I will begin with a quick introduction to the Cheeger-Gromov rho invariants from a topological viewpoint, and then present recent quantitative results on how they are related to triangulations and Heegaard splittings of 3-manifolds. I will also discuss quantitative bordism theory and an algebraic notion of controlled chain homotopy, which are the key ingredients of the proofs. Applications to topology of dimension 3 and 4 will be discussed if time permits.