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In , 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, -theory of certain geometric -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 -theory of certain -algebra induced by higher rho invariant associated to orientation-preserving homotopy equivalence.
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Baojie Jiang, Hongzhi Liu, Additivity of higher rho invariant for the topological structure group from a differential point of view. J. Noncommut. Geom. 14 (2020), no. 2, pp. 441–486