Topological Rigidity Problems

TL;DR

This survey discusses recent advances and open problems in topological rigidity for closed aspherical manifolds, emphasizing the Farrell-Jones Conjecture and its implications for major conjectures like Borel, Novikov, and Whitehead group vanishing.

Contribution

It reviews the current status of topological rigidity problems and the Farrell-Jones Conjecture, highlighting their connections to key conjectures in topology and algebraic K- and L-theory.

Findings

Summary of recent results on topological rigidity

Status updates on the Farrell-Jones Conjecture

Open problems and conjectures in the field

Abstract

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.