Isoperimetric inequalities in simplicial complexes

Abstract

In graph theory there are intimate connections between the expansion properties of a graph and the spectrum of its Laplacian. In this paper we define a notion of combinatorial expansion for simplicial complexes of general dimension, and prove that similar connections exist between the combinatorial expansion of a complex, and the spectrum of the high dimensional Laplacian defined by Eckmann. In particular, we present a Cheeger-type inequality, and a high-dimensional Expander Mixing Lemma. As a corollary, using the work of Pach, we obtain a connection between spectral properties of complexes and Gromov's notion of geometric overlap. Using the work of Gunder and Wagner, we give an estimate for the combinatorial expansion and geometric overlap of random Linial-Meshulam complexes.

High Dimensional Expanders

“…The language of L p -expanders described in §2 gives a systematic way to express this. (Compare also to [PRT16]). The fact that we have an ε = ε(d) which is independent of q, provided q > q 0 , (which is more than one needs in order to answer Gromov's geometric question) is due to the fact that for a fixed d ∈ N, one has the same p in the table in §2.3.…”

“…Moreover as pointed out in §2, over R, ∆ up i f, f f, f = δ i f f and so the definition of h i here is "the characteristic 2 analogue" of the spectral gap defined in Definition 2.2. For the connection between the spectral gap and the coboundary expansion -see [SKM14], [PRT16] and [GS15]. (iii) Also here it is easy to see that h i (X) > 0 iff H i (X, F 2 ) = {0}.…”

Local Spectral Expansion Approach to High Dimensional Expanders Part II: Mixing and Geometrical Overlapping

“…• In the general case, mixing for random complexes was proven by Parzanchevski, Rosenthal and Tessler [PRT16]. The error term in that work was of the form |U 0 ||U n ||U 1 |...|U n−1 |.…”

Local Spectral Expansion Approach to High Dimensional Expanders Part I: Descent of Spectral Gaps

“…In [Lub14] two main approaches are suggested: The first is through the F 2 -coboundary expansion of X originated in [Gro10], [LM06] and [MW09] . The second is through studying the spectral gap of the (n − 1)-Laplacian of X (where n is the dimension of X) or the spectral gaps of all 0, .., (n − 1)-Laplacians of X (see [Par17], [PRT16]). One of the difficulties with both approaches are that both the F 2 -coboundary expansion and the spectral gap of the (n − 1)-Laplacian are usually hard to calculate or even bound in examples.…”