Miraculous Cancellation and Pick's Theorem

TL;DR

This paper links Pick's theorem for lattice polytopes to a broader topological relation involving characteristic classes and cobordism, revealing a deep connection between combinatorics and topology.

Contribution

It demonstrates that Cappell-Shaneson's version of Pick's theorem follows from a general topological relation derived from the action of the Landweber-Novikov algebra.

Findings

Pick's theorem is a consequence of characteristic number relations.

A topological framework explains combinatorial lattice polytope properties.

The relation is analogous to the miraculous cancellation formula.

Abstract

We show that the Cappell-Shaneson version of Pick's theorem for simple lattice polytopes is a consequence of a general relation between characteristic numbers of virtual submanifolds dual to the characteristic classes of a stably almost complex manifold. This relation is analogous to the miraculous cancellation formula of Alvarez-Gaume and Witten, and is imposed by the action of the Landweber-Novikov algebra in the complex cobordism ring of a point.