Geometric realization via unoriented bordism and a counterexample to Singer's conjecture for the sixth algebraic transfer

Abstract

Let $\mathcal{A}$ be the mod-2 Steenrod algebra acting in the usual way on $P_q={\mathbb{F}}_2[x_1,\dots ,x_q]$, and let $Q{P}_q={\mathbb{F}}_2{\otimes }_{\mathcal{A}}{P}_q$. Singer's algebraic transfer $T{r}_q$ sends the dual of $[(Q{P}_q{)}_n{]}^{GL(q,{\mathbb{F}}_2)}$ to ${\mathrm{Ext}}_{\mathcal{A}}^{q,q+n}({\mathbb{F}}_2,{\mathbb{F}}_2)$; Singer conjectured that $T{r}_q$ is always injective. We disprove this nearly forty-year-old conjecture at rank $q=6$, degree $n=36$. Verifying this requires computing $[(Q{P}_6{)}_{36}{]}^{GL(6,{\mathbb{F}}_2)}$ exactly; to handle the resulting combinatorial complexity, we build a new Julia package \texttt{AlgebraicTransfer.jl}, coupling modular invariant theory with bit-level linear algebra over ${\mathbb{F}}_2$ via Steenrod-hit reductions and Kameko homomorphisms. We prove this source space is two-dimensional, strictly exceeding the known one-dimensional target ${\mathrm{Ext}}_{\mathcal{A}}^{6,42}({\mathbb{F}}_2,{\mathbb{F}}_2)$, so $T{r}_6$ is not injective. We also interpret the transfer kernel geometrically via unoriented bordism: $T{r}_q$ factors through bordism classes over $B(\mathbb{Z}/2{)}^q$ whose Thom images are primitive, characterized by the vanishing of all mixed Wu numbers. Thom's representability theorem guarantees closed $36$-manifolds realizing the homological duals of the source generators, yet we show that standard models (such as the indecomposable Milnor hypersurface $H_{4,33}$, projective products, and Dold manifolds) cannot represent them. We further interpret the inverse Kameko map via Thom spaces of universal real line bundles. Validated by recovering classical Dickson invariant dimensions, this work delivers both a counterexample to Singer's conjecture and a scalable methodology for the Peterson hit problem.

arXiv comment31 pages. Both the approach and the title have been updated in this new version. We would be glad to receive comments. Any comments would be appreciated