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International Journal of Pure and Applied Mathematics ———————————————————————— V International Journal of Pure and Applied Mathematics ————————————————————————– Volume 16 No. 3 2004, 285-296 THE BUNDLED AND PARTITIONED COMB IN THE MOD-p STEENROD ALGEBRA 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 cohomology 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 John Milnor [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 |.
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