Additivity of higher rho invariants and nonrigidity of topological manifolds

Abstract

Let $X$ be a closed oriented connected topological manifold of dimension $n\geq 5$. The structure group of $X$ is the abelian group of equivalence classes of all pairs $(f, M)$ such that $M$ is a closed oriented manifold and $f\colon M \to X$ is an orientation-preserving homotopy equivalence. The main purpose of this article is to prove that a higher rho invariant defines a group homomorphism from the topological structure group of $X$ to the $C^*$-algebraic structure group of $X$. In fact, we introduce a higher rho invariant map on the homology manifold structure group of a closed oriented connected $\textit{topological}$ manifold, and prove its additivity. This higher rho invariant map restricts to the higher rho invariant map on the topological structure group. More generally, the same techniques developed in this paper can be applied to define a higher rho invariant map on the homology manifold structure group of a closed oriented connected $\textit{homology}$ manifold. As an application, we use the additivity of the higher rho invariant map to study non-rigidity of topological manifolds. More precisely, we give a lower bound for the free rank of the $\textit{algebraically reduced}$ structure group of $X$ by the number of torsion elements in $\pi_1 X$. Here the algebraic reduced structure group of $X$ is the quotient of the topological structure group of $X$ modulo a certain action of self-homotopy equivalences of $X$. We also introduce a notion of homological higher rho invariant, which can be used to detect many elements in the structure group of a closed oriented topological manifold, even when the fundamental group of the manifold is torsion free. In particular, we apply this homological higher rho invariant to show that the structure group is not finitely generated for a class of manifolds.