### THE HIT PROBLEM AND THE ALGEBRAIC TRANSFER IN SOME DEGREES

#### Abstract

Denote by

We study the

*P**k*the graded polynomial algebra F2[*x*1*, x*2*, . . . , x**k*], with the degree of each generator*x**i*being 1, and let*GL**k*be the general linear group over the prime field F2 of two elements which acts naturally on*P**k*by matrix substitution.We study the

*Peterson hit problem*of determining a minimal set of generators for*P**k*as a module over the mod-2 Steenrod algebra,*A*. In this paper, we study the hit problem in terms of the admissible monomials at the degree (*k**−*1)(2*d**−*1). These results are used to verify Singer’s conjecture for the algebraic transfer, which is a homomorphism from the homology of the mod-2 Steenrod algebra, Tor*A**k,k*+*n*(F2*,*F2), to the subspace of F2*⊗**A**P**k*consisting of all the*GL**k*-invariant classes of degree*n*. More precisely, using the results on the hit problem, we prove that Singer’s conjecture for the algebraic transfer is true in the case*k*= 5 and the degree 4(2*d**−*1) with*d*an arbitrary positive integer.#### Keywords

Steenrod algebra, Peterson hit problem, algebraic transfer, polynomial algebra

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