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

Abstract

We show the existence of linear bounds on Wall $\rho$-invariants of PL manifolds, employing a new combinatorial concept of $G$-colored polyhedra. As application, we show that how the number of h-cobordism classes of manifolds simple homotopy equivalent to a lens space with $V$ simplices and the fundamental group of $\mathbb{Z}_n$ 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 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.