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

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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.