Closed flat affine 3-manifolds are prime

2014/07/16 by Suhyoung Choi, Choi, Suhyoung
Mathematics · #57M50 #FOS: Mathematics #Geometric Topology (math.GT) #math.GT #msc:57M50

paper · pdf · doi:10.48550/arxiv.1407.4264

This paper has been withdrawn by the author. The second crucial part of the proof of Theorem 1.2 is not correct. I cannot prove Theorem 1.2. The other correct parts will be published in other papers

arxiv created 2014/11/01 · arxiv updated 2014/11/04

Abstract

An (flat) affine 3-manifold is a 3-manifold with an atlas of charts to an affine space \mathbf R³ with transition maps in the affine transformation group Aff(\mathbf R³). Equivalently an affine 3-manifold is a 3-manifold with a flat torsion-free affine connection. We show that a closed affine 3-manifold is either irreducible or is finitely covered by an affine Hopf manifold. A real projective 3-manifold is a manifold with an atlas of charts to a real projective space \mathbf R P³ with transition maps in the projective transformation group PGL(4, \mathbf R). Using the convex concave decomposition of real projective manifolds, we will show that a closed real projective 3-manifold decomposes into concave affine submanifolds, toral π-submanifolds and 2-convex real projective manifolds.

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