A Genus-Two Algebraic Family with Vanishing Real Toledo Numbers and Nonzero Generic Witt Class
Year: 2026
Type: preprint
Abstract: We give four explicit matrices defining a genus-two surface-group representation over R[s,t,1/(st(1+t^2))], where R denotes the real numbers. Every real specialization has zero Witt class and zero Toledo number. Over R(s,t), however, the Witt class is the anisotropic Pfister form <1,-(1+t^2)> tensor <1,-s>, of additive order two. A nonzero second residue and a trace pole detect bad reduction at the nonreal divisor 1+t^2=0. The calculation shows concretely that the real Toledo function does not determine the generic Witt-valued invariant. General realization results of Dymara and Januszkiewicz (Math. Ann. 390 (2024), 4463-4496, Theorem 11.6) already imply existence of such examples. This note provides a compact matrix presentation and a direct proof, not a new existence or range theorem. It does not resolve the full open-ended programme of AIM-TOPOLOGY-0276 (Burger's Question 4.5); no priority claim is made and novelty of the explicit presentation remains undetermined. The four-page English preprint is AI-assisted and unrefereed. A written self-audit and twelve exact symbolic checks accompany it. No independent review or proof-assistant verification is claimed.
Author: Ferudun, Alper
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7219715092
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