同调指数的拓扑不变性
Topological invariance of the homological index
摘要：
R. W. Carey和J. Pincus在[CaPi 86]中提出了可分希尔伯特空间H上的非Fredholm有界算子T的指标理论，使得TT* - T*T在迹族中.在[CGK 13]中，我们证明了利用作用在R^{2n}上的丛的截面上的Dirac型算子，我们可以构造出满足更一般条件的有界算子T：（1-TT*）^n -（1-T*T）^n是迹族.我们提出了一个由Tr（（1-TT*）^n -（1-T*T）^n）给出的Dirac型算子的“同调”指标.本文证明了在[CGK 13]中引入的指数表示由T和T* 生成的代数的循环同调理论与它的对偶上同调理论配对的结果。这导致我们建立我们的同调指数的同伦不变性（在循环理论的意义上）。然后，我们能够在一个非常普遍的方式定义一个同调指数为某些无界运营商，并证明不变性的这一指数下一类无界扰动。
英文摘要：
R. W. Carey and J. Pincus in [CaPi86] proposed and index theory for non-Fredholm bounded operators T on a separable Hilbert space H such that TT* - T*T is in the trace class. We showed in [CGK13] using Dirac-type operators acting on sections of bundles over R^{2n} that we could construct bounded operators T satisfying the more general condition that (1-TT*)^n - (1-T*T)^n is trace class. We proposed there a "homological" index for these Dirac-type operators given by Tr( (1-TT*)^n - (1-T*T)^n ). In this paper we show that the index introduced in [CGK13] represents the result of a pairing between a cyclic homology theory for the algebra generated by T and T* and its dual cohomology theory. This leads us to establish homotopy invariance of our homological index (in the sense of cyclic theory). We are then able to define in a very general fashion a homological index for certain unbounded operators and prove invariance of this index under a class of unbounded perturbations.
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