Additivity of Higher Rho Invariants and Nonrigidity of Topological Manifolds
Additivity of Higher Rho Invariants and Nonrigidity of Topological Manifolds
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高阶ρ不变量的可加性与拓扑流形的非刚性
DOI：
10.1002/cpa.21962
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发表时间：
2016-08
影响因子：
3
通讯作者：
S. Weinberger;Zhizhang Xie;Guoliang Yu
中科院分区：
文献类型：
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作者：
S. Weinberger;Zhizhang Xie;Guoliang Yu
设X是维数n ≥ 5的闭定向连通拓扑流形.结构群STOPX是所有对（f，M）的等价类的阿贝尔群，使得M是闭定向流形，f：M → X是保定向同伦等价。本文的主要目的是证明一个高ρ不变映射定义了一个从X的拓扑结构群STOPX到X的解析结构群KnCL，0*XΓ的群同态.这里X是X的泛覆盖，Γ = π1X是X的基本群，CL，0*XΓ是某个C*-代数。实际上，我们在闭定向连通拓扑流形的同调流形结构群上引入了一个高ρ不变映射，并证明了它的可加性。这种高ρ不变映射仅限于拓扑结构群上的高ρ不变映射。更一般地说，本文中开发的相同的技术可以应用于定义一个高rho不变映射上的同调流形结构组的一个封闭的定向连通的同调流形。作为应用，我们利用高ρ不变映射的可加性研究了拓扑流形的非刚性。更精确地说，我们给出了X的代数约化结构群的自由秩的一个下界，这个下界是由π1X中挠元素的个数决定的.这里X的代数约化结构群是STOPX模X的自同伦等价的某个作用的商。我们还引入了一个概念的同调高rho不变，它可以用来检测许多元素的结构组中的一个封闭的定向拓扑流形，即使当基本组的流形是挠自由。特别地，我们应用这个同调高ρ不变量来证明结构群对于一类流形是非线性生成的。© 2020威利期刊有限责任公司
Let X be a closed oriented connected topological manifold of dimension n ≥ 5. The structure group STOPX is the abelian group of equivalence classes of all pairs (f, M) such that M is a closed oriented manifold and f : M → X is an orientation‐preserving homotopy equivalence. The main purpose of this article is to prove that a higher rho invariant map defines a group homomorphism from the topological structure group STOPX of X to the analytic structure group KnCL,0*XΓ of X. Here X is the universal cover of X, Γ = π1X is the fundamental group of X, and CL,0*XΓ is a certain C*‐algebra. In fact, we introduce a higher rho invariant map on the homology manifold structure group of a closed oriented connected 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 homology manifold. As an application, we use the additivity of the higher rho invariant map to study nonrigidity of topological manifolds. More precisely, we give a lower bound for the free rank of the algebraically reduced structure group of X by the number of torsion elements in π1X. Here the algebraically reduced structure group of X is the quotient of STOPX 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. © 2020 Wiley Periodicals LLC