Cosystoles and cheeger constants of the simplex

Abstract

The central interest of this thesis is to develop tools to get hands on the cosystolic norm and the coboundary expansion of a cochain, values which are important to determine the Cheeger constants of a simplicial complex. We develop some structural theory about the cosystolic norm of a cochain, in which we, among other small results, study an interesting connection between that norm and the hitting number of a certain set system (see Chapter 2). In Chapter 3 we restrict our research to 1- dimensional cosystoles of a simplex which are slightly easier to understand, so we can provide more explicit results for that case, including the explicit determination of the largest 1-dimensional cosystoles of a simplex and a rough insight, how all 1-cosystoles of a simplex in a certain dimension can be arranged in the so-called cosystolic complex. Furthermore, we prove the strict inequality h1(D[n]) > n3 for the case n = 16, which strengthens our conjecture that this strict inequality holds in general if n is a power of 2. In Chapter 4 we study some alternative ways to generalize the classical Cheeger constant which might be easier to access and prove that these different constants equal for a large family of simplicial complexes. In Chapter 5 we solve a beautiful combinatorial ordering problem, which is not directly related to the main subject of this thesis but arose during considerations about that and should be worth to be provided to the reader as well. In Appendix A we give an algorithm for the exact calculation of the solutions of the ordering problem from Chapter 5.