Additivity of higher rho invariant for the topological structure group from a differential point of view (Q2032755)

Summary: In [Commun. Pure Appl. Math. 74, No. 1, 3--113 (2021; Zbl 1476.57044)], \textit{S. Weinberger} et al. 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 \(C^*\)-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 \(C^*\)-algebra induced by higher rho invariant associated to orientation-preserving homotopy equivalence.