Perelman's invariant and collapse via geometric characteristic   splittings
  Any closed orientable and smooth non-positively curved manifold M is known to admit a geometric characteristic splitting, analogous to the JSJ decomposition in three dimensions. We show that when this splitting consists of pieces which are Seifert fibered or pieces each of whose fundamental group has non-trivial centre, M collapses with bounded curvature and has zero Perelman invariant. 
Pablo Su\'arez-Serrato