DETERMINATION OF THE FIFTH SINGER ALGEBRAIC TRANSFER IN SOME DEGREES

Abstract

Let Pk be the graded polynomial algebra F2[x1, x2, . . . , xk] over the prime field F2 with two elements and the degree of each variable xi being 1, and let GLk be the general linear group over F2 which acts on Pk as the usual manner. The algebra Pk is considered as a module over the mod-2 Steenrod algebra A. In 1989, Singer [19] defined the k-th homological algebraic transfer, which is a homomorphism

ϕk : TorA k,k+d(F2, F2) → (F2 ⊗A Pk)GLk d

from the homological group of the Steenrod algebra TorA k,k+d(F2, F2) to the subspace (F2 ⊗A Pk)GLk d of F2⊗APk consisting of all the GLkinvariant classes of degree d.

In this paper, by using the results of the Peterson hit problem we present the proof of the fact that the Singer algebraic transfer of rank five is an isomorphism in the internal degrees d = 20 and d = 30. Our result refutes the proof for the case of d = 20 in Phúc [15].