2017/10/22 by Sum, Nguyen · 4 citations
#55S05 #Algebraic Topology (math.AT) #FOS: Mathematics #Primary 55T15 #Secondary 55S10
paper · doi:10.48550/arxiv.1710.07895
Let Pk be the graded polynomial algebra \mathbb F2[x1,x2,… ,xk] with the degree of each generator xi being 1, where \mathbb F2 denote the prime field of two elements, and let GLk be the general linear group over \mathbb F2 which acts regularly on Pk. We study the algebraic transfer Trk^* constructed by Singer using the technique of the Peterson hit problem. This transfer is a homomorphism from the homology of the mod-2 Steenrod algebra \mathcal A, Tor\mathcal Ak,k+d (\mathbb F2,\mathbb F2), to the subspace of \mathbb F2⊗\mathcal 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 \leqslant 3. We also explicitly determine the fourth Singer algebraic transfer in some degrees. The new results in the paper are different from the ones of Bruner, Ha and Hung [5], Chon and Ha [6,7,8], Ha [9], Hung and Quynh [12], Nam [16]. To illustrate the fact that d0 ∈ Im(Tr4), we present the computations of Ha [9, Page 102] for this result. We can easily verify that these computations are correct. So, it is possible the algorithm in Phuc [29] is flawed.