The Negation Of Singer's Conjecture For The Sixth Algebraic Transfer
Dang Vo Phuc
TL;DR
The paper disproves Singer's conjecture that the algebraic transfer $Tr_q(\mathbb F_2)$ is always a monomorphism by constructing an explicit counterexample at $q=6$ in bidegree $ (6,6+36)$. It introduces a novel OSCAR-based algorithm to compute $GL(q)$-invariants of the kernel of the Kameko map and of the domain $QP_q$, enabling precise dimension counts and explicit invariant generators $[\zeta_1]$ and $[\zeta_2]$ in degree $36$. The results yield $\dim (\mathbb F_2\otimes_{GL(6)} \mathcal P_{\mathscr A}(H_*(\mathcal V^6)))_{36} = 2$ while $\mathrm{Ext}^{6,6+36}_{\mathscr A}(\mathbb F_2,\mathbb F_2) = \mathbb F_2$, showing the transfer is not injective in this case. This demonstrates a concrete limitation of Singer's conjecture and showcases a scalable, high-performance computational approach to the hit problem and modular invariant theory, with explicit data and code availability for reproducibility.
Abstract
Let $\mathscr A$ be the Steenrod algebra over the field of characteristic two, $\mathbb F_2.$ Denote by $GL(q)$ the general linear group of rank $q$ over $\mathbb F_2.$ The algebraic transfer, introduced by W. Singer [Math. Z. 202 (1989), 493-523], is a rather effective tool for unraveling the intricate structure of the (mod-2) cohomology of the Steenrod algebra, ${\rm Ext}_{\mathscr A}^{q,*}(\mathbb F_2, \mathbb F_2).$ The Kameko homomorphism is one of the useful tools to study the dimension of the domain of the Singer transfer. Singer conjectured that the algebraic transfer is always a monomorphism, but this remains open in general case. In this work, by constructing a novel algorithm implemented in the computer algebra system OSCAR for computing $GL(q)$-invariants of the kernel of the Kameko homomorphism, we disprove Singer's conjecture for bidegree $(6,6+36).$