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Quantitative aspects of acyclicity

  • Dmitry N. Kozlov
  • Roy MeshulamEmail author
Research
  • 32 Downloads

Abstract

The Cheeger constant is a measure of the edge expansion of a graph and as such plays a key role in combinatorics and theoretical computer science. In recent years, there is an interest in k-dimensional versions of the Cheeger constant that likewise provide quantitative measure of cohomological acyclicity of a complex in dimension k. In this paper, we study several aspects of the higher Cheeger constants. Our results include methods for bounding the cosystolic norm of k-cochains and the k-th Cheeger constants, with applications to the expansion of pseudomanifolds, Coxeter complexes and homogenous geometric lattices. We revisit a theorem of Gromov on the expansion of a product of a complex with a simplex and provide an elementary derivation of the expansion in a hypercube. We prove a lower bound on the maximal cosystole in a complex and an upper bound on the expansion of bounded degree complexes and give an essentially sharp estimate for the cosystolic norm of the Paley cochains. Finally, we discuss a non-abelian version of the one-dimensional expansion of a simplex, with an application to a question of Babson on bounded quotients of the fundamental group of a random 2-complex.

Keywords

Simplicial complexes High dimensional expansion Random complexes 

Mathematics Subject Classification

55U10 05E45 

Notes

Acknowlegements

This research was supported by the German-Israeli Foundation for Scientific Research, under Grant number 1261/14.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BremenBremenFederal Republic of Germany
  2. 2.Department of MathematicsTechnion - Israel Institute of TechnologyHaifaIsrael

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