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The geometry and topology of three-manifolds is a set of widely circulated notes for a graduate course taught at Princeton University by William Thurston from 1978 to 1980 describing his work on 3-manifolds. They were written by Thurston, assisted by students William Floyd and Steven Kerchoff. [1]
Thurston, William P. (1982). "Three-dimensional manifolds, Kleinian groups and hyperbolic geometry". Bulletin of the American Mathematical Society. New Series. 6 (3): 357– 381. doi: 10.1090/S0273-0979-1982-15003-0. ISSN 0002-9904. MR 0648524. This gives the original statement of the conjecture. William Thurston. Three-dimensional geometry and ...
William Paul Thurston (October 30, 1946 – August 21, 2012) was an American mathematician.He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds.
Gromov's topology utilizes the Gromov-Hausdorff metric and is defined on pointed hyperbolic 3-manifolds. One essentially considers better and better bi-Lipschitz homeomorphisms on larger and larger balls. This results in the same notion of convergence as above as the thick part is always connected; thus, a large ball will eventually encompass ...
Thurston's contributions to the theory allow one to also consider, in many cases, the additional structure given by a particular Thurston model geometry (of which there are eight). The most prevalent geometry is hyperbolic geometry. Using a geometry in addition to special surfaces is often fruitful.
In hyperbolic geometry, the ending lamination theorem, originally conjectured by William Thurston () as the eleventh problem out of his twenty-four questions, states that hyperbolic 3-manifolds with finitely generated fundamental groups are determined by their topology together with certain "end invariants", which are geodesic laminations on some surfaces in the boundary of the manifold.
The Geometry and Topology of Three-Manifolds, 1980 Princeton lecture notes on geometric structures on 3-manifolds, that states his elliptization conjecture near the beginning of section 3. This Riemannian geometry -related article is a stub .
Robion Kirby, Problems in low-dimensional topology, (see problem 1.77, due to Cameron Gordon, for exceptional slopes) Marc Lackenby and Robert Meyerhoff, The maximal number of exceptional Dehn surgeries, arXiv:0808.1176; William Thurston, The geometry and topology of 3-manifolds, Princeton lecture notes (1978–1981).