On polyhedral 3-manifolds of nonnegative curvature

Vladimir Matveev  (Jena), mardi 10 mars 2009, 11h15, salle CM1113

Abstract:

The main theorem of this joint result with V. Shevchishin is as follows:

Theorem: Any closed polyhedral 3-manifold such that the sum of the dihedral angles around each edge does not exceed 2 \pi can be approximated by a Riemannian metric on nonnegative sectional curvature and therefore can be covered by S^3, S^2× S^1, or S^1× S^1× S^1.

Recall that a polyhedron is the convex hull of finitely many points in R^3 equipped with the induced metric. Polyhedral manifold are manifolds glued from polyhedra along isometries of their faces.

During the talk I explain the proof of the result stated above and its two corollaries:

Corollary 1 (suggested by Sullivan). Suppose a closed 3-manifolds admits a triangulation such that for every edge the number of simplices containing this edge is less than 6. Then, the manifold is covered by the 3-sphere.

Corollary 2 (Independently obtained by Lutz and Sullivan). Every triangulation of a closed 3-manifold satisfying the assumptions of corollary 1 consists of less than 600 simplices. I will also discuss relations of these statements to the theory of Ricci-flow and to the Poincare conjecture, and the difficulties of possible generalizations to higher dimensions.