Problems in Differential and Algebraic Topology

微分和代数拓扑问题

• 批准号: 0306616
• 负责人: Lowell Jones
• 金额: $ 12万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0306616&HistoricalAwards=false
• 关键词: Problems; Differential; Algebraic; Topology

DMS-0306616Lowell E. JonesMany problems in "higher dimensional" topology have been reformulated in terms of one or more of the following theories: homotopy theory; algebraic K-theory; surgery L-theory; stable pseudoisotopy theory.In an effort to better understand the latter three theories, Jones (in collaboration with F.T. Farrell) has formulated an "Isomorphism Conjecture" for each of these three theories. For example, the Isomorphism Conjecture for algebraic K-theory gives a simple recipe for computing the algebraic K-groups for the integral group ring of a group G in terms of the algebraic K-groups for the integral group rings of all the infracyclic subgroups of G. (A group is "infracyclic" if it is a finite extension of a cyclic group.) Quite a bit is know about the algebraic K-groups for integral group rings of infracyclic groups; so the truth of the preceeding conjecture would help considerably in understanding the algebraic K-theory of all integral group rings. As part of this proposal, Jones intends to continue working towards verifying the Isomorphism Conjecture for algebraic K-theory of the integral group ring of G, under the condition that the group G acts properly, discontinuously by isometries on a complete, Riemannian manifold which has non-positive sectional curvature values everywhere.Mathematicians study the structure of spaces by trying to find the "genetic code" for each space. The "genetic code" for a space usually comes in the form of an algebraic gadget associated to the space; typically there is a simple recipe for constructing the algebraic gadget from the space. The hope is that two spaces having equal or congruent "genetic codes" should be equal or congruent as spaces. One of the first algebraic gadgets to be associated to a space is called the "fundamental group" of the space. The French mathematician Henri Poincare introduced this idea about a 100 years ago. The fundamental group of a space is certainly an important ingredient in its "genetic code", but there are examples of many different spaces which have the same fundamental group; in these cases the fundamental group is not a complete "genetic code" for the space. About 50 years ago the mathematician Armand Borel (of the Institute for Advanced Study in Princeton, N.J.) conjectured that for an important class of spaces (called "aspherical spaces") the fundamental group does give the complete "genetic code" for the space; that is, two aspherical spaces with the same fundamental group are forced to be the same space. Jones has made some progress towards proving Borel's Conjecture, and will continue his work in that direction.

DMS-0306616 Lowell E. Jones在“高维”拓扑学中的许多问题已经被重新表述为以下一个或多个理论：同伦理论、代数K-理论、手术L-理论、稳定伪合痕理论。Farrell）为这三个理论中的每一个都提出了一个“同构猜想”。 例如，代数K-理论的同构猜想给出了一个计算群G的整群环的代数K-群的简单方法，它是根据G的所有非循环子群的整群环的代数K-群来计算的。 （一个群是“infracyclic”，如果它是一个循环群的有限扩张。 关于非循环群的整群环的代数K-群已经有相当多的了解;因此，上述猜想的真实性将大大有助于理解所有整群环的代数K-理论。 作为这个提议的一部分，琼斯打算继续致力于验证G的整群环的代数K理论的同构猜想，条件是群G通过等距不连续地正确地作用在一个处处具有非正截面曲率值的完整黎曼流形上。数学家通过试图找到每个空间的“遗传密码”来研究空间的结构。一个空间的“遗传密码”通常以与该空间相关联的代数小工具的形式出现;通常有一个简单的方法来从该空间构造代数小工具。 希望是两个具有相等或全等“遗传密码”的空间应该是相等或全等的空间。 第一个与空间相关联的代数小工具之一被称为空间的“基本群”。 法国数学家亨利·庞加莱（Henri Poincare）在大约100年前提出了这个想法。空间的基本群当然是其“遗传密码”的重要组成部分，但有许多不同空间具有相同基本群的例子;在这些情况下，基本群不是空间的完整“遗传密码”。 大约50年前，数学家阿曼德·博雷尔（新泽西州普林斯顿高等研究院）证明了对于一类重要的空间（称为“非球面空间”），基本群确实给出了空间的完整“遗传密码”;也就是说，具有相同基本群的两个非球面空间被迫成为相同的空间。 琼斯在证明波莱尔猜想方面取得了一些进展，并将继续朝着这个方向努力。