Combinatorial Stokes formulas via minimal resolutions

  • Authors:

  • Bernhard Hanke;Raman Sanyal;Carsten Schultz;Günter M. Ziegler

  • Affiliations:

  • Institute of Mathematics, LMU München, 80333 München, Germany;Institute of Mathematics, MA 6-2, TU Berlin, 10623 Berlin, Germany;Institute of Mathematics, MA 6-2, TU Berlin, 10623 Berlin, Germany;Institute of Mathematics, MA 6-2, TU Berlin, 10623 Berlin, Germany

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2009

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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).