Some results on the fifth Singer transfer (Q2786892)

In [Math. Z. 202, No. 4, 493--523 (1989; Zbl 0687.55014)] \textit{W. M. Singer} defined the algebraic transfer, which is a homomorphism \(\varphi_{k}: Tor_{k, k+n}^{A} \rightarrow (\mathbb{F}_{2}\otimes_{A}P_{k})_{n}^{GL_{k}}\) and he conjectured that it is an epimorphism for any \(k \geq 0\) and any \(n \geq 0\); where \(A\) is the mod \(2\) Steenrod algebra, \(P_{k}=\mathbb{F}_{2}[x_{1},...,x_{k}]\) is the polynomial algebra over \(\mathbb{F}_{2}\) in \(k\) variables of degree one and \(GL_{k}\) is the general linear group over \(\mathbb{F}_{2}\).NEWLINENEWLINEThe aim of this paper is to prove this conjecture for \(k = 5\) in degree \(n=r.2^{s} -5\), \(r = 3, 4\) and \(s \geq 0\) is an integer.