Triangulated 3-Manifolds: from Hakens normal surfaces to Thurstons algebraic equation

Feng Luo

Geometric Analysis and Geometric Topology mathscidoc:1912.43813

Interactions between hyperbolic geometry, quantum topology and number theory, Contemp. Math, 541, 183-204
We give a brief summary of some of our work and our joint work with Stephan Tillmann on solving Thurstons equation and Haken equation on triangulated 3-manifolds in this paper. Several conjectures on the existence of solutions to Thurstons equation and Haken equation are made. Resolutions of these conjecture will lead to a new proof of the Poincar 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 Thurstons gluing equation and Hakens 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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@inproceedings{fengtriangulated,
  title={Triangulated 3-Manifolds: from Hakens normal surfaces to Thurstons algebraic equation},
  author={Feng Luo},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224205941710196377},
  booktitle={Interactions between hyperbolic geometry, quantum topology and number theory, Contemp. Math},
  volume={541},
  pages={183-204},
}
Feng Luo. Triangulated 3-Manifolds: from Hakens normal surfaces to Thurstons algebraic equation. Vol. 541. In Interactions between hyperbolic geometry, quantum topology and number theory, Contemp. Math. pp.183-204. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224205941710196377.
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