非交换拓扑及其在指标理论中的应用
Noncommutative Topology and Prospects in Index Theory
摘要：
这篇文章是对恩佐·马蒂内利教授的悼念，怀着深深的敬意和敬意。尼古拉·泰勒曼指数公式是一个局部的声明，全球和当地的数据，为此，我们介绍了当地的亚历山大- Spanier上同调，当地的周期循环同调，当地陈特征和当地的$T^{\ast}$ -理论。指标理论应该做：情况1：对任意环，情况2：对环的功能在拓扑流形。例1产生一般的指标定理，例如，在伪流形。例2给出了经典和非交换指标定理的一般处理.现有的指标定理都属于第二类。该理论的工具包括：局部$T^{\ast}$ -理论，局部双周期循环同调，局部Chern特征标.这些工具被推广到非交换拓扑中.指数公式有三个阶段：第一阶段是在$T^{loc}\_{i}$理论中完成的，第二阶段是在局部周期循环同调中完成的，第三阶段涉及分布的乘积或对对角的限制。对于每个阶段，有相应的拓扑指数和分析指数。$T^{\ast}$ -理论的构造涉及到T-完备化。它还涉及到需要处理半整数;这应该有重要的后果。
英文摘要：
This article is a tribute to the memory of Professor Enzo Martinelli, with deep respect and reconesance. Nicolae Teleman. The index formula is a local statement made on global and local data; for this reason we introduce local Alexander - Spanier co-homology, local periodic cyclic homology, local Chern character and local $T^{\ast}$-theory. Index theory should be done: Case 1: for arbitrary rings, Case 2: for rings of functions over topo- logical manifolds. Case 1 produces general index theorems, as for example, over pseudo-manifolds. Case 2 gives a general treatment of classical and non- commutative index theorems. All existing index theorems belong to the second category. The tools of the theory would contain: local $T^{\ast}$ -theory, local peri- odic cyclic homology, local Chern character. These tools are extended to non- commutative topology. The index formula has three stages : Stage I is done in $T^{loc}\_{i}$-theory, Stage II is done in the local periodic cyclic homology and Stage III involves products of distributions, or restriction to the diagonal. For each stage there corresponds a topological index and an analytical index. The construction of $T^{\ast}$-theory involves the T-completion. It involves also the need to work with half integers; this should have important consequences.
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