A Chain-Level Borsuk--Ulam Obstruction Proof of Norine's Antipodal-Coloring Conjecture

Hehui Wu; Ningyuan Yang

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arXiv · 2607.19276

A Chain-Level Borsuk--Ulam Obstruction Proof of Norine's Antipodal-Coloring Conjecture

Hehui Wu·Ningyuan Yang

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Abstract

We prove Norine's conjecture: every red--blue edge-coloring of the $n$-dimensional hypercube $Q_n$, $n\geq2$, in which antipodal edges have opposite colors contains a monochromatic path joining some vertex to its antipode. From a hypothetical counterexample we construct an antipodally equivariant, augmentation-preserving chain map from the cellular chains of the cubical boundary of a cube to subdivision-invariant polyhedral chains on a sphere of one lower dimension. A purely algebraic chain-level Borsuk--Ulam obstruction rules out this map.

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Hehui Wu, Ningyuan Yang. 2026-07-21. A Chain-Level Borsuk--Ulam Obstruction Proof of Norine's Antipodal-Coloring Conjecture. https://arxiv.org/abs/2607.19276

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