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Combinatorial Stacks and the Four-Colour Theorem

Romain Attal
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Abstract

We interpret the number of good four-colourings of the faces of a trivalent, spherical polyhedron as the 2-holonomy of the 2-connection of a fibered category, phi, modeled on Rep(sl(2)) and defined over the dual triangulation, T. We also build an sl(2)-bundle with connection over T, that is a global, equivariant section of phi, and we prove that the four-colour theorem is equivalent to the fact that the connection of this sl(2)-bundle vanishes nowhere. This interpretation may be a first step toward a cohomological proof of the four-colour theorem.

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Romain Attal. 2005-01-14. Combinatorial Stacks and the Four-Colour Theorem. https://arxiv.org/abs/math/0501231

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