A counter-example to Singer's conjecture for the algebraic transfer

Write $P_k:={\mathbb{F}}_2[x_1,x_2,\dots ,x_k]$ for the polynomial algebra over the prime field ${\mathbb{F}}_2$ with two elements, in $k$ generators $x_1,x_2,\dots ,x_k$, each of degree 1. The polynomial algebra $P_k$ is considered as a module over the mod-2 Steenrod algebra, $\mathcal{A}$. Let $G{L}_k$ be the general linear group over the field ${\mathbb{F}}_2$. This group acts naturally on $P_k$ by matrix substitution. Since the two actions of $\mathcal{A}$ and $G{L}_k$ upon $P_k$ commute with each other, there is an inherit action of $G{L}_k$ on ${\mathbb{F}}_2{\otimes }_{\mathcal{A}}{P}_k$. Denote by $({\mathbb{F}}_2{\otimes }_{\mathcal{A}}{P}_k{)}_n^{G{L}_k}$ the subspace of ${\mathbb{F}}_2{\otimes }_{\mathcal{A}}{P}_k$ consisting of all the $G{L}_k$-invariant classes of degree $n$. In 1989, Singer [24] defined the homological algebraic transfer $${\phi }_k:To{r}_{k,k+n}^{\mathcal{A}}({\mathbb{F}}_2,{\mathbb{F}}_2)⟶({\mathbb{F}}_2{\otimes }_{\mathcal{A}}{P}_k{)}_n^{G{L}_k},$$ where $To{r}_{k,k+n}^{\mathcal{A}}({\mathbb{F}}_2,{\mathbb{F}}_2)$ is the dual of Ext${}_{\mathcal{A}}^{k,k+n}({\mathbb{F}}_2,{\mathbb{F}}_2)$, the $E_2$ term of the Adams spectral sequence of spheres. In general, the transfer ${\phi }_k$ is not a monomorphism and Singer made a conjecture that ${\phi }_k$ is an epimorphism for any $k⩾0$. The conjecture is studied by many authors. It is true for $k⩽3$ but unknown for $k⩾4$. In this paper, by using a technique of the Peterson hit problem we prove that Singer’s conjecture is not true for $k=5$ and the internal degree $n=108$. This result also refutes a one of Phúc in [19].