Triangulated 3-Manifolds: from Haken's normal surfaces to Thurston's algebraic equation

Abstract

We give a brief summary of some of our work and our joint work with Stephan Tillmann on solving Thurston's equation and Haken equation on triangulated 3-manifolds in this paper. Several conjectures on the existence of solutions to Thurston's equation and Haken equation are made. Resolutions of these conjecture will lead to a new proof of the Poincaré conjecture without using the Ricci flow. We approach these conjectures by a finite dimensional variational principle so that its critical points are related to solutions to Thurston's gluing equation and Haken's normal surface equation. The action functional is the volume. This is a generalization of an earlier program by Casson and Rivin for compact 3-manifolds with torus boundary.

arXiv comment24 pages, 2 figures, substantial change in the introduction part of the original paper due to the recent work of Segerman-Tillmann. We also added section 5. The paper will appear in the Contemporary Mathematics volume based on "Interactions Between Hyperbolic Geometry, Quantum Topology and Number Theory"