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

Abstract: Write $P_k:= \mathbb F_2[x_1,x_2,\ldots ,x_k]$ for the polynomial algebra over the prime field $\mathbb F_2$ with two elements, in $k$ generators $x_1, x_2, \ldots , x_k$, each of degree 1. The polynomial algebra $P_k$ is considered as a module over the mod-2 Steenrod algebra, $\mathcal A$. Let $GL_k$ 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 $GL_k$ upon $P_k$ commute with each other, there is an inherit action of $GL_k$ on $\mathbb F_2{\otimes}{\mathcal A}P_k$. Denote by $(\mathbb F_2{\otimes}{\mathcal A}P_k)n^{GL_k}$ the subspace of $\mathbb F_2{\otimes}{\mathcal A}P_k$ consisting of all the $GL_k$-invariant classes of degree $n$. In 1989, Singer [24] defined the homological algebraic transfer $$\varphi_k :\mbox{Tor}^{\mathcal A}{k,k+n}(\mathbb F_2,\mathbb F_2) \longrightarrow (\mathbb F_2{\otimes}{\mathcal A}P_k)n^{GL_k},$$ where $\mbox{Tor}^{\mathcal{A}}{k, k+n}(\mathbb{F}2, \mathbb{F}_2)$ 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 $\varphi_k$ is not a monomorphism and Singer made a conjecture that $\varphi_k$ is an epimorphism for any $k \geqslant 0$. The conjecture is studied by many authors. It is true for $k \leqslant 3$ but unknown for $k \geqslant 4$. 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 Ph\'uc in [19].