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

[j] .设G为任意离散群。Jones和Farrell推测（“同构猜想”）G， L(G)的手术L群，以及整数群环Z(G)， K（Z(G)）的代数K群，应该可以从所有群L(H)的集合中以一种简单的方式计算，K(Z(H))，其中H是g的有限子群的任意循环。这些猜想的成立不仅暗示了非球流形的刚性结果（“Borel猜想”），而且还提供了关于非球流形的同胚和微分同态空间的许多信息。Jones与Farrell合作，试图通过在非正曲率的完全黎曼流形上的等距来验证任意不连续的G群的同构猜想。现代的几何学家和拓扑学家关心的是“空间”和它们的样子。甜甜圈的表面、球体和平面都是二维空间的例子，从几何和拓扑的角度来看，它们都是不同的。拓扑学家研究空间（不仅是二维空间，还有高维空间）的一种方法是将一些代数小工具与每个空间联系起来。据推测，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”，因为它们告诉了我们关于空间的几乎所有我们想知道的东西。因此，这里有两个基本问题：验证这个“遗传密码猜想”；并破译空间的遗传密码这是几何学家和拓扑学家感兴趣的。

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.