The main result of this paper is the $G$-homotopy invariance of the $G$-index of the signature operator of proper co-compact $G$-manifolds. If proper co-compact $G$-manifolds $X$ and $Y$ are $G$-homotopy equivalent, then we prove that the images of their signature operators by the $G$-index map are the same in the $K$-theory of the $C^*$-algebra of the group $G$. Neither discreteness of the locally compact group $G$ nor freeness of the action of $G$ on $X$ are required, so this is a generalization of the classical case of closed manifolds. Using this result, we can deduce the equivariant version of Novikov conjecture for proper co-compact $G$-manifolds from the strong Novikov conjecture for $G$.

中文翻译：

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$G$-同伦的正确协紧$G$-流形和等变Novikov猜想的解析签名的不变性

本文的主要结果是适当协紧$G$-流形的签名算子的$G$-index的$G$-同伦不变性。如果适当的协紧$G$-流形$X$ 和$Y$ 是$G$-同伦等价的，那么我们证明$G$-index 映射的它们的签名算子的图像在$K 中是相同的群 $G$ 的 $C^*$-代数的 $-理论。既不需要局部紧群 $G$ 的离散性，也不需要 $G$ 对 $X$ 的作用的自由性，所以这是封闭流形的经典情况的推广。使用这个结果，我们可以从 $G$ 的强 Novikov 猜想中推导出 Novikov 猜想的等变版本，用于适当的 co-compact $G$-流形。