Twisted face-pairing 3-manifolds

Authors:
J. W. Cannon, W. J. Floyd and W. R. Parry

Journal:
Trans. Amer. Math. Soc. **354** (2002), 2369-2397

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 face-pairing 3-manifolds. Just as every edge-pairing of a 2-dimensional disk yields a closed 2-manifold, so also every face-pairing of a faceted 3-ball yields a closed 3-dimensional pseudomanifold. In dimension 3, the pseudomanifold may suffer from the defect that it fails to be a true 3-manifold at some of its vertices. The method of *twisted face-pairing* 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 orientation-reversing face-pairing of by twisting, so as to yield an infinite parametrized family of face-pairings whose quotient complexes are all closed orientable 3-manifolds. The method is so efficient that, starting even with almost trivial face-pairings , it yields a rich family of highly nontrivial, yet relatively simple, 3-manifolds.

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 face-pairing is a closed 3-manifold.** 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 0-handles with 3-handles, natural 1-handles with 2-handles. This duality result of (2) is central to our further studies of twisted face-pairings.

We also relate the fundamental groups and homology groups of the twisted face-pairing 3-manifolds and of the original pseudomanifold (with vertices removed).

We conclude the paper by giving examples of twisted face-pairing 3-manifolds. These examples include manifolds from five of Thurston's eight 3-dimensional geometries.

**1.**James W. Cannon,*The combinatorial Riemann mapping theorem*, Acta Math.**173**(1994), no. 2, 155–234. MR**1301392**, 10.1007/BF02398434**2.**J. W. Cannon, W. J. Floyd, and W. R. Parry,*Squaring rectangles: the finite Riemann mapping theorem*, The mathematical legacy of Wilhelm Magnus: groups, geometry and special functions (Brooklyn, NY, 1992) Contemp. Math., vol. 169, Amer. Math. Soc., Providence, RI, 1994, pp. 133–212. MR**1292901**, 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****4.**J. W. Cannon, W. J. Floyd, and W. R. Parry, Introduction to twisted face-pairings,*Math. Res. Lett.***7**(2000), 477-491. CMP**2001:01****5.**J. W. Cannon, W. J. Floyd, and W. R. Parry, Ample twisted face-pairing 3-manifolds, preprint.**6.**J. W. Cannon, W. J. Floyd, and W. R. Parry, Heegaard diagrams and surgery descriptions for twisted face-pairing 3-manifolds, preprint.**7.**J. W. Cannon, W. J. Floyd, and W. R. Parry, A survey of twisted face-pairing 3-manifolds, 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**, 10.1090/S0002-9947-98-02107-2**9.**H. Seifert and W. Threlfall,*Lehrbuch der Topologie*, (Chelsea Publishing Company, New York 1947).**10.**W. P. Thurston,*The Geometry and Topology of 3-Manifolds*, Princeton lecture notes, http://www.msri.org/gt3m, 1979.**11.**William P. Thurston,*Three-dimensional geometry and topology. Vol. 1*, Princeton Mathematical Series, vol. 35, Princeton University Press, Princeton, NJ, 1997. Edited by Silvio Levy. MR**1435975****12.**J. Weeks,*SnapPea: A computer program for creating and studying hyperbolic 3-manifolds*, 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:
https://doi.org/10.1090/S0002-9947-02-02955-0

Keywords:
3-manifold constructions,
surgeries on 3-manifolds,
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 DMS-9803868, DMS-9971783, and DMS-10104030

Article copyright:
© Copyright 2002
American Mathematical Society