Secondary large-scale index theory and positive scalar curvature
Secondary large-scale index theory and positive scalar curvature
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发表时间：
2016-09
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通讯作者：
Rudolf Zeidler
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作者：
Rudolf Zeidler
我们发展了与自旋流形上指定子集外一致正标量曲率的完备黎曼度量相关的次不变量理论。我们在大规模(或“粗略”)指数理论的背景下工作。这些不变量可以用来区分这样的黎曼度量，直到与规定的子集相一致。给出了部分次不变量的一般外积公式，并由此推导出具有一致正标量曲率的度量的Rho不变量以及具有一致正数量曲率的两个度量的粗指数差的乘积公式。我们的方法给出了二次分块流形指数定理的一个新的概念证明和Piazza-Schick的离域APS指数定理的一个精化版本。我们建立了粗指标差的一个分块流形指标定理。此外，我们还证明了实K-理论中从Stolz的正标量曲率序列到Higson-Roe的解析运算序列的变换的存在性。作为我们理论的应用，我们在非紧自旋流形上构造了一致正标量曲率的几个完备度量，这些度量相对于某些子集可以区分到相容。此外，我们通过子流形上的指数不变量建立了正标量曲率度量的存在障碍和相容的变体。从技术的角度来看，本文的中心创新之处在于，我们将Yu的局部化代数与K-理论的描述结合起来，用于由于Trout的分次C*-代数。这种形式允许直接定义我们在狄拉克算符的泛函演算方面考虑的所有不变量，并使我们能够给出乘积公式的简明证明。它还允许我们在实K理论的背景下始终如一地工作。
We develop a theory of secondary invariants associated to complete Riemannian metrics of uniformly positive scalar curvature outside a prescribed subset on a spin manifold. We work in the context of large-scale (or "coarse") index theory. These invariants can be used to distinguish such Riemannian metrics up to concordance relative to the prescribed subset. We exhibit a general external product formula for partial secondary invariants, from which we deduce product formulas for the Rho-invariant of a metric with uniformly positive scalar curvature as well as for the coarse index difference of two metrics with uniformly positive scalar curvature. Our methods yield a new conceptual proof of the secondary partitioned manifold index theorem and a refined version of the delocalized APS-index theorem of Piazza--Schick for the spinor Dirac operator in all dimensions. We establish a partitioned manifold index theorem for the coarse index difference. Moreover, we reprove the existence of a transformation from the positive scalar curvature sequence of Stolz to the analytic surgery sequence of Higson--Roe for real K-theory. As applications of our theory, we construct several complete metrics of uniformly positive scalar curvature on non-compact spin manifolds which can be distinguished up to concordance relative to certain subsets. Moreover, we establish variants of obstructions to existence and concordance of positive scalar curvature metrics via index invariants on submanifolds. From a technical standpoint, the central novelty of this thesis is that we use Yu's localization algebras in combination with the description of K-theory for graded C*-algebras due to Trout. This formalism allows direct definitions of all the invariants we consider in terms of the functional calculus of the Dirac operator and enables us to give concise proofs of the product formulas. It also allows us to consistently work in the setting of real K-theory.