On the fifth Singer algebraic transfer in a generic family of internal degree characterized by μ(n)=4

Abstract

Let A denote the Steenrod algebra over the field of characteristic two, F2. Singer's algebraic transfer, introduced by Singer in his work (Singer (1989) [27]), is a rather effective tool for unraveling the intricate structure of the mod-two cohomology of the Steenrod algebra, ExtA⁎,⁎(F2,F2). In the present study, we aim to investigate the behavior of this algebraic transfer for rank five in the generic family of internal degree n:=ℓ(2s−1)+k⋅2s, wherein ℓ=4,8≤k≤11,k≠9, and s is any positive integer. The principal results obtained lead to interesting conclusions regarding the image of algebraic transfers in ranks 5, 6, 9, and 10. As direct consequences, within the bidegrees (5,5+n), Singer's conjecture on the monomorphism of algebraic transfers remains upheld.