Topological rigidity problems

We survey the recent results and current issues on the topological rigidity problem for closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. A number of open problems and conjectures are presented during the course of the discussion. We also review the status and applications of the Farrell-Jones Conjecture for algebraic \(K\)-and \(L\)-theory for a group ring $RG$ and coefficients in an additive category. These conjectures imply many other well-known and important conjectures. Examples are the Borel Conjecture about the topological rigidity of closed aspherical manifolds, the Novikov Conjecture about the homotopy invariance of higher signatures and the Conjecture for vanishing of the Whitehead group. We here present the status of the Borel, Novikov and vanishing of the Whitehead group Conjectures.

Homotopy invariance of non-stable K1-functors

Let G be a reductive algebraic group over a field k, such that every semisimple normal subgroup of G has isotropic rank ≥ 2, i.e. contains (Gm)2. Let K1G be the non-stable K1-functor associated to G, also called the Whitehead group of G. We show that K1G(k) = K1G (k[X1 ,…, Xn]) for any n ≥ 1. If k is perfect, this implies that K1G (R) = K1G (R[X]) for any regular k-algebra R. If k is infinite perfect, one also deduces that K1G (R) → K1G (K) is injective for any local regular k-algebra R with the fraction field K.

Algebraic K-theory and cyclic homology

The following are personal reminiscences of my research years in algebraic K-theory and cyclic homology during which Dan Quillen was everyday present in my professional life.In the late sixties (of the twentieth century) the groups K0;K1;K2 were known and well-studied. The group K0 had been introduced by Alexander Grothendieck, then came K1 by Hyman Bass [2] (as a variation of the Whitehead group), permitting one to generalize the notion of determinant, and finally K2 by John Milnor [9] and Michel Kervaire. The big problem was: how about Kn? Having in mind topological K-theory and all the other generalized (co)homological theories, one was expecting higher K-groups which satisfy similar axioms, in particular the Mayer-Vietoris exact sequence. The discovery by Richard Swan of the existence of an obstruction for this property to hold shed some embarrassment. What kind of properties should we ask of Kn? There were various attempts, for instance by Max Karoubi and Orlando Villamayor [4]. And suddenly Dan Quillen came with a candidate sharing a lot of nice properties. He had even two different constructions of the same object: the “+” construction and the “Q” construction [14, 15]. Not only did he propose a candidate but he already got a computation: the higher K-theory of finite fields. This was a fantastic step forward and Hyman Bass organized a two week conference at the Battelle Institute in Seattle during the summer of 1972, which was attended by Bass, Borel, Husemoller, Karoubi, Priddy, Quillen, Segal, Stasheff, Tate, Waldhausen, Wall and sixty other mathematicians. The Proceedings appeared as Springer Lecture Notes 341, 342 and 343. I met Quillen for the first time on this occasion.