  We establish enhanced bounds on Cheeger-Gromov rho-invariants for general
3-manifolds and yet stronger bounds for special classes of 3-manifold. As key
ingredients, we construct chain null-homotopies whose complexity is linearly
bounded by its boundary's. This result can be regarded as an algebraic
topological analogue of Gromov's conjecture for quantitative topology. The
author hopes for applications to various fields including the smooth knot
concordance group, quantitative topology, and complexity theory.
