Almost normal Heegaard splittings


The study of three-manifolds via their Heegaard splittings was initiated by Poul Heegaard in 1898 in his thesis. Our approach to the subject is through almost normal surfaces, as introduced by Hyam Rubinstein [Geo. Top. (Athens), 1993] and distance, as introduced by John Hempel [Topology, 2001].

Among the results presented is a proof that every closed, orientable three-manifold has only finitely many Heegaard splittings with distance greater than 4, a new recognition algorithm for surface bundles over the circle, and a series of results which bound the distance of a splitting in terms of its structure as an almost normal surface.

  • page 4, line 10: Comma missing at the end of the line.
  • page 6, line 7: Except for the genus one splitting of the three-sphere.
  • page 16, line 1: "Suppose _that_ F and G..."
  • page 27, line 1: A is not compressible.
  • page 30, line 8: There is an extra space after the y at the end of the line.
  • page 30, line -8: "...a simple curve _is_ a properly..."
  • page 30, line -4: The equation at the beginning of the line should be "D \cap \Tii = D \cap \Ti = \beta" and there is a missing comma after the first \beta.
  • page 31, line 9: There is an extra space after "tightening disks".
  • page 31, line -6: "blockedsubmanifolds" should be two words.
  • page 32, line 1: Should be "D' = D - image(F_0), which does not..."
  • page 32, line 6: Should be "D \subset \Tii"
  • page 52, line 15: "as we shall see" should be "as we have seen."
  • page 58, line -8 and -1: Punctuation goes outside a ending paren (unless the parens enclose a complete thought). This error occurs throughout the thesis.
  • page 68, line 1: Wandering "proof box".
  • page 68, line 12: The surfaces A and B should have disjoint boundary.
  • page 72, line 7: Theorem 7.1.1 is poorly phrased. The constant d_1 is a linear function of |T|, the number of tetrahedra in the given triangulation of M.

    If you have any questions or corrections please contact me via email.


    The copyright on this thesis is held by Saul Schleimer.

    last touched - Jan. 2005
    To maintainer's homepage.
    UIC: Up to the math department webpage.