Higher Dimensional Discrete Cheeger Inequalities

4 For graphs there exists a strong connection between spectral and combinatorial 5 expansion properties. This is expressed, e.g., by the discrete Cheeger inequality, the 6 lower bound of which states that λ(G) ≤ h(G), where λ(G) is the second smallest 7 eigenvalue of the Laplacian of a graph G and h(G) is the Cheeger constant measuring 8 the edge expansion of G. We are interested in generalizations of expansion properties 9 to finite simplicial complexes of higher dimension (or uniform hypergraphs). 10 Whereas higher dimensional Laplacians were introduced already in 1945 by Eck11 mann, the generalization of edge expansion to simplicial complexes is not straight12 forward. Recently, a topologically motivated notion analogous to edge expansion 13 that is based on Z2-cohomology was introduced by Gromov and independently by 14 Linial, Meshulam and Wallach and by Newman and Rabinovich. It is known that 15 for this generalization there is no higher dimensional analogue of the lower bound 16 of the Cheeger inequality. A different, combinatorially motivated generalization of 17 the Cheeger constant, denoted by h(X), was studied by Parzanchevski, Rosenthal 18 and Tessler. They showed that indeed λ(X) ≤ h(X), where λ(X) is the smallest 19 non-trivial eigenvalue of the ((k − 1)-dimensional upper) Laplacian, for the case of 20 k-dimensional simplicial complexes X with complete (k − 1)-skeleton. 21 Whether this inequality also holds for k-dimensional complexes with non-complete 22 (k − 1)-skeleton has been an open question. We give two proofs of the inequality 23 for arbitrary complexes. The proofs differ strongly in the methods and structures 24 employed, and each allows for a different kind of additional strengthening of the 25 original result. 26 ∗Institut für Theoretische Informatik, ETH Zürich, CH-8092 Zürich, Switzerland. [email protected]. Research supported by the Swiss National Science Foundation (SNF Projects 200021-125309 and 200020-138230). †Institut für Theoretische Informatik, ETH Zürich, CH-8092 Zürich, Switzerland. [email protected].