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arXiv · 0710.0050

Combinatorial Stokes formulas via minimal resolutions

Bernhard Hanke·Raman Sanyal·Carsten Schultz·Günter M. Ziegler
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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).

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Bernhard Hanke, Raman Sanyal, Carsten Schultz, Günter M. Ziegler. 2007-09-29. Combinatorial Stokes formulas via minimal resolutions. https://doi.org/10.1016/j.jcta.2008.06.009

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