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

Local spectral expansion approach to high dimensional expanders