Twisted facepairing 3manifolds
Authors:
J. W. Cannon, W. J. Floyd and W. R. Parry
Journal:
Trans. Amer. Math. Soc. 354 (2002), 23692397
MSC (2000):
Primary 57Mxx
Published electronically:
February 4, 2002
MathSciNet review:
1885657
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Abstract: This paper is an enriched version of our introductory paper on twisted facepairing 3manifolds. Just as every edgepairing of a 2dimensional disk yields a closed 2manifold, so also every facepairing of a faceted 3ball yields a closed 3dimensional pseudomanifold. In dimension 3, the pseudomanifold may suffer from the defect that it fails to be a true 3manifold at some of its vertices. The method of twisted facepairing shows how to correct this defect of the quotient pseudomanifold systematically. The method describes how to modify by edge subdivision and how to modify any orientationreversing facepairing of by twisting, so as to yield an infinite parametrized family of facepairings whose quotient complexes are all closed orientable 3manifolds. The method is so efficient that, starting even with almost trivial facepairings , it yields a rich family of highly nontrivial, yet relatively simple, 3manifolds. This paper solves two problems raised by the introductory paper: (1) Replace the computational proof of the introductory paper by a conceptual geometric proof of the fact that the quotient complex of a twisted facepairing is a closed 3manifold. We do so by showing that the quotient complex has just one vertex and that its link is the faceted sphere dual to . (2) The twist construction has an ambiguity which allows one to twist all faces clockwise or to twist all faces counterclockwise. The fundamental groups of the two resulting quotient complexes are not at all obviously isomorphic. Are the two manifolds the same, or are they distinct? We prove the highly nonobvious fact that clockwise twists and counterclockwise twists yield the same manifold. The homeomorphism between them is a duality homeomorphism which reverses orientation and interchanges natural 0handles with 3handles, natural 1handles with 2handles. This duality result of (2) is central to our further studies of twisted facepairings. We also relate the fundamental groups and homology groups of the twisted facepairing 3manifolds and of the original pseudomanifold (with vertices removed). We conclude the paper by giving examples of twisted facepairing 3manifolds. These examples include manifolds from five of Thurston's eight 3dimensional geometries.
 1.
James
W. Cannon, The combinatorial Riemann mapping theorem, Acta
Math. 173 (1994), no. 2, 155–234. MR 1301392
(95k:30046), http://dx.doi.org/10.1007/BF02398434
 2.
J.
W. Cannon, W.
J. Floyd, and W.
R. Parry, Squaring rectangles: the finite Riemann mapping
theorem, functions (Brooklyn, NY, 1992) Contemp. Math.,
vol. 169, Amer. Math. Soc., Providence, RI, 1994,
pp. 133–212. MR 1292901
(95g:20045), http://dx.doi.org/10.1090/conm/169/01656
 3.
J.
W. Cannon, W.
J. Floyd, and W.
R. Parry, Sufficiently rich families of planar rings, Ann.
Acad. Sci. Fenn. Math. 24 (1999), no. 2,
265–304. MR 1724092
(2000k:20057)
 4.
J. W. Cannon, W. J. Floyd, and W. R. Parry, Introduction to twisted facepairings, Math. Res. Lett. 7 (2000), 477491. CMP 2001:01
 5.
J. W. Cannon, W. J. Floyd, and W. R. Parry, Ample twisted facepairing 3manifolds, preprint.
 6.
J. W. Cannon, W. J. Floyd, and W. R. Parry, Heegaard diagrams and surgery descriptions for twisted facepairing 3manifolds, preprint.
 7.
J. W. Cannon, W. J. Floyd, and W. R. Parry, A survey of twisted facepairing 3manifolds, in preparation.
 8.
J.
W. Cannon and E.
L. Swenson, Recognizing constant curvature
discrete groups in dimension 3, Trans. Amer.
Math. Soc. 350 (1998), no. 2, 809–849. MR 1458317
(98i:57023), http://dx.doi.org/10.1090/S0002994798021072
 9.
H. Seifert and W. Threlfall, Lehrbuch der Topologie, (Chelsea Publishing Company, New York 1947).
 10.
W. P. Thurston, The Geometry and Topology of 3Manifolds, Princeton lecture notes, http://www.msri.org/gt3m, 1979.
 11.
William
P. Thurston, Threedimensional geometry and topology. Vol. 1,
Princeton Mathematical Series, vol. 35, Princeton University Press,
Princeton, NJ, 1997. Edited by Silvio Levy. MR 1435975
(97m:57016)
 12.
J. Weeks, SnapPea: A computer program for creating and studying hyperbolic 3manifolds, http://www.northnet.org/weeks.
 1.
 J. W. Cannon, The combinatorial Riemann mapping theorem, Acta Math. 173, (1994) 155234. MR 95k:30046
 2.
 J. W. Cannon, W. J. Floyd, and W. R. Parry, Squaring rectangles: the finite Riemann mapping theorem, in The Mathematical Heritage of Wilhelm MagnusGroups, Geometry and Special Functions, Contemporary Mathematics 169, (Amer. Math. Soc., Providence 1994) 133212. MR 95g:20045
 3.
 J. W. Cannon, W. J. Floyd, and W. R. Parry, Sufficiently rich families of planar rings, Ann. Acad. Sci. Fenn. 24 (1999) 265304. MR 2000k:20057
 4.
 J. W. Cannon, W. J. Floyd, and W. R. Parry, Introduction to twisted facepairings, Math. Res. Lett. 7 (2000), 477491. CMP 2001:01
 5.
 J. W. Cannon, W. J. Floyd, and W. R. Parry, Ample twisted facepairing 3manifolds, preprint.
 6.
 J. W. Cannon, W. J. Floyd, and W. R. Parry, Heegaard diagrams and surgery descriptions for twisted facepairing 3manifolds, preprint.
 7.
 J. W. Cannon, W. J. Floyd, and W. R. Parry, A survey of twisted facepairing 3manifolds, in preparation.
 8.
 J. W. Cannon and E. L. Swenson, Recognizing constant curvature discrete groups in dimension 3, Trans. Amer. Math. Soc. 350 (1998) 809849. MR 98i:57023
 9.
 H. Seifert and W. Threlfall, Lehrbuch der Topologie, (Chelsea Publishing Company, New York 1947).
 10.
 W. P. Thurston, The Geometry and Topology of 3Manifolds, Princeton lecture notes, http://www.msri.org/gt3m, 1979.
 11.
 W. P. Thurston, ThreeDimensional Geometry and Topology, Vol. 1, (Princeton University Press, Princeton 1997). Edited by S. Levy. MR 97m:57016
 12.
 J. Weeks, SnapPea: A computer program for creating and studying hyperbolic 3manifolds, http://www.northnet.org/weeks.
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Additional Information
J. W. Cannon
Affiliation:
Department of Mathematics, Brigham Young University, Provo, Utah 84602
Email:
cannon@math.byu.edu
W. J. Floyd
Affiliation:
Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061
Email:
floyd@math.vt.edu
W. R. Parry
Affiliation:
Department of Mathematics, Eastern Michigan University, Ypsilanti, Michigan 48197
Email:
walter.parry@emich.edu
DOI:
http://dx.doi.org/10.1090/S0002994702029550
PII:
S 00029947(02)029550
Keywords:
3manifold constructions,
surgeries on 3manifolds,
Thurston's geometries
Received by editor(s):
December 8, 2000
Received by editor(s) in revised form:
October 5, 2001
Published electronically:
February 4, 2002
Additional Notes:
This research was supported in part by NSF grants DMS9803868, DMS9971783, and DMS10104030
Article copyright:
© Copyright 2002 American Mathematical Society
