Complutense University Library

On minimal Heegaard splittings

Montesinos Amilibia, José María and Birman, Joan S. (1980) On minimal Heegaard splittings. Michigan Mathematical Journal, 27 (1). pp. 47-57. ISSN 0026-2285

[img] PDF
Restricted to Repository staff only until 31 December 2020.

1MB

Official URL: http://www.math.columbia.edu/~jb/b-montesinos-min-HS.pdf

View download statistics for this eprint

==>>> Export to other formats

Abstract

This paper deals with Heegaard splittings and Heegaard diagrams (denoted H-diagrams). Two interesting examples are given which shed light on certain questions about "minimality'' of H-diagrams. An H-diagram is a quadruple (M,F,v,w), where M is a closed orientable 3-manifold, F is a surface embedded in M that separates it into two handlebodies V and W, and v and w are complete systems of meridian discs for V and W. The complexity of the H-diagram, c(M,F,v,w), is the cardinality of ∂v∩∂w. An H-diagram (M,F,v,w) is pseudominimal if c(M,F,v,w)≤c(M,F,v,w′) for all w′ and c(M,F,v,w)≤c(M,F,v′,w) for all v′. It is minimal if c(M,F,v,w)≤c(M,F,v′,w′) for all v′ and w′. In the first example, two H-diagrams of the lens space M=L(7,2) are given with different complexity. This shows that pseudominimality does not imply minimality. In this example, the H-diagram has a pair of cancelling handles. F. Waldhausen asked the question: "In an H-diagram which is pseudominimal but not minimal is there always a pair of cancelling handles?'' The second example shows that either (a) there is a 3-manifold with two minimal H-splittings of different genus or (b) there is an H-diagram that is pseudominimal but not minimal and has no pair of cancelling handles. The authors conjecture that (a) holds.
This is an enlightening paper to read for anyone wishing to learn some of the methods and techniques of Heegaard splittings.

Item Type:Article
Uncontrolled Keywords:minimal Heegaard splittings
Subjects:Sciences > Mathematics > Topology
ID Code:17215
References:

W. Haken, Various aspects of the three-dimensional Poincaré problem. Topology ofManifolds

(Proc. Inst., Univ. of Georgia, Athens, Ga., 1969), pp. 140-152. Markham, Chicago, 111., 1970.

P. J. Higgins and R. C. Lyndon, Equivalence of elements under automorphisms of a free group. J. London Math. Soc. (2) 8 (1974), 254-258.

J. M. Montesinos, Surgery on links and double branched covers of S3. Knots, groups and 3-manifolds (Papers dedicated to the memory of R. H. Fox), pp. 227-259. Ann. of Math. Studies, No. 84, Princeton Univ. Press, Princeton, N.J., 1975.

H. Poincaré, Cinquieme complément a l'analysis situs. Rend. Circo Mat. Palermo 18 (1904), 45-110.

F. Waldhausen, Heegaard-Zerlegungen der 3-Sphiire. Topology 7 (1968), 195-203.

F. Waldhausen, Some problems on 3-manifolds. Proceedings of Symposia in Pure Mathematics, 32 (1978), 313-322.

J. H. C. Whitehead, On certain sets of elements in a free group. Proc. London Math. Soc., 11. s. 41 (1936), 45-56.

H. Zieschang, On simple systems of paths on complete pretzels, Amer. Math. Soco Transl. (2) 92 (1970), 127-137.

Deposited On:27 Nov 2012 09:06
Last Modified:07 Feb 2014 09:44

Repository Staff Only: item control page