Bounds on Cheeger-Gromov invariants and simplicial complexity of triangulated manifolds

We show the existence of linear bounds on Wall rho-invariants of PL manifolds, employing a new combinatorial concept of G-colored polyhedra. As an application, we show how the number of h-cobordism classes of manifolds simple homotopy equivalent to a lens space with V simplices and the fundamental group of Z(n) grows in V. Furthermore, we count the number of homotopy lens spaces with bounded geometry in V. Similarly, we give new linear bounds on Cheeger-Gromov rho-invariants of PL manifolds endowed with a faithful representation also. A key idea is to construct a cobordism with a linear complexity whose boundary is pi(1)-injectively embedded, using relative hyperbolization. As an application, we study the complexity theory of high-dimensional lens spaces. Lastly, we show the density of rho-invariants over manifolds homotopy equivalent to a given manifold for certain fundamental groups. This implies that the structure set is not finitely generated.