On Singer’s conjecture for the fourth algebraic transfer in certain generic degrees

Abstract

Let A be the Steenrod algebra over the finite field \(k:= {\mathbb {F}}_2\) and G(q) be the general linear group of rank q over k. A well-known open problem in algebraic topology is the explicit determination of the cohomology groups of the Steenrod algebra, \(\textrm{Ext}^{q, *}_A(k, k),\) for all homological degrees \(q \geqslant 0.\) The Singer algebraic transfer of rank q,  formulated by William Singer in 1989, serves as a valuable method for describing that Ext groups. This transfer maps from the coinvariants of a certain representation of G(q) to \(\textrm{Ext}^{q, *}_A(k, k).\) Singer predicted that the algebraic transfer is always injective, but this has gone unanswered for all \(q\geqslant 4.\) This paper establishes Singer’s conjecture for rank four in the generic degrees \(n = 2^{s+t+1} +2^{s+1} - 3\) whenever \(t\ne 3\) and \(s\geqslant 1,\) and \(n = 2^{s+t} + 2^{s} - 2\) whenever \(t\ne 2,\, 3,\, 4\) and \(s\geqslant 1.\) In conjunction with our previous results, this completes the proof of the Singer conjecture for rank four.