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

In [14], Weinberger, Xie and Yu proved that higher rho invariant associated to homotopy equivalence defines a group homomorphism from the topological structure group to 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 induced by higher rho invariant associated to homotopy equivalence on topological structure group. Mathematics Subject Classification (2010). 58J22.