Transactions of the American Mathematical Society

Author(s): Feng Luo; Stephan Tillmann
Journal: Trans. Amer. Math. Soc. 360 (2008), 2849-2866.
MSC (2000): Primary 57M25, 57N10
Posted: January 7, 2008
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Abstract: Let

be the interior of a compact 3-manifold with boundary, and let $\mathcal{T}$ be an ideal triangulation of

This paper describes necessary and sufficient conditions for the existence of angle structures, semi-angle structures and generalised angle structures on $(M; \mathcal{T})$ respectively in terms of a generalised Euler characteristic function on the solution space of the normal surface theory of $(M; \mathcal{T}).$ This extends previous work of Kang and Rubinstein, and is itself generalised to a more general setting for 3-dimensional pseudo-manifolds.

1.: M. Boileau, J. Porti: Geometrization of 3-orbifolds of cyclic type, Astérisque No. 272 (2001). MR 1844891 (2002f:57034)
2.: A.J. Casson: private communication.
3.: Y. Colin de Verdière: Un principe variationnel pour les empilements de cercles, Invent. Math., 104, no. 3, 655-669 (1991). MR 1106755 (92h:57020)
4.: R. Guo: Geometric angle structures on triangulated surfaces, math.GT/0601486.
5.: A. Hatcher: Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354 (2002k:55001)
6.: W. Jaco, J.H. Rubinstein: 0-efficient triangulations of 3-manifolds, J. Differential Geom. 65, no. 1, 61-168 (2003). MR 2057531 (2005d:57034)
7.: E. Kang, J.H. Rubinstein: Ideal triangulations of 3-manifolds I; spun normal surface theory, Geometry and Topology Monographs, Vol. 7 , Proceedings of the Casson Fest, 235-265 (2004). MR 2172486 (2006g:57035)
8.: E. Kang, J.H. Rubinstein: Ideal triangulations of 3-manifolds II; taut and angle structures, Algebr. Geom. Topol., 5, 1505-1533 (2005). MR 2186107 (2006g:57036)
9.: M. Lackenby: Word hyperbolic Dehn surgery, Invent. Math., 140, no. 2, 243-282 (2000). MR 1756996 (2001m:57003)
10.: I. Rivin: Combinatorial optimization in geometry, Adv. in Appl. Math. 31, no. 1, 242-271 (2003). MR 1985831 (2004i:52005)
11.: W.P. Thurston: The geometry and topology of 3-manifolds, Princeton Univ. Math. Dept. (1978). Available from http://msri.org/publications/books/gt3m/.
12.: S. Tillmann: Normal surfaces in topologically finite 3-manifolds, math.GT/0406271.
13.: J.L. Tollefson: Normal surface -theory, Pacific J. of Math., 183, 359-374 (1998). MR 1625962 (99c:57047)
14.: G.M. Ziegler: Lectures on polytopes, Springer-Verlag, New York, 1995. MR 1311028 (96a:52011)

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Feng Luo
Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08854
Email: fluo@math.rutgers.edu

Stephan Tillmann
Affiliation: Départment de mathématiques, Université du Québec à Montréal, Case postale 8888, Succursale Centre-Ville, Montréal, Québec, Canada H3C 3P8
Email: tillmann@math.uqam.ca

DOI: 10.1090/S0002-9947-08-04301-8
PII: S 0002-9947(08)04301-8
Keywords: 3--manifold, ideal triangulation, angle structure
Received by editor(s): December 5, 2005
Posted: January 7, 2008
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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