This is a survey on known results and open problems about closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. Many examples come from certain kinds of non-positive curvature conditions. The property aspherical, which is a purely homotopy theoretical condition, has many striking implications about the geometry and analysis of the manifold or its universal covering, and about the ring theoretic properties and the K- and L-theory of the group ring associated to its fundamental group.
Primary: 57N99, 19A99, 19B99, 19D99, 19G24, 20C07, 20F25, 57P10
|Received:||25th November 2010|
|Revised:||8th July 2012|
|Accepted:||24th August 2012|
|Published:||27th September 2012|