Let $ G $ be a Lie group with finitely many connected components, $ K $ a maximal compact subgroup, and $ \Gamma $ a torsion free cocompact lattice. Then $$ \Gamma \backslash G/K $$ is an aspherical closed manifold with fundamental group $ \Gamma $.
This construction gives, for example, all closed manifolds which are locally symmetric spaces of noncompact type or of Euclidean type.
In general this construction gives all aspherical locally Riemannian homogeneous manifolds. This includes all compact solvmanifolds as well as all the 3-manifolds with the six aspherical Thurston geometries ($ S^3 $ and $ S^2 \times E^1 $ geometries are excluded since they are not aspherical).
In dimensions $ n \leq 2 $ every aspherical closed manifold is a locally symmetric space of noncompact type or of Euclidean type.
However for $ n=3 $ this is no longer true since the closed manifolds of the form $$ \Gamma \backslash G/K $$ given above are exactly the manifolds admitting a Thurston geometry. The only nontrivial connected sum with a geometric structure is $ \mathbb{R}P^3 \# \mathbb{R}P^3 $. In particular that means any connected sum of two aspherical closed 3-manifolds gives an example of a 3-manifold which cannot be expressed as $$ \Gamma \backslash G/K $$ Is it true for all dimensions $ n \geq 3 $ that a connected sum of two aspherical closed manifolds is never locally Riemannian homogeneous? In other words there is no way to write $ M_1 \# M_2 $ as $$ \Gamma \backslash G/K $$ where $ G $ is a Lie group with finitely many connected components, $ K $ a maximal compact subgroup, and $ \Gamma $ a torsion free cocompact lattice.
