A topological approach to Cheeger-Gromov universal bounds for von Neumann rho-invariants

Using deep analytic methods, Cheeger and Gromov showed that for any smooth (4k-1)-manifold there is a universal bound for the von Neumann $L^2$ $\rho$-invariants associated to arbitrary regular covers. We present a proof of the existence of a universal bound for topological (4k-1)-manifolds, using $L^2$-signatures of bounding 4k-manifolds. For 3-manifolds, we give explicit linear universal bounds in terms of triangulations, Heegaard splittings, and surgery descriptions respectively. We show that our explicit bounds are asymptotically optimal. As an application, we give new lower bounds of the complexity of 3-manifolds which can be arbitrarily larger than previously known lower bounds. As ingredients of the proofs which 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.