On minimal Heegaard splittings

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.