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

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

2010-03-23

arxiv.org/abs/1003.4413v2

Mathematics

Geometric Topology

24 pages, 2 figures, substantial change in the introduction part of the original paper due to the recent work of Segerman-Till

Scientific paper

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 Poincar\'e 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.

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