2025/09/11 by Dang Vo Phuc, Phuc, Dang Vo
#math.AT #math.RA #math.RT
paper · pdf · doi:10.48550/arxiv.2509.09455
Let \mathscrA be the mod-2 Steenrod algebra acting in the usual way on Pq = \mathbbF2[x1, …, xq], and let QPq = \mathbbF2 ⊗_\mathscrA Pq. Singer's algebraic transfer Trq sends the dual of [(QPq)n]^GL(q, \mathbbF2) to Ext_\mathscrAq,q+n(\mathbbF2,\mathbbF2); Singer conjectured that Trq is always injective. We disprove this nearly forty-year-old conjecture at rank q=6, degree n=36. Verifying this requires computing [(QP6)36]^GL(6, \mathbbF2) exactly; to handle the resulting combinatorial complexity, we build a new Julia package AlgebraicTransfer.jl, coupling modular invariant theory with bit-level linear algebra over \mathbbF2 via Steenrod-hit reductions and Kameko homomorphisms. We prove this source space is two-dimensional, strictly exceeding the known one-dimensional target Ext_\mathscrA6,42(\mathbbF2,\mathbbF2), so Tr6 is not injective. We also interpret the transfer kernel geometrically via unoriented bordism: Trq factors through bordism classes over B(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 H4,33, 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.