Combinatorial Stokes formulas via minimal resolutions

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:116

Article

Hanke, Bernhard¹  Sanyal, Raman²  Schultz, Carsten²  Ziegler, Guenter M.² 

[1] LMU Munchen, Inst Math, D-80333 Munich, Germany

[2] Tech Univ Berlin, Inst Math, D-10623 Berlin, Germany

关键词: Resolutions of cyclic groups;    Chain maps;    Tucker's lemma;    Dold's theorem;    Combinatorial Stokes formulas;   

DOI  :  10.1016/j.jcta.2008.06.009

来源: Elsevier

【 摘 要 】

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 trial) 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 all 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 Fail ( 1967). and Meunier's work (2006). (C) 2008 Elsevier Inc. All rights reserved.

【 授权许可】

Free