math_groups_discgeom

Combinatorial Stokes formulas via minimal resolutions

Bernhard Hanke and Raman Sanyal and Carsten Schultz and Günter M. Ziegler— 2009

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).

TitelCombinatorial Stokes formulas via minimal resolutions
VerfasserBernhard Hanke and Raman Sanyal and Carsten Schultz and Günter M. Ziegler
Datum2009
Quelle/n
Erschienen inJ. Combinatorial Theory, Ser.~A, volume 116, pages 404-420
ArtText