A Topological Approach to Cheeger‐Gromov Universal Bounds for von Neumann ρ‐Invariants

Using deep analytic methods, Cheeger and Gromov showed that for any smooth (4 k ‐1)‐manifold there is a universal bound for the von Neumann L 2 ρ ‐invariants associated to arbitrary regular covers. We present a proof of the existence of a universal bound for topological (4 k ‐1)‐manifolds, using L 2 ‐signatures of bounding 4 k ‐manifolds. We give explicit linear universal bounds for 3‐manifolds in terms of triangulations, Heegaard splittings, and surgery descriptions. We show that our explicit bounds are asymptotically optimal. As an application, we give new lower bounds of the complexity of 3‐manifolds that can be arbitrarily larger than previously known lower bounds. As ingredients of the proofs that seem interesting on their own, we develop a geometric construction of efficient 4‐dimensional bordisms of 3‐manifolds over a group and develop an algebraic topological notion of uniformly controlled chain homotopies.© 2016 Wiley Periodicals, Inc.