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

这项研究的主要重点将是关系， 通过不同定义的迹映射，在K理论和自由 循环空间 几何、代数和数学中的几个基本问题 微分拓扑学的关键在于更好地理解这一点， 连接. 其中包括P. Baum的同构猜想 和A.康纳斯和它所暗示的那些（例如，诺维科夫号和 Gromov-Lawson-Rosenberg猜想），Bass猜想 群环的Hattori-Stallings迹图的图像，以及 关于Banach空间K-理论的Milnor-Friedlander猜想 系数在有限交换群中的代数。 自由 循环空间在理解 的稳定伪同位素空间的同伦型 流形，和变量似乎是密切相关的积分 幂零理想的相对K-理论 计划工作建立 这些领域的进展情况，特别是调查员的 KABI和Karoubi结构的解决方案（使用Weibel），新 关于空间的积分相对K理论的信息，以及 空间的算子代数K-理论（与P. Baum和A. Connes）。 在代数理论中， 计算空间拓扑问题的答案是 各种所谓的K理论 该项目的实质是 建立这些K理论之间的关系，以及它们之间的关系， 一些其他的代数理论来处理拓扑学。

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.