Collapsing irreducible 3-manifolds with nontrivial fundamental group

Sylvain Maillot (1) , Laurent Bessières (2) , Gérard Besson (2) , Michel Boileau (3) , Joan Porti (4)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier
2 IF - Institut Fourier
3 IMT - Institut de Mathématiques de Toulouse UMR5219
4 UAB - Departament de Matemàtiques [Barcelona]

Let M be a closed, orientable, irreducible, non-simply connected 3-manifold. We prove that if M admits a sequence of Riemannian metrics which volume-collapses and whose sectional curvature is locally controlled, then M is a graph manifold. This is the last step in Perelman’s proof of Thurston’s Geometrisation Conjecture.

Domaines: Topologie géométrique [math.GT]

Sylvain Maillot, Laurent Bessières, Gérard Besson, Michel Boileau, Joan Porti. Collapsing irreducible 3-manifolds with nontrivial fundamental group. Inventiones Mathematicae, 2010, 179 (2), pp.435-460. ⟨10.1007/s00222-009-0222-6⟩. ⟨hal-00796507⟩