Perelman's invariant and collapse via geometric characteristic splittings

TL;DR

The paper demonstrates that certain non-positively curved manifolds with specific geometric splittings collapse with bounded curvature and have zero Perelman invariant, linking geometric decomposition to topological and curvature properties.

Contribution

It establishes a connection between geometric characteristic splittings and collapse behavior, showing manifolds with these splittings have zero Perelman invariant and collapse with bounded curvature.

Findings

Manifolds with Seifert fibered or central fundamental group pieces collapse with bounded curvature.

Such manifolds have zero Perelman invariant.

Collapse behavior is linked to the geometric characteristic splitting structure.

Abstract

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.