Quantitative aspects of acyclicity

  • Dmitry N. Kozlov
  • Roy MeshulamEmail author


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.


Simplicial complexes High dimensional expansion Random complexes 

Mathematics Subject Classification

55U10 05E45 



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


  1. 1.
    Alon, N., Spencer, J.: The Probabilistic Method, 2nd edn. Wiley-Intescience, New York (2000) CrossRefGoogle Scholar
  2. 2.
    Babson, E., Hoffman, C., Kahle, M.: The fundamental group of random 2-complexes. J. Am. Math. Soc. 24, 1–28 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chung, F.R.K.: Several generalizations of Weil sums. J. Number Theory 49, 95–106 (1994)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Dotterer, D., Kahle, M.: Coboundary expanders. J. Topol. Anal. 4, 499–514 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Edelsbrunner, H.: A short course in computational geometry and topology. In: SpringerBriefs in Applied Sciences and Technology. Springer (2014)Google Scholar
  6. 6.
    Farber, M., Gonzalez, J., Schütz, D.: Oberwolfach Arbeitsgemeinschaft: Topological Robotics. Report No. 47/2010 2010 47.pdf. Accessed 1 Jan 2013
  7. 7.
    Folkman, J.: The homology groups of a lattice. J. Math. Mech. 15, 631–636 (1966)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Gromov, M.: Singularities, expanders and topology of maps. Part 2: From combinatorics to topology via algebraic isoperimetry. Geom. Funct. Anal. 20, 416–526 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hoffman, C., Kahle, M., Paquette, E.: The threshold for integer homology in random d-complexes. Discrete Comput. Geom. 57, 810–823 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hoory, S., Linial, N., Wigderson, A.: Expander graphs and their applications. Bull. Am. Math. Soc. (N.S.) 43, 439–561 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Humphreys, J.E.: Reflection groups and Coxeter groups. In: Cambridge Studies in Advanced Mathematics, vol. 29. Cambridge University Press, Cambridge (1990)Google Scholar
  12. 12.
    Kozlov, D.N.: Combinatorial algebraic topology. In: Algorithms and Computation in Mathematics, vol. 21. Springer, Berlin (2008)Google Scholar
  13. 13.
    Kozlov, D.N.: The first Cheeger constant of a simplex. Graphs Combin. To appear arXiv:1610.07136
  14. 14.
    Linial, N., Meshulam, R.: Homological connectivity of random 2-complexes. Combinatorica 26, 475–487 (2006)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lubotzky, A.: Expander graphs in pure and applied mathematics. Bull. Am. Math. Soc. (N.S.) 49, 113–162 (2012)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lubotzky, A., Meshulam, R.: Random Latin squares and 2-dimensional expanders. Adv. Math. 272, 743–760 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Lubotzky, A., Meshulam, R., Mozes, S.: Expansion of building-like complexes. Groups Geom. Dyn. 10, 155–175 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Łuczak, T., Peled, Y.: Integral homology of random simplicial complexes. arXiv:1607.06985
  19. 19.
    McDiarmid, C.: On the method of bounded differences. Surv. Comb. 141, 148–188 (1989)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Meshulam, R., Wallach, N.: Homological connectivity of random \(k\)-dimensional complexes. Random Struct. Algorithms 34, 408–417 (2009)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Olum, P.: Non-abelian cohomology and van Kampen’s theorem. Ann. Math. 68, 658–668 (1958)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ronan, M.: Lectures on Buildings. Academic Press Inc., Boston (1989)zbMATHGoogle Scholar
  23. 23.
    Steenbergen, J., Klivans, C., Mukherjee, S.: A Cheeger-type inequality on simplicial complexes. Adv. Appl. Math. 56, 56–77 (2014)MathSciNetCrossRefGoogle Scholar
  24. 24.
    West, D.B.: Introduction to Graph Theory. Prentice Hall Inc., Upper Saddle River (1996)zbMATHGoogle Scholar

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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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