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

We prove Norine’s conjecture: every red–blue edge-coloring of the $n$-dimensional hypercube $Q_n$, $n\ge 2$, 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.