Algebraic and Geometric Topology 3 (2003),
paper no. 10, pages 235-285.
Heegaard diagrams and surgery descriptions for twisted face-pairing 3-manifolds
J.W. Cannon, W.J. Floyd, W.R. Parry
Abstract.
The twisted face-pairing construction of our earlier papers gives an
efficient way of generating, mechanically and with little effort,
myriads of relatively simple face-pairing descriptions of interesting
closed 3-manifolds. The corresponding description in terms of surgery,
or Dehn-filling, reveals the twist construction as a carefully
organized surgery on a link.
In this paper, we work out the relationship between the twisted
face-pairing description of closed 3-manifolds and the more common
descriptions by surgery and Heegaard diagrams. We show that all
Heegaard diagrams have a natural decomposition into subdiagrams called
Heegaard cylinders, each of which has a natural shape given by the
ratio of two positive integers. We characterize the Heegaard diagrams
arising naturally from a twisted face-pairing description as those
whose Heegaard cylinders all have integral shape. This
characterization allows us to use the Kirby calculus and standard
tools of Heegaard theory to attack the problem of finding which
closed, orientable 3-manifolds have a twisted face-pairing
description.
Keywords.
3-manifold constructions, Dehn surgery, Heegaard diagrams
AMS subject classification.
Primary: 57N10.
DOI: 10.2140/agt.2003.3.235
E-print: arXiv:math.GT/0303081
Submitted: 12 November 2001.
(Revised: 5 February 2003.)
Accepted: 14 February 2003.
Published: 5 March 2003.
Notes on file formats
J.W. Cannon, W.J. Floyd, W.R. Parry
Department of Mathematics, Brigham Young University
Provo, UT
84602, USA
Department of Mathematics, Virginia Tech
Blacksburg, VA 24061, USA
Department of Mathematics,
Eastern Michigan University
Ypsilanti, MI 48197, USA
Email: cannon@math.byu.edu, floyd@math.vt.edu, walter.parry@emich.edu
URL:
http://www.math.vt.edu/people/floyd
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