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

Abstract: Let $A$ be the Steenrod algebra over the finite field $k := \mathbb Z_2$ 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, ${\rm Ext}^{q, *}_A(k, k),$ for all homological degrees $q \geq 0.$ The Singer algebraic transfer of rank $q,$ formulated by William Singer in 1989, serves as a valuable method for the description of such Ext groups. This transfer maps from the coinvariants of a certain representation of $G(q)$ to ${\rm Ext}^{q, *}_A(k, k).$ Singer predicted that the algebraic transfer is always injective, but this has gone unanswered for all $q\geq 4.$ This paper establishes Singer's conjecture for rank four in the generic degrees $n = 2^{s+t+1} +2^{s+1} - 3$ whenever $t\neq 3$ and $s\geq 1,$ and $n = 2^{s+t} + 2^{s} - 2$ whenever $t\neq 2,\, 3,\, 4$ and $s\geq 1.$ In conjunction with our previous results, this completes the proof of the Singer conjecture for rank four.