The Isomorphism Conjectures for surgery L-groups, algebraic K-groups, and stable pseudo-isotopy spaces

手术 L 群、代数 K 群和稳定赝同位素空间的同构猜想

• 批准号: 0072349
• 负责人: Lowell Jones
• 金额: $ 14.72万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-06-15 至 2004-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072349&HistoricalAwards=false
• 关键词: Isomorphism; Conjectures; surgery; groups; algebraic

DMS-0072349Lowell E. JonesLet G be an arbitrary discrete group. Jones and Farrell have conjectured ("Isomorphism Conjectures") that the surgery L-groups of G, L(G), and the algebraic K-groups of the integral group ring Z(G), K(Z(G)), should be computable in a simple way from the collections of all the groups L(H), K(Z(H)) where H is any cyclic by finite subgroup of G. The truth of these conjecutures would imply rigidity results for aspherical manifolds ("Borel Conjecture") and also would yield much information about the spaces of homeomorphisms and diffeomorphisms of aspherical manifolds. Jones,in collaboration with Farrell, is trying to verify the isomorphism conjectures for any group G which acts properly discontinuously via isometries on a complete Riemannian manifold having non-positive curvature. Modern day geometers and topologists are concerned with "spaces" and what they look like. The surface of a donut, the sphere and the plane are examples of 2-dimensional spaces which are all different from one another from both the perspective of geometry and topology. One way that topologists study spaces (not just of dimension 2 but ofhigher dimension also) is to associate to each space some algebraic gadgets. It is conjectured that for many interesting spaces these algebraic gadgets act like a "genetic code" for the space, in that they tell us most everything we want to know about the space. Thus there are two fundamental problems here: to verify this "genetic code conjecture"; and to decifer the genetic code for the spaces which are of interest to geometers and topologists.

DMS-0072349 Lowell E.设G是任意离散群. Jones和Farrell证明了（“同构猜想”）G的外科L-群，L（G），整群环Z（G），K（Z（G））的代数K-群应该可以用一种简单的方法从所有群L（H），K（Z（H））的集合中计算出来，其中H是G的任意有限循环子群。 这些猜想的真实性将意味着非球面流形的刚性结果（“波莱尔猜想”），也将产生许多关于非球面流形的同胚和复同胚空间的信息。 琼斯，在合作与法雷尔，正试图验证同构aesthetures为任何一组G的行为适当不连续通过isometries上一个完整的黎曼流形具有非正曲率。 现代几何学家和拓扑学家关心的是“空间”和它们的样子。 圆环面、球面和平面都是二维空间的例子，从几何和拓扑的角度来看，它们都是彼此不同的。 拓扑学家研究空间的一种方法（不仅是二维空间，也是高维空间）是将每个空间与一些代数小工具联系起来。 它是明确的，对于许多有趣的空间，这些代数小工具就像空间的“遗传密码”，因为它们告诉我们大多数我们想知道的关于空间的一切。 因此，这里有两个基本问题：验证这个“遗传密码猜想”;和decifer的遗传密码的空间感兴趣的几何学家和拓扑学家。