International Journal of Pure and Applied Mathematics ————————————————————————– Volume 16 No. 3 2004, 285-296

THE BUNDLED AND PARTITIONED COMB IN THE MOD-p

Ismet Karaca1, Bekir Tanay2 § 1Department of Mathematics Faculty of Science Ege University Izmir, 35100, TURKEY e-mail: [email protected] 2Department of Mathematics Faculty of Science and Literature University of Mugla Mugla, TURKEY e-mail: [email protected]

Abstract: Judith H. Silverman has constructed a different method, named bundled and partitioned comb by using combinatorics in her paper [5], that whether a given Milnor element appears as a summand in the product two oth- ers. Here, the method and some results are generalized in the mod-p Steenrod algebra.

AMS Subject Classification: 55S10, 55S05 Key Words: Steenrod algebra, Milnor product, combinatorics, bundled and partitioned comb

1. Introduction

Let p an odd prime number throughout this paper. The study of mod-p Steen- rod algebra began with Norman Steenrod’s work constructing the Steenrod square operations

Received: June 12, 2004 c 2004, Academic Publications Ltd. §Correspondence author 286 I. Karaca, B. Tanay

i k k+2i(p−1) P : H (X; Zp) → H (X; Zp),

k where H is the k-th group of X with coefficient Zp. By composi- tion, these Steenrod square operations and Bockstein operations

q q+1 β : H (X; Zp) → H (X; Zp), associated with the exact sequence 0 → Zp → Zp2 → Zp → 0 with the property

β2 = 0 and β(x.y)= β(x)y + (−1)qxβ(y), dim(x)= q give rise to an algebra of operations acting on the Zp-cohomology groups of topological spaces. The structure of this algebra, Mod-p Steenrod Algebra, is eluciated by Adem [1], Cartan [2] and Serre [4]. Since the Bockstain operations does not have any role in this work we will study on the sub-algebra generated only by the elements P T . Throughout this paper the name Steenrod algebra will mean this sub-algebra. There are several bases in the mod-p Steenrod algebra. One of this basis, whose structure was given by [3], is the Milnor basis. The mod-p Steenrod algebra structure in Milnnor basis is given by the Milnor Product Formula which is mentioned in Section 4. The bundled and partitioned comb method was costructed by J.H. Silverman in her article [5] by using the prop- erties, named dimension, excess, of the Milnor elements in the mod-2 Steenrod algebra. This method decides whether a given Milnor element appears as a summand in the product of two others. We adopt hier some results into mod-p Steenrod algebra.

2. Preliminaries

Let T = (t1,t2,...) be a sequence of non negative integers almost all of which are zero and the Milnor element associated with this sequence is P (T ). If T is a sequence for which tl = 0, for l>m, we denote the corresponding basis element by P (T ) = (t1,t2,...,tm). The dimension |P (T )| of the Milnor element m k P (T ) is 2 |T |, where |T | = (p − 1)tk and the excess ex(P (T )) of the Milnor k=1 element P (T ) associated with this sequence is T = (t1,t2,...,tm) is .2ex(T ) m where ex(T )= tk. k=1 For Milnor basis, it is known how to express of two generators as a sum of other generators by Milnor product formula. This product formula involve THE BUNDLED AND PARTITIONED COMB... 287 binomial or multinomial coefficients taken mod-p. There are several criterion to compute this coefficient. Let pnσ pn2 pn1 and prσ pr2 pr1 be the p-adic represantation of the integers n and r respectively. We write n ⊲i r to mean ni ≥ ri and we say n dominates r (n ⊲ r) if n ⊲i r for all i. It is known that n the coefficient = 0 (mod p) ⇔ n ⊲ r (n ⊲ n − r). In other words, each r power of p appearing in the p-adic representation of n appears in exactly one of the p-adic representations of r and n − r. More genarally, if m ≥ 3 and m ri = n the multinomial coefficient giving the number of ways to divide a set k=1 of n elements into m subsets of orders r1,...,rm is written n s s s s = 1 2 3 m , r1 |r2||rm r1 r2 r3 rm where sl = rl + rl+1 + + rm. n The criterion mentioned above and inductive argument imply that = r1 |r2||rm 0 (mod p) ⇔ each power of p in the p-adic representation of n occurs in exactly one of the p-adic representation of r1,...,rm. For more details about the mod-p Steenrod algebra, see [6].

3. Structures on Combs

3.1. Combs

m k We interpret |T | = (p −1)tk as the value of the sequence T = (t1,t2,...,tm) k=1 in the system in which the l-th term counts for pl − 1 times its face value. Since pl − 1= p∗(pl−1 + pl−2 + + p1 + p0) with p∗ = p − 1 we can represent |T | in a different way:

1 2 m−1 m |T | = (p − 1)t1 + (p − 1)t2 + + (p − 1)tm−1 + (p − 1)tm ∗ 0 ∗ 1 0 ∗ m−1 m−2 1 0 = p p t1 + p (p + p )t2 + + p (p + p + + p + p )tm ∗ 0 ∗ 0 ∗ 0 ∗ 0 = p p t1 + p p t2 + p p t3 + + p p tm ∗ 1 ∗ 1 ∗ 1 . + p p t2 + p p t3 + + p p tm ∗ 2 ∗ 2 . + p p t3 + + p p tm . . . ∗ m−1 . + p p tm . 288 I. Karaca, B. Tanay

Therefore we can represent |T | with the picture below where i-th row is associ- ated with pi for all i = 0, 1,...,m − 1 p∗p∗ p∗ p∗p∗ p∗ p∗p∗ p∗ → p0 p∗p∗ p∗ p∗p∗ p∗ → p1 . . .. . p∗p∗ p∗ → pm−1 . This picture, or any obtained from it by a permutation of colums, will be called the comb of T and denoted C(T). A column of l p∗’s is called a tooth of length l−1 l, denoted τ l, and its weight is W (T )= p∗ pk = pl −1. The excess of C(T) is k=1 the number of teeth which is equal to ex(T ) and the weight of C(T), W (C(T )), is the sum of the weights of the teeth which is equal |T |. Example. Let p = 5 and find the comb C(T ) for the sequence T = (4, 3, 2): |T | = (p1 − 1)4 + (p2 − 1)3 + (p3 − 1)2 = p∗p04+ p∗p03+ p∗p13+ p∗p22+ p∗p12+ p∗p02. So the picture of the comb C(T ) is p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ .

3.2. Bundles

A bundle of size pσ is a collection of pσ teeth of the same length and represented in column form as a sort of generalized tooth having same number of p∗’s as the teeth including it but preceded by σ times zeros as the teeth. Example. Let p = 5. The bundle of 52 teeth of length 3 (according to the definition above) is 0 0 p∗ p∗ p∗ The sum of the weights of 25 teeth of length 3 is 25(53 − 1). The 5-adic representation of this number is: 25(53 − 1) = 3100 = 0.50 + 0.51 + 4.52 + 4.53 + 4.54, p∗ = 5 − 1. THE BUNDLED AND PARTITIONED COMB... 289

If we represent the coefficients of this representation vertically 0 0 p∗ p∗ p∗ we will have the bundle of 52 teeth of length 3 again. From this point, the number of the form pσ(pl − 1) identified as a bundle of pσ teeth of length l. Let T = (t1,t2,...,tm) be a sequence. We can write tl, 1 ≤ l ≤ m, as sums of powers of p and this writing gives rise a bundle structure on C(T ): if

n1,l n2,l ns,l tl = αn1 (tl)p + αn2 (tl)p + + αns (tl)p , then the teeth of length l are arranged in bundles of sizes pn1,l,pn2,l,..., pns,l. The orders of the bundles is not important. The comb having the bundles with cofficient αn1 (tl), αn2 (tl),...,αns (tl) as columns is called canonically bundled comb of T and denoted Cb(T ).

Example. Let p = 5 and find canonically bundled comb Cb(T ) of T = (13, 25, 10, 4):

|T | = (51 − 1)13 + (52 − 1)25 + (53 − 1)10 + (54 − 1)4 = (51 − 1)(3.50 + 2.51) + (52 − 1)(1.52) + (53 − 1)(2.51) + (54 − 1)(4.50) = 3.50(51 − 1) + 2.51(51 − 1) + 1.52(52 − 1) + 2.51(53 − 1) + 4.50(54 − 1) , therefore the picture of the canonically bundled comb of T is below 0 3.p∗ 0 0 4.p∗ 2.p∗ 0 2.p∗ 4.p∗ 1.p∗ 2.p∗ 4.p∗ 1.p∗ 2.p∗ 4.p∗

We can find same information about P (T ) in Cb(T ) as does the comb C(T ): A bundle of teeth pσ of length l has its topmost p∗ in the σ-th row and the σ,l l number ασ(tl).p .(p − 1) is called weight of this bundle. The sum of the σ,l numbers ασ(tl).p is the excess of Cb(T ) which equals to ex(T ) and the sum of the weights of bundles in the Cb(T ) is the weight of Cb(T ) which equals to |T |. Example. Let p = 5, T = (13, 25, 10, 4) and find ex(T ) and |T |. The topmost rows, in which the p∗’s are seen first and the coefficient, which arise 290 I. Karaca, B. Tanay

in the p-adic reprsentation of tl for all 1 ≤ l ≤ m, are found easily from the picture of Cb(T ) above. So,

ex(T ) = 2.p1 + 3.p0 + 1.p2 + 2.p1 + 4.p0 = 52 , |T | = 2.p1(51 − 1) + 3.p0(52 − 1) + 1.p2 + 2.p1 + 4.p01.

3.3. Partitions

We will define the partitioned comb, PC(T ), as a comb whose each tooth τ is split in two horizantally by choosing a partition number 0 ≤ π(τ) ≤ l indicating that the tooth is to be split above the π(τ)-th row. Grafically we represent the partition number of each tooth as a horizantal partition line across the tooth. Example. A partition can be given as below on C(T ) of the sequence T = (4, 3, 2): p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗

We can also costruct bundle structure on the partitioned comb too as before. The generalized teeth of a bundled partitioned comb, denoted PCb(T ), represent pσ teeth with the same length and same partition number. However, we can give a partition to a bundled comb such a way that each generalized tooth τ of size pσ and length l is assigned a partition number 0 ≤ π(τ) ≤ l indicating that the generalized tooth is to be split obove the (σ + π(τ))-th row.

3.4. Compatibility

Let T be a sequence. A partition of the comb C(T ) is compatible with a bundle structure on Cb(T ), if one can indicate the partition on a picture of the bundled l comb. That is, given integers 0 ≤ π ≤ l, let Nπ be the number of teeth of length l by the partition structure to have partition number π. The partition is compatible with the bundle structure if the generalized teeth of each length l in the bundled comb can be arranged in l groups in such away that the number l of ordinary teeth represented in the π-th group is Nπ. Each generalized tooth in the π-th group is then assigned the partition number π. Example. Let p = 5 and T = (0, 7) then the partition of C(T ) given by

p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ THE BUNDLED AND PARTITIONED COMB... 291 is compatible with the bundle structure below, with the assigment of partition number to teeth as indicated,

p∗ 4.p∗ 2.p∗ p∗ 4.p∗ 2.p∗ but is not compatible with the bundle structure below

0 2.p∗ p∗ 2.p∗ . p∗

If we change the partition of C(T ) as indicated below

p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ then this partition will be compatible with the bundle structure below

0 2.p∗ p∗ 2.p∗ p∗

The first part (resp. second part) of a partitioned bundled comb of a se- quence T , is the bundled comb obtained by replacing all the p∗’s of C(T ) below (resp. above) the partition lines with blanks (resp. 0’s).

4. Milnor Product Formula

Let R = (r1, r2,...,rm), S = (s1,s2,...,sm) and T = (t1,t2,...,tv) be se- quences of non negative integers with |R| + |S| = |T |. Now we will find the condition if P (T ) is a summand in the product P (R) P (S). In [3] Milnor describes the product in terms of certain matrices. Let MR,S be the set of infinite matrices

∗ x01 x02  x10 x11 x12  X = x x x  20 21 22   . . . .   . . . ..    292 I. Karaca, B. Tanay

of non negative integers with x00 = ∗, such that

∞ j p xij = ri, for all i ≥ 1 , (4.1) j=1 ∞ xij = sj, for all j ≥ 1 . (4.2) i=1

For each matrix X ∈ MR,S, define the sequence T (X) = (t1,t2,...) by l tl = xi(l−i) and let bl(X) be multinomial coefficient i=1 n . x0l x1(l−1) |xl0 Then the product P (R) P (S) is given by

P (R) P (S)= [b1(X)b2(X) ]P (T (X)).

X∈MR,S

Thus: P (T ) is a summand of P (R)P (S) ⇔ a(T ) = 0 (mod p) and bl(X) = 0 (mod p) for al l, where a(T ) is the number of matrices X ∈ MR,S with T (X)= T . Rather than trying to construct such matrices mentioned in the product formula one by one, it is often advantageous to translate the question into the language of combs. Now our goal is to find the condition if P (T ) is a summand in the product P (R) P (S) by using the bundled and partitioned comb structures. To be able to do this we must translate the Milnor formula into the language of bundle and partition structures. Fix R and S and suppose X ∈ MR,S with T (X) = T . The PX C(T ) is the partitioned comb induced with the matrix X which for all i, j has xij teeth of length i + j and partition number j. The matrix X is associated not only to the comb C(T ) but also to the partitioned comb PX C(T ). Example. Let p = 5, R = (12, 32), S = (5, 6), and the matrix

∗ 3 5  7 1 0  , 2 1 1   with T (X) = (10, 8, 1, 1). THE BUNDLED AND PARTITIONED COMB... 293

Is the Milnor element P (T ) a summand of the Milnor product P (R)P (S)? Firstly, let us answer this question by using formula described above: Since the first multinomial coefficient in the product b1 ≡ 0 (mod 5) the answer of the question is no. Now, let us answer the question by the bundle and partition structures. The comb PX C(T ) is

p∗ p∗ p∗ p∗ p∗ p∗ p∗p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ and bundled partitioned comb PX Cb(T ) is

0 2p∗ 3p∗ 2p∗ p∗ 0 p∗ p∗ p∗ 2p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗

But the partition on PX Cb(T ) is not compatible with the bundle structure on Cb(T ) below 0 0 3p∗ p∗ p∗ 2p∗ p∗ 3p∗ p∗ p∗ p∗ p∗ p∗ p∗

Because, the 1-st tooth of length 1 in Cb(T ) can not be arranged in three groups such that each group will be the one of the teeth of length 1 in PX Cb(T ). If we examine the comb PX C(T ) carefully we can se that the first part of it is the C(S), after arranging the teeth up to their lengths,

p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ ⇓ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ p∗ and the second part of it is the Cb(R), after arranging bundles up to their length 294 I. Karaca, B. Tanay and mod-p,

p∗ p∗ p∗ p∗ p∗ p∗ p∗ 0 p∗ p∗ 0 0 p∗ p∗ p∗ p∗ 0 p∗ p∗ p∗ ⇓ 0 2p∗ 2p∗ 0 0 p∗ 2p∗ p∗ 0 p∗ p∗ p∗

It is the fact that, we view PX C(T ) as the result of suspending bundles of teeth of C(R) below the teeth of C(S) of appropriate of length. That is, a tooth of length i + j and partition number j is obtained from a tooth of C(S) of length j by appending a bundle of pi teeth of length i of C(R). Equations (4.1) and (4.2) imply that the first part of PX C(T ) is exactly the comb C(S) and second part of PX C(T ) is a bundled comb for R. Conversely, any partition of C(T ) whose first and second parts are combs for S and R respectively is readily seen to be PX C(T ) for some Y ∈ MR,S with T (Y )= T .

5. Main Theorem

Let n(T )denote the number of partitions of canonical bundled comb Cb(T ) whose first parts are combs for S and whose second parts are bundled combs for R Theorem. The Milnor element P(T) is a summand in the product P (R) P (S) ⇔ n(T ) = 0 (mod p). Proof. From the properties of Milnor product we know that P (T ) is a summand of P (R) P (S) ⇔ a(T ) = 0 (mod p) and bl(X) = 0 (mod p) for al l, where a(T ) is the number of matrices X ∈ MR,S with T (X) = T . As discussed in Section 2, bl(X) = 0 (mod p) when each power of p in the p- l adic representation of the tl = xi(l−i) occurs in the binary representation i=1 of exactly one of the xi(l−i). But these powers of p are exactly the sizes of the generalized teeth of length l in canonical bundle comb Cb(T ). Therefore the above condition may be rephrased as the requirement that the generalized teeth of length l of Cb(T ) can be divided into l groups in such a way that the sizes of the teeth in the i-th group add up to xi(l−i). This is the case for for THE BUNDLED AND PARTITIONED COMB... 295

all l ⇔ the partition structure of PX C(T ) is compatible with the canonical bundle structure on C(T ). Accordingly, the number a(T )is exactly the number of partition of canonical bundle comb Cb(T ) whose first parts are combs for S and whose second parts are bundled combs for R. That is, a(T )= n(T ).

6. Product of n-Times Milnor Elements

In this section we will characterize the Milnor element which appear as sum- mand in a product P (Rn) P (Rn−1) P (R1). For n ≥ 2 the n-partitioned comb K is one in which each tooth is divided into n parts, some possibly empty, by n − 1 horizantal lines. That is, to each generalized tooth τ of length l and σ size p is assigned an (n − 1)-tuple of integers with 0 ≤ π1(τ) ≤ π2(τ) ≤≤ πn−1(τ) ≤ 0. Let π0 = 0 and πn = l for all τ. The i-th part of K, 1 ≤ i ≤ n, ∗ is obtained by replacing all the p ’s in each τ except those in rows σ + πi−1(τ) through σ + πi(τ) − 1 with 0’s. Thus 2-partitions are the familiar partitions of Section 3.4 Example. A picture of 5-partitioned bundled comb and its 4-th part are pictured below. p∗ 0 0 p∗ 0 p∗ 0 p∗ 0 0 0 p∗ p∗ p∗ 0 ⇒ 0 0 p∗ 0 p∗ p∗ p∗

Let m(T ) be the number of n-partitions of Cb(T ) whose i-th parts are (bundled) combs for Ri for all i. Now we can establish the following theorem which can be proved by induction on n. Theorem. The Milnor basis element P(T) is a summand of a product P (Rn) P (Rn−1) P (R1) ⇔ m(T ) = 0 (mod p).

References

[1] J. Adem, The iteration of the steenrod squares in , Proc. Nat. Acad. Sci. USA, 38 (1952), 720-726.

[2] H. Cartan, Sur l’iteration des operations de steenrod, Comm. Math. Hel- vet., 29 (1955), 40-58. 296 I. Karaca, B. Tanay

[3] J. Milnor, The Steenrod algebra and its dual, Ann. of Math., 67 (1958), 150-171.

[4] J.P. Serre, Cohomologie modulo 2 des complexes d’Eilenberg-Maclane, Comm. Math. Helvet., 29 (1956), 198-232.

[5] J.H. Silverman, Multiplication and the combina toricsin the Steenrod al- gebra, Journal of Pure and Applied Algebra, 111 (1996), 303-323.

[6] N.E. ve Epstein Steenrod, Cohomology Operations, Press (1962).