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

Let A be the Steenrod algebra over the finite field \(k:= {\mathbb {F}}_2\) $k:={\mathbb{F}}_2$ 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, \(\textrm{Ext}^{q, *}_A(k, k),\) $\text{Ext}_A^{q,\ast }(k,k),$ for all homological degrees \(q \geqslant 0.\) $q⩾0.$ 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 \(\textrm{Ext}^{q, *}_A(k, k).\) $\text{Ext}_A^{q,\ast }(k,k).$ Singer predicted that the algebraic transfer is always injective, but this has gone unanswered for all \(q\geqslant 4.\) $q⩾4.$ This paper establishes Singer’s conjecture for rank four in the generic degrees \(n = 2^{s+t+1} +2^{s+1} - 3\) $n=2^{s+t+1}+2^{s+1}-3$ whenever \(t\ne 3\) $t\ne 3$ and \(s\geqslant 1,\) $s⩾1,$ and \(n = 2^{s+t} + 2^{s} - 2\) $n=2^{s+t}+2^s-2$ whenever \(t\ne 2,\, 3,\, 4\) $t\ne 2,3,4$ and \(s\geqslant 1.\) $s⩾1.$ In conjunction with our previous results, this completes the proof of the Singer conjecture for rank four.