On the May spectral sequence and the algebraic transfer II

Abstract

We study the algebraic transfer constructed by Singer [19] using the May spectral sequence technique. We show that the two squaring operators defined by Kameko [8] and Nakamura [16] on the domain and range respectively of our E 2 version of the algebraic transfer are compatible. We also prove that the two Sq 0 -families n i ∈ Ext 5,36·2 i A (Z/2, Z/2), i ≥ 0, and k i ∈ Ext 7,36·2 i A (Z/2, Z/2), i ≥ 1, are in the image of the algebraic transfer.

On the determination of the Singer transfer

“…By applying this technique for k = 4, we show that the fourth Singer transfer is also an isomorphism in certain internal degrees. This result is new and it is different from the ones of Bruner, Ha and Hung [3], Chon and Ha [4], Ha [5], Hung and Quynh [6], Nam [7]. In those works it is shown only that the fourth Singer transfer detects certain families of elements in Ext 4, * A (F 2 , F 2 ), and fails to detect others.…”

“…We also compute this space, however we use the admissible monomial basis for QP 3 that is different from the one of Boardman in [2]. Our approach can be apply for k = 4 by using the admissible monomial basis for QP 4 which is given in [14,15].…”

“…It is a useful tool in describing the homology groups of the Steenrod algebra, Tor A k,k+d (F 2 , F 2 ). This transfer was studied by Boardman [2], Bruner, Ha and Hung [3], Ha [5], Hung [9], Chon and Ha [4,10,11], Minami [12], Nam [7], Hung and Quynh [6], the present author [13] and others.…”

The "hit" problem of five variables in the generic degree and its application

“…However, when s 5, it is an open problem. Recently, some authors have been studied the conjecture for s = 4, 5 (see Bruner-Hà-Hưng [4], Hưng [11], Chơn-Hà [7,8], Hà [9], Nam [22], the present author [26], [28]- [35], Sum [45,47,48,49] and others). In the present work, by using techniques of the hit problem of five variables, we investigate Conjecture 1.2 in bidegree (5, d + 5), where d is determined as in Theorem 1.1.…”

“…We explicitly compute p (i;I) (S) in terms u j , 1 j 23. By a direct computation using Lemma 2.2.9,Theorem 3.1.3, and from the relations p (i;j) (S) ≡ ω (5,2) 0 with either i = 1, j = 2, 3 or i = 2, j = 3, 4, one gets Here J = {1, 2, 3,4,5,6,7,8,9,10,11,12,15,16,17,18,19,21,22,23,24,25,26,28,29,31,38,39,40,41,42,45,46,48,50,51,57 [46]). By a simple computation, we get…”