# Collapsing irreducible 3-manifolds with nontrivial fundamental group

@article{Bessires2009CollapsingI3, title={Collapsing irreducible 3-manifolds with nontrivial fundamental group}, author={Laurent Bessi{\`e}res and G'erard Besson and Michel Boileau and Sylvain Maillot and Joan Porti}, journal={Inventiones mathematicae}, year={2009}, volume={179}, pages={435-460} }

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.

## 21 Citations

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We prove that a 3-dimensional compact Riemannian manifold which is locally collapsed, with respect to a lower curvature bound, is a graph manifold. This theorem was stated by Perelman and was used in…

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Author(s): Theerakarn, Thunwa | Advisor(s): Lott, John | Abstract: Perelman stated without proof that a 3-dimensional compact Riemannian manifold which is locally volume collapsed, with respect to a…

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We will simplify the earlier proofs of Perelman's collapsing theorem of 3-manifolds given by Shioya-Yamaguchi and Morgan-Tian.
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