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

DMS-0072349 Lowell E.Jones设G是任意离散群。Jones和Farrell猜想(同构猜想)G，L(G)的外科Borel-群和积分群环Z(G)，K(Z(G))的代数K-群应该是由所有群的集合L(H)，K(Z(H))简单地计算出来的，其中H是G的有限子群的任意循环。这些猜想的真实性将意味着非球面流形的刚性结果(“Borel猜想”)，并将提供关于非球面流形的同胚和微分同胚空间的许多信息。Jones和Farrell合作，试图验证任何群G的同构猜想，群G在具有非正曲率的完备黎曼流形上通过等距不连续地适当地作用。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，它们都是彼此不同的。拓扑学家研究空间(不仅是2维的，还有更高维的)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来说，这些代数小工具就像是空间的“遗传密码”，因为它们告诉了我们关于这个空间的几乎所有我们想知道的事情。因此，这里有两个基本问题：验证这一“遗传密码猜想”；以及确定地质学家和拓扑学家感兴趣的空间的遗传密码。

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.