2006/05/23 by Roman Sauer, Sauer, Roman · 1 citation
Computer Science · Mathematics · #22D20 #53C20 #58J22 #Algebraic Topology (math.AT) #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.math/0605627
openalex publication_date 2006/05/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We provide a proof for an inequality between volume and L2-Betti numbers of aspherical manifolds for which Gromov outlined a strategy based on general ideas of Connes. The implementation of that strategy involves measured equivalence relations, Gaboriau's theory of L2-Betti numbers of R-simplicial complexes, and other themes of measurable group theory. Further, we prove new vanishing theorems for L2-Betti numbers that generalize a classical result of Cheeger and Gromov. As one of the corollaries, we obtain a gap theorem which implies vanishing of L2-Betti numbers of an aspherical manifold when its minimal volume is sufficiently small.