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

Abstract

Let $\mathscr{A}$ be the mod-2 Steenrod algebra acting in the usual way on $P_q = \mathbb{F}_2[x_1, \ldots, x_q]$, and let $QP_q = \mathbb{F}_2 \otimes_{\mathscr{A}} P_q$. Singer's algebraic transfer $Tr_q$ sends the dual of $[(QP_q)_n]^{GL(q, \mathbb{F}_2)}$ to $\operatorname{Ext}_{\mathscr{A}}^{q,q+n}(\mathbb{F}_2,\mathbb{F}_2)$; Singer conjectured that $Tr_q$ is always injective. We disprove this nearly forty-year-old conjecture at rank $q=6$, degree $n=36$. Verifying this requires computing $[(QP_6)_{36}]^{GL(6, \mathbb{F}_2)}$ 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 $\operatorname{Ext}_{\mathscr{A}}^{6,42}(\mathbb{F}_2,\mathbb{F}_2)$, so $Tr_6$ is not injective. We also interpret the transfer kernel geometrically via unoriented bordism: $Tr_q$ 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 standard models -- 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.