Additivity of higher rho invariant for the topological structure group from a differential point of view

Abstract

In [16], Weinberger, Xie, and Yu proved that higher rho invariant associated to homotopy equivalence defines a group homomorphism from the topological structure group to the analytic structure group, $K$-theory of certain geometric $C^{\ast }$-algebras, by piecewise-linear approach. In this paper, we adapt part of Weinberger, Xie, and Yu’s work, to give a differential geometry theoretic proof of the additivity of the map from the topological structure group to $K$-theory of certain $C^{\ast }$-algebra induced by higher rho invariant associated to orientation-preserving homotopy equivalence.