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On the determination of the singer transfer

Let Pk be the graded polynomial algebra F2[x1, x2,.,xk] with the degree of each generator xi being 1, where F2 denote the prime field of two elements, and let GLk be the general linear group over F2 which acts regularly on Pk. We study the algebraic transfer constructed by Singer [1] using the technique of the Peterson hit problem. This transfer is a homomorphism from the homology of the mod-2 Steenrod algebra A, TorA k,k+d(F2, F2), to the subspace of F2⊗APk consisting of all the GLk-invariant classes of degree d. In this paper, by using the results on the Peterson hit problem we present the proof of the fact that the Singer algebraic transfer is an isomorphism for k ¬ 3. This result has been proved by Singer in [1] for k ¬ 2 and by Boardman in [2] for k = 3. 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] | MATHEMATICS AND COMPUTER SCIENCE MATHEMATICS On the determination of the Singer transfer Sum Nguyen Department of Mathematics Quy Nhon University Received 23 October 2017 accepted 24 January 2018 Abstract Let Ph be the graded polynomial algebra F2 x1 x2 . . xh with the degree of each generator xi being 1 where F2 denote the prime field of two elements and let GLh be the general linear group over F2 which acts regularly on Ph. We study the algebraic transfer constructed by Singer 1 using the technique of the Peterson hit problem. This transfer is a homomorphism from the homology of the mod-2 Steenrod algebra A TorAh d F2 F2 to the subspace of F2 APh consisting of all the GLh-invariant classes of degree d. In this paper by using the results on the Peterson hit problem we present the proof of the fact that the Singer algebraic transfer is an isomorphism for k c 3. This result has been proved by Singer in 1 for k c 2 and by Boardman in 2 for k 3. 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 . Keywords algebraic transfer polynomial algebra steenrod algebra. Classification number 1.1 1. Introduction Denote by Pk F2 xi x2 . xk the polynomial algebra over the field of two elements F2 in k generators x1 x2 . xk each of degree 1. This algebra arises as the cohomology with coefficients in F2 of an elementary abelian 2-group of rank k. Therefore Pk is a module over the mod-2 Steenrod algebra A. The action of A on Pn is determined by the elementary properties of the Steenrod squares Sqi and subject to the Cartan formula Sqk fg Ek Sqi f Uqk-iU for f g e Pk 8 . The Peterson hit problem is to find a minimal generating set for Pk regarded as a module over the mod-2 Steenrod algebra. Equivalently this problem is to find a vector space basis for QPk F2 A Pk in each degree d. Such a basis may be represented by a list of .