Local spectral expansion approach to high dimensional expanders

Abstract

This paper introduces the notion of local spectral expansion of a simplicial complex as a possible analogue of spectral expansion defined for graphs. We show the condition of local spectral expansion has several nice implications. For example, for a simplicial complex with local spectral expansion we show vanishing of cohomology with real coefficients, Cheeger type inequalities and mixing type results and geometric overlap results.

96 pages. This version has minor corrections regarding the result of mixing for partite complexes