maysurvive on the left, and it equals the survivor D
2j-2-1x2j-1 on the right.This pattern continues.
Proof. We show bicontainment for the generators of the two left ideals.
For k <= j - 2, we consider the generator Sq2
k of I
j. Now Sq2
kx
2j-1 =
D2j-2k-1x2j-1, and 2j - 2k - 1 is not of the form 2l - 1, so D2j-2k-1 is agenerator of J .
In the other direction, let r < 2j - 1 be not of the form 2l - 1 for any l, and consider Drx2j-1 in Fx2j-1 /A iSq2
k | k <= j - 2j x
2j-1 = M(j, 1).
To show that Drx2j-1 = 0 2 M(j, 1), we will induct downwards on r. If 2j-1 - 1 < r < 2j - 1, then Drx2j-1 = Sqmx2j-1 with 0 < m < 2j-1,which is zero from the defining relations of M(
j, 1). Continuing downwards,suppose that r < 2j-1 - 1. Then we can write r = \Gamma 2j - 1\Delta - \Gamma 2ac + 2b\Delta ,with c odd and 0 <= b < a <= j - 1. We consider two cases.Case 1. Let \Gamma 2
j - 1\Delta - 2ac 6= 2l - 1, for any l. A calculation with theK
Adem relations, combined with the unstable condition and the inductivehypothesis, yields
Dr+2a+1cD(2j-1)-2acx2j-1 = DrD2j-1x2j-1 = Drx2j-1. Since \Gamma 2j - 1\Delta - 2ac > r, inductively we have Drx2j-1 = 0.
Case 2. Let \Gamma 2j - 1\Delta - 2ac = 2l - 1 for some l. It follows that l = a and that c = 2j-a - 1. Again using the Adem relations in K, combined with theunstable condition, the inductive hypothesis, and the assumption that
r isnot of the form 2 l - 1 for any l, we get
0 = D2j+1-2a-3*2b-1D2a+2b-1x2j-1 = DrD2j-1x2j-1 = Drx2j-1.
\Lambda
Now we recall a remarkable feature of the left ideal J in K, for which weknow no analog in A.
Theorem 3.4 (Quasi-two-sided ideal in K; no apparent A analog).The left ideal J is quasi-two-sided in K, i.e.,
{admissible Di1 * * * Dir * * * Diq | at least one Dir is in J (i.e., ir 6= 2l - 1)} is contained in J .
In other words, any admissible in K with a generator of J anywhere inits product can be written in terms of elements with generators of J on the
right.This property of K was proven from the K-Adem relations in Theorem 2.9 of [6].The preceding two theorems now come together with Theorem 2.1 to prove Theorem 3.2.
12 DAVID J. PENGELLEY AND FRANK WILLIAMS Proof of Theorem 3.2. First note that nSq2
k | k <= j - 2o acts trivially on
t2
j-1 2 F
j/Fj-1, since Fj/Fj-1 is zero in the target degrees. Thus there is
a (unique) A-map j : M(j, 1) ! Fj/Fj-1 with j (x2j-1) = t2
j-1. From
Theorems 3.3 and 2.1 we have a module epimorphism
Fx2j-1/K iDi | i 6= 2l - 1j x2j-1 = Fx2j-1 /A iSq2
k | k <= j - 2j x
2j-1
= M(j, 1)
j! F
j/Fj-1
= F2{De00 * * * Dei2i-1 * * * Dej-12j-1-1t2
j-1 | e
i ? 0}.
Moreover, from Theorem 3.4, every admissible in K (assumed ending in Drwith
r < 2j - 1 for unstable nontriviality and nonredundancy on a class indegree 2
j - 1), other than those appearing in the basis for Fj/Fj-1, actually
produces zero on x2j-1. So the map is an isomorphism. \Lambda
From the proof of the theorem we also have Corollary. For j >= 0, all admissibles in K (assumed ending in Dr with r < 2j - 1 for unstable nontriviality and nonredundancy on a class in degree2
j - 1), other than those shown in the basis above, are zero on both the
fundamental class t2
j-1 in the cyclic module F
jH*RP (1)/Fj-1H*RP (1)and the fundamental class
x2j-1 in the abstract cyclic module M(j, 1).
We end this section by proving the minimal presentations for FsH*RP (1). Proof of Theorem 3.1. First we create notation for the claimed module pre-sentation, and observe that desired maps exist between the presentation and
the filtrations in H*RP (1).Let N
s (s ? 0) denote the free unstable module on the set
x2j-1, for 0 <= j <= s (degree of x2j-1 is 2j - 1), subject to the relations
Sq2
k x
2j-1 = Dj-k2k-1x2k+1-1, for 0 <= j <= s, 0 <= k <= j - 2.
We verified earlier that the images of these relations are satisfied in Fs, sothere are epimorphisms of unstable modules
's : Ns ! Fs with 's (x2j-1) =
t2
j-1 for all 0 <= j <= s, such that the diagram
's-1 : Ns-1 ! Fs-1# #
's : Ns ! Fs commutes, where Ns-1 ! Ns is the obvious map between these presenta-tions.
Inductively assume that 's-1 is an isomorphism. It follows from thediagram that N
s-1 ! Ns is monic. We shall prove that 's is an isomorphism
PRESENTATIONS FOR COHOMOLOGY OF REAL PROJECTIVE SPACES 13 by showing that the induced map Ns/Ns-1 ! Fs/Fs-1 is an isomorphism.Clearly an unstable presentation for N
s/Ns-1 consists of the single generator x2s-1 and relations Sq2
k x
2s-1 = 0, for 0 <= k <= s - 2, yielding M(s, 1). Sothe induced map is an isomorphism by Theorem 3.2.
Minimality of the set of relations is clear from looking at filtration quotients, since the Sq2
k minimally generate A. \Lambda
4. Minimal module presentations of H*RP (m) With Theorem 3.1 in hand we can turn to minimal A-module presenta-tions for the finite projective spaces. We illustrate first with a special case,
presenting H*RP (m) for the particular values m of the form 2s+1 - 2. Proposition 4.1 (Presentation for a special family of finite projec-tive spaces). A minimal presentation of
H*RP (2s+1 - 2) as an unstableA-module is given by generators
x2j-1 in degree 2j - 1, for 0 <= j <= s, and relations
Sq2
kx
2j-1 = Dj-k2k-1x2k+1-1, for 0 <= j <= s and 0 <= k <= j - 2,
and
0 = Ds+1-k2k-1 x2k+1-1, for 0 <= k <= s - 1.
Proof. Clearly H*RP (2s+1 - 2) is the quotient of Fs = FsH*RP (1) ob-tained by killing all
tr in Fs with r ? 2s+1 - 1. So the generators andrelations from Theorem 3.1 for
Fs must all necessarily appear, since they liein degrees lower than 2 s+1 - 1. It is not immediately clear what minimal set
of A-relations should then additionally be imposed in Fs to achieve trivialityin degrees above 2
s+1 - 1 (in degree 2s+1 - 1, Fs is zero). We consider the
least such degrees nonzero in Fs with given numbers of trailing ones, namely
(0 $ 0 1s+1)
* * * (1 $ 1 0k $ 0 1s+1)
* * * (1 $ 1 0s-1 0 1s+1).
It is clear that each of these degrees holds a necessary relation, being thelowest degree above 2
s+1 - 1 with its number of trailing ones, and thus
unreachable from any others by shoveling. But clearly also any elementof
Fs in any degree above 2s+1 - 1 can be reached from one of these byshoveling, i.e. (using Theorem 2.1), from the set of elements of the form
Ds+1-k2k-1 x2k+1-1 in Fs. This is precisely the second set of relations. \Lambda
14 DAVID J. PENGELLEY AND FRANK WILLIAMS
A minimal presentation of H*RP (m) for arbitrary m is more complicated.It will need new relations accounting for the deviation of
m below the nearesthigher number of the form 2 s+1-2, but will still utilize a subset of the second
set of relations used above to present H*RP (2s+1 - 2). Theorem 4.2 (Minimal presentation for finite projective spaces).Let 2
s - 1 <= m <= 2s+1 - 2. Let ff be the number of ones in the binary
expansion of m + 1, and write the expansion as m + 1 = Pff-1i=0 2e0+***+ei+i(each
ei >= 0). Also let fi be the number of leading ones in this expansion;in other words, if there is a nonzero
ei, then ff - fi is the largest index i forwhich ei is nonzero.Then a minimal presentation for
H*RP (m) is given by generators
x2j-1 in degree 2j - 1, for 0 <= j <= s, and the three sets of relations (a) Sq2
kx
2j-1 = Dj-k2k-1x2k+1-1, for 0 <= j <= s and 0 <= k <= j - 2;
0 = Dei-l2i+r-1Dei+12i+1+r-1 * * * Deff-12ff-1+r-1x2ff+r-1,(b) for 0 <= i <= ff - 1 and 0 <= l < ei, where r = e0 + * * * + ei-1 + l; and
(c) 0 = Ds+1-k2k-1 x2k+1-1, for s + 1 - fi <= k <= s - 1. Remark. These relations may be demystified by observing, as will be ex-plained in the proof, that the second set of relations consists of elements in
degrees obtained by beginning with degree m+1, then successively replacingzeros in its binary expansion by ones, starting from the left, until only one zero remains, so as to stay inside Fs. And the third set then continues abovethe next power of two in all the lowest degrees with more trailing ones than the second set, but stopping with two zeros, again to stay inside Fs. Theproof spells this out in detail.
Example. If m + 1 = (1010011100111) , the second set of relations is in de-grees (1110011100111), (1111011100111), (1111111100111), and (1111111110111), and the third set picks up more trailing ones in the degrees (11111111110001)and (11111111111001) above the next power of two.
Remark. The first and third sets of relations arise from the status of H*RP (m) as the quotient of H*RP (2s+1 - 2), which latter was minimallypresented in Proposition 4.1. The second set of relations in the general
theorem truncates the cohomology down further from H*RP (2s+1 - 2) to H*RP (m), and the minimal third set of relations is only a subset of the sec-ond set in Proposition 4.1, since the addition of the second set of relations
for general m makes some of the second set of relations for H*RP (2s+1 - 2)from Proposition 4.1 redundant for minimally presenting
H*RP (m), basedon how many trailing ones can appear in the second set for
H*RP (m).The extreme case m = 2s+1 - 2 corresponds precisely to fi = s + 1, inwhich case all the ei are zero, there are no relations in the second set, and
PRESENTATIONS FOR COHOMOLOGY OF REAL PROJECTIVE SPACES 15 the third set is the full set from Proposition 4.1. At the other extreme, if fi = 1, then the third set is empty; this corresponds to the first half of eachdegree range for
m.
Proof of Theorem 4.2. Initially we impose the first set of relations in thetheorem. From Theorem 3.1, the resulting quotient provides a minimal presentation for the filtration Fs of H*RP (1). Hence we may now use theformulas for the unstable action in
H*RP (1), in particular we may shovelones. And furthermore, in degrees less than 2
s+1, this initial quotient, isomorphic to Fs, is also isomorphic to H*RP (2s+1 - 2).The remaining two sets of relations are to minimally ensure that the final
result is zero in all degrees greater than or equal to m + 1. Note that thethird set of relations in the theorem all lie in degrees at least 2
s+1, as seen
in the proof of Proposition 4.1. Thus we proceed first with relations in therange 2
s - 1 <= m <= 2s+1 - 2 within which m lies, and in which range the
presentation so far agrees with both Fs and H*RP (2s+1 - 2), having a singleelement in each degree.
Consider the expansion
m + 1 =
ff-1X
i=0 2
e0+***+ei+i = (0 $ 0-- -z ""
e0
1 * * * 1 0 $ 0-- -z ""
ei
1 * * * 1 0 $ 0-- -z ""
eff-1
1),
with ff-many ones. We need only identify sequentially by degree which ele-ments need to be killed in this range to produce
H*RP (m), simultaneouslytracking which elements will be zero in the developing quotient as we inductively impose relations in lower degrees. This will all be determined by howthe A-action, generated by shoveling, does or does not connect one degree to another in Fs. Then we can write the explicit relations to kill these interms of our canonical basis in terms of K for all elements of
Fs.Note first that if m = 2s+1 -2, then m+1 = 2s+1 -1 is already outside ourrange, so no relations in this range will be needed. However, if
m < 2s+1 - 2,then after killing the element in degree m + 1 itself, it is clear that thelist of additional necessary and sufficient relations in this range must be in
those degrees obtained from the binary expansion of m + 1 by successivelyreplacing its zeros by ones, from left to right. This is because any given element in the range is clearly in the A-image of the element in this list withdegree closest below the given element. Moreoever, none of the elements of increasing degree in this list can be reached from each other by shoveling,since their degrees' reversed binary expansions begin with ever increasing numbers of trailing ones. Finally, note that this list stops when there is asingle zero remaining, namely the rightmost zero in the original expansion. Otherwise the degree would become 2s+1 - 1, outside the range. Using thecorrespondence of Theorem 2.1 between degrees and a basis in terms of K for H*RP (1), these relations are precisely those in the second set in thetheorem. Note that in the degree of each relation,
r represents the number ofzeros in the expansion of the degree m + 1 being replaced by ones. Note too
16 DAVID J. PENGELLEY AND FRANK WILLIAMS that some of the ei may be zero, reflecting consecutive ones in the expansionof
m + 1, producing corresponding empty factors in the basis descriptionsin terms of K, except for the leading factor, for which the exponent
ei - l isalways positive.
Finally we consider what relations are still needed above degree 2s+1 -1 in order to ensure that the quotient is zero there. Clearly this will be some subset of the set of relations added in this range for H*RP (2s+1 - 2)in Proposition 4.1. The final relation we just added in the second set of relations has the largest number of trailing ones in its binary expansion.Recalling the definition of
fi in the theorem, this final relation in the secondset is in degree
(1 $ 1 0s-fi 1 $ 1s). Clearly any element of Fs in higher degree than this, and with no moretrailing ones, can be reached by shoveling from the element in this degree. So it remains only to deal with elements in degrees with more trailing ones,for which we consider the elements in degrees
(1 $ 1 0s-fi+1 $ 0 1s+1)
* * * (1 $ 1 0k $ 0 1s+1)
* * * (1 $ 1 0s-1 0 1s+1)
in Fs. These are precisely the lowest degrees above 2s+1 - 1 in Fs withmore trailing ones, so they are all necessary relations. (Note that this is also correct for the case when there are no type two relations, i.e., when m = 2s+1 -2, fi = s+1, and the relations above include all possible numbersof trailing ones, as necessary.) But they are also sufficient, since clearly any
element of Fs in any degree above 2s+1 - 1 with as many trailing ones asone of these can be reached from one of these by shoveling. So again using the correspondence of Theorem 2.1, we set the elements in these degrees tozero to write the third set of relations in the theorem. \Lambda
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PRESENTATIONS FOR COHOMOLOGY OF REAL PROJECTIVE SPACES 17 [5] D. Pengelley, F. Williams, Sheared algebra maps and operation bialgebras for mod 2
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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