A note on the hit problem for the polynomial algebra of six variables and the sixth algebraic transfer

Abstract

Let A be the classical, singly-graded Steenrod algebra over the prime order field F2 and let P⊗h:=F2[t1,…,th] denote the polynomial algebra on h generators, each of degree 1, viewed as a module over A. Write GLh for the usual general linear group of rank h over F2. As well known, the (mod 2) cohomology groups of the Steenrod algebra, ExtAh,h+⁎(F2,F2) is still largely mysterious for all homological degrees h≥6. The h-th algebraic transferTrhA:(F2⊗GLhAnnA‾[P⊗h]⁎)n⟶ExtAh,h+n(F2,F2), defined by William Singer [28], is a helpful tool to describe that Ext groups. Singer conjectured that this transfer is a monomorphism, but it remains open for any h≥5. There is currently no information on the conjecture for h=6. In this paper, we verify Singer's conjecture for homological degree h=6 and the internal degree of the general form ns:=6(2s−1)+10⋅2s. This result is important, since it tells us that the non-zero elements h22g1=h4Ph2∈ExtA6,6+n1(F2,F2), and D2∈ExtA6,6+n2(F2,F2) are not in the image of the sixth algebraic transfer. This Note is a continuation of our previous one [22], which will refer to as Part I. Our approach is based on explicitly solving the hit problem for the Steenrod algebra in the case h=6 and degree ns. This extends a result in Mothebe, Kaelo and Ramatebele [13]. At the same time, we also use this obtained result to establish the dimension result for the space of the unhit elements, F2⊗AP⊗7 in a certain general degree.