Combinatorial Stokes formulas via minimal resolutions

TL;DR

This paper develops an explicit chain map from the standard to the minimal resolution for cyclic groups, leading to a combinatorial Stokes theorem that offers a new combinatorial approach to problems in topology and geometry.

Contribution

It introduces a new explicit chain map for cyclic groups that induces a combinatorial Stokes theorem, connecting algebraic topology with combinatorics and discrete geometry.

Findings

Derived a chain map from standard to minimal resolution for Z_k

Established a combinatorial Stokes theorem for Z_k

Provided a combinatorial proof of Dold's theorem

Abstract

We describe an explicit chain map from the standard resolution to the minimal resolution for the finite cyclic group Z_k of order k. We then demonstrate how such a chain map induces a "Z_k-combinatorial Stokes theorem", which in turn implies "Dold's theorem" that there is no equivariant map from an n-connected to an n-dimensional free Z_k-complex. Thus we build a combinatorial access road to problems in combinatorics and discrete geometry that have previously been treated with methods from equivariant topology. The special case k=2 for this is classical; it involves Tucker's (1949) combinatorial lemma which implies the Borsuk-Ulam theorem, its proof via chain complexes by Lefschetz (1949), the combinatorial Stokes formula of Fan (1967), and Meunier's work (2006).