Higher rho invariants and the moduli space of positive scalar curvature metrics
Higher rho invariants and the moduli space of positive scalar curvature metrics
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DOI：
10.1016/j.aim.2016.11.030
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发表时间：
2013-10
期刊：
影响因子：
--
通讯作者：
Zhizhang Xie;Guoliang Yu
中科院分区：
文献类型：
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作者：
Zhizhang Xie;Guoliang Yu
给定一个带有正数量曲率度量的闭光滑流形M，我们可以将阿贝尔群P（M）与该流形上的正数量曲率度量空间相关联。流形的所有超同态的群自然作用在P（M）上。正数量曲率度量的模群P ∈（M）被定义为该作用的商阿贝尔群，即该作用的共不变量。P（M）度量M上正数量曲率度量的模空间的大小。本文利用π 1（M）的群C-代数的高ρ不变量和K-理论的有限部分，给出了模群P-代数（M）的秩的一个下界.我们证明的关键思想是使用更高的ρ不变量，这是一个二级不变量与狄拉克算子，特别是取决于黎曼度量的选择。我们证明了高ρ不变量在某个K-理论群中在群同态群的作用下保持不变，从而使我们能够区分P <$（M）中的元素。
Given a closed smooth manifold M which carries a positive scalar curvature metric, one can associate an abelian group P (M) to the space of positive scalar curvature metrics on this manifold. The group of all diffeomorphisms of the manifold naturally acts on P (M). The moduli group P˜(M) of positive scalar curvature metrics is defined to be the quotient abelian group of this action, ie the coinvariant of the action. P˜(M) measures the size of the moduli space of positive scalar curvature metrics on M. In this paper, we use the higher rho invariant and the finite part of the K-theory of the group C⁎-algebra of π 1 (M) to give a lower bound of the rank of the moduli group P˜(M). The key idea of our proof is the use of higher rho invariant; this is a secondary invariant associated to Dirac operators and in particular depends on the choice of Riemannian metric. We show that the higher rho invariant remains unchanged in a certain K-theory group under the action of the diffeomorphism group, allowing us to distinguish elements in P˜(M).