Let $X$ be a closed oriented connected topological manifold of dimension $n\geq 5$. The structure group of $X$ is the abelian group of equivalence classes of all pairs $(f, M)$ such that $M$ is a closed oriented manifold and $f\colon M \to X$ is an orientation-preserving homotopy equivalence. The main purpose of this article is to prove that a higher rho invariant defines a group homomorphism from the topological structure group of $X$ to the $C^*$-algebraic structure group of $X$. In fact, we introduce a higher rho invariant map on the homology manifold structure group of a closed oriented connected $\textit{topological}$ manifold, and prove its additivity. This higher rho invariant map restricts to the higher rho invariant map on the topological structure group. More generally, the same techniques developed in this paper can be applied to define a higher rho invariant map on the homology manifold structure group of a closed oriented connected $\textit{homology}$ manifold. As an application, we use the additivity of the higher rho invariant map to study non-rigidity of topological manifolds. More precisely, we give a lower bound for the free rank of the $\textit{algebraically reduced}$ structure group of $X$ by the number of torsion elements in $\pi_1 X$. Here the algebraic reduced structure group of $X$ is the quotient of the topological structure group of $X$ modulo a certain action of self-homotopy equivalences of $X$. We also introduce a notion of homological higher rho invariant, which can be used to detect many elements in the structure group of a closed oriented topological manifold, even when the fundamental group of the manifold is torsion free. In particular, we apply this homological higher rho invariant to show that the structure group is not finitely generated for a class of manifolds.

中文翻译：

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较高 Rho 不变量的可加性和拓扑流形的非刚性

令 $X$ 是维数为 $n\geq 5$ 的闭向连通拓扑流形。$X$ 的结构群是所有对 $(f, M)$ 的等价类的阿贝尔群，使得 $M$ 是一个闭定向流形，而 $f\colon M \to X$ 是一个方向保持同伦等价。本文的主要目的是证明更高的rho不变量定义了从$X$的拓扑结构群到$X$的$C^*$-代数结构群的群同态。事实上，我们在闭向连通的$\textit{topological}$流形的同调流形结构群上引入了一个更高的rho不变映射，并证明了它的可加性。这个更高的rho 不变映射限制在拓扑结构组上的更高rho 不变映射。更普遍，本文中开发的相同技术可用于在封闭定向连接 $\textit{homology}$ 流形的同源流形结构群上定义更高的 rho 不变映射。作为一个应用，我们使用更高的 rho 不变映射的可加性来研究拓扑流形的非刚性。更准确地说，我们通过 $\pi_1 X$ 中扭转元素的数量给出了 $X$ 的 $\textit{algebraically reduction}$ 结构群的自由秩的下界。这里$X$的代数约简结构群是$X$的拓扑结构群以$X$的自同伦等价的某个作用为模的商。我们还引入了同调更高rho不变量的概念，它可以用来检测封闭定向拓扑流形的结构群中的许多元素，即使流形的基本群是无扭转的。特别是，我们应用这个同调更高的 rho 不变量来表明结构群不是为一类流形有限生成的。