Combinatorial Stokes formulas via minimal resolutions

We describe an explicit chain map from the standard resolution to the minimal resolution for the finite cyclic group Zk of order k. We then demonstrate how such a chain map induces a “Zk-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 Zk-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).