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 $C^*$-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.

中文翻译：

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从差分角度看拓扑结构组的较高rho不变量的加性

在[16]中，Weinberger，Xie和Yu证明与同伦同等性相关的较高rho不变性定义了从拓扑结构组到解析结构组的组同态，即$ K $-某些几何$ C ^ * $-代数的理论，采用分段线性方法。在本文中，我们改编了Weinberger，Xie和Yu的部分工作，以给出从拓扑结构组到某些$ C ^ * $-代数的$ K $-理论的可加性的微分几何理论证明。较高的rho不变性与保持取向的同位体等效性有关。