Mathematical Sciences: K-theory, Trace Maps and the Free Loop Space

本研究的主要重点将是通过各种定义的轨迹映射，k理论与自由循环空间之间的关系。几何、代数和微分拓扑中的几个基本问题取决于对这种联系的更好理解。这些猜想包括P. Baum和a . Connes的同构猜想及其所暗示的猜想（例如，Novikov猜想和Gromov-Lawson-Rosenberg猜想），关于群环的Hattori-Stallings迹映射象的Bass猜想，以及关于有限阿贝尔群中带系数的Banach代数的k理论的Milnor-Friedlander猜想。自由环空间在理解流形稳定伪同位素空间的同伦型方面也起着重要的作用，其变体似乎与幂零理想的积分相对k理论密切相关。计划开展的工作以这些领域的进展为基础；特别是研究者对KABI猜想和Karoubi猜想的解（与Weibel合作），关于空间的积分相对k理论的新信息，以及空间的算子代数k理论的表述（与P. Baum和a . Connes合作）。在使计算关于空间的拓扑问题的答案成为可能的代数理论中，有各种所谓的k理论。本课题的实质是建立这些k理论之间的关系，以及它们与其他一些处理拓扑的代数理论之间的关系。

The main focus of this research will be on the relationship, via variously defined trace maps, between K-theory and the free loop space. Several basic problems in geometry, algebra and differential topology hinge on a better understanding of this connection. These include the isomorphism conjecture of P. Baum and A. Connes and those that it implies (e.g., the Novikov and Gromov-Lawson-Rosenberg conjectures), the Bass conjecture on the image of the Hattori-Stallings trace map for group rings, and the Milnor-Friedlander conjecture concerning the K-theory of a Banach algebra with coefficients in a finite abelian group. The free loop space also plays an important role in understanding the homotopy-type of the space of stable pseudo-isotopies of a manifold, and variants seem to be closely related to the integral relative K-theory of nilpotent ideals. The planned work builds on progress in these areas; in particular, the investigator's solution of the KABI and Karoubi conjectures (with Weibel), new information on the integral relative K-theory of spaces, and a formulation of the operator algebra K-theory of spaces (with P. Baum and A. Connes). Among the algebraic theories which make it possible to compute answers to topological questions about spaces are the various so-called K-theories. The essence of this project is to establish relations among these K-theories and between them and some other algebraic theories for dealing with topology.