An aspherical space is a connected CW-complex whose homotopy groups vanish in every degree greater than 1. Equivalently, its universal cover is contractible (Hatcher 2002). It is therefore a classifying space for its fundamental group. This equivalence is stated for CW-complexes, and applies in particular to manifolds through their homotopy models.
The circle and all tori are examples. Their universal covers are Euclidean
spaces. An aspherical space need not itself be contractible,
since its fundamental group may be nontrivial.
The Borel conjecture asks for topological rigidity
of closed aspherical manifolds, while the Singer
conjecture concerns their 
-homology.
