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

Write Pₖ:= F₂[x₁,x₂,… ,xₖ] for the polynomial algebra over the prime field F₂ with two elements, in k generators x₁, x₂, … , xₖ, each of degree 1. The polynomial algebra Pₖ is considered as a module over the mod-2 Steenrod algebra, A. Let GLₖ be the general linear group over the field F₂. This group acts naturally on Pₖ by matrix substitution. Since the two actions of A and GLₖ upon Pₖ commute with each other, there is an inherit action of GLₖ on F₂⊗APₖ. Denote by ( F₂⊗APₖ)ₙGLₖ the subspace of F₂⊗APₖ consisting of all the GLₖ-invariant classes of degree n. In 1989, Singer [24] defined the homological algebraic transfer φₖ :TorAk,k+n( F₂, F₂) ( F₂⊗APₖ)ₙGLₖ, where TorAk, k+n(F₂, F₂) is the dual of ExtAk,k+n( F₂, F₂), the E₂ term of the Adams spectral sequence of spheres. In general, the transfer φₖ is not a monomorphism and Singer made a conjecture that φₖ 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 Ph\'uc in [19].