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

TL;DR

This paper proves the existence of universal bounds for von Neumann rho-invariants of topological manifolds using topological and geometric methods, providing explicit bounds for 3-manifolds and applications to manifold complexity.

Contribution

It introduces a topological approach to Cheeger-Gromov bounds, including explicit bounds for 3-manifolds and new techniques for constructing efficient 4-dimensional bordisms.

Findings

Universal bounds for von Neumann rho-invariants are established for topological manifolds.

Explicit linear bounds are provided for 3-manifolds based on triangulations, Heegaard splittings, and surgery.

New lower bounds for 3-manifold complexity are derived, surpassing previous estimates.

Abstract

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.