Perelman's Invariant and collapse via geometric characteristic splittings'

Any compact 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, more generally, pieces whose
fundamental group has non-trivial centre, M collapses with bounded curvature
and has zero Perelman invariant.