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 Z k of order k. We then demonstrate how such a chain map induces a “ Z k -combinatorial Stokes theorem,” which in turn implies “Doldʹs 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 Tuckerʹs (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 Meunierʹs work (2006).