A second counterexample to Singer's conjecture on the algebraic transfer

Abstract

In this paper, we provide a second explicit counterexample to Singer's injectivity conjecture for the algebraic transfer in bidegree $(6, 6+40) = (6,46).$ This result corrects one of the main theorems in our recent preprint \cite[Theorem 2.5]{Phuc}. Exact computation gives a two-dimensional transfer domain and a two-dimensional target, so a comparison of dimensions alone is inconclusive. We construct Steenrod-annihilated divided-power polynomials $U$ and $V$, identify the normalized dual coinvariant basis as $[U+V]$ and $[V]$, and exhibit a lambda-algebra element $B$ satisfying $\psi_6(U+V)=\delta B$. A nonzero invariant pairing proves that $[U+V]\neq0$. The other basis element maps to $h_0h_3p_0$, and a complete boundary-space calculation excludes $h_5Ph_1$ from the image. Thus the transfer has one-dimensional kernel, image, and cokernel. We specify the finite matrices and verification procedures, and print every coefficient of the primitives, invariants, boundary, and separating functional in the appendices.