Let $\widetilde{X}$ be a smooth Riemannian manifold equipped with a proper, free, isometric, and cocompact action of a discrete group $\Gamma$. In this paper, we prove that the analytic surgery exact sequence of Higson–Roe for $\widetilde{X}$ is isomorphic to the exact sequence associated to the adiabatic deformation of the Lie groupoid $\widetilde{X}\times_\Gamma\widetilde{X}$. We then generalize this result to the context of smoothly stratified manifolds. Finally, we show, by means of the aforementioned isomorphism, that the $\varrho$-classes associated to a metric with a positive scalar curvature defined by Piazza and Schick (2014) correspond to the $\varrho$-classes defined by Zenobi (2019).

中文翻译：

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绝热群状体和 Higson-Roe 精确序列

设 $\widetilde{X}$ 是一个光滑的黎曼流形，它配备了离散群 $\Gamma$ 的适当、自由、等距和协紧作用。在本文中，我们证明了Higson-Roe 对$\widetilde{X}$ 的解析手术精确序列与与Lie groupoid $\widetilde{X}\times_\Gamma\ 的绝热变形相关的精确序列同构宽波浪线{X}$。然后我们将这个结果推广到平滑分层流形的背景。最后，我们通过上述同构证明，与 Piazza 和 Schick (2014) 定义的具有正标量曲率的度量相关联的 $\varrho$-类对应于 Zenobi 定义的 $\varrho$-类（ 2019）。