Low Dimensional Topology

December 5, 2008

The Rectangle Condition

Filed under: 3-manifolds,Heegaard splittings — Jesse Johnson @ 11:05 am

It seems that there are certain ideas in mathematics that,  after some period of importance and activity, are superseded by more general and more powerful ideas.  In the world of Heegaard splittings, one such idea is what’s called the rectangle condition.  Soon after Casson and Gordon discovered the dichotomoy between weakly-reducible and strongly-irreducible Heegaard splittings, they defined the rectangle condition as a sufficient condition to show that a Heegaard diagram represents a strongly irreducible splitting.  While the condition “strongly irreducible” involves all disks in the two handlebodies, the rectangle condition is based on only a finite number of disks in each handlebody.  (For the definition, I’ll give a reference below.)

But two things happened:  First, for the only class of examples in which Casson and Gordon were planning on using the rectangle condition, Casson discovered a shorter and more general way to prove that the Heegaard splittings in question are strongly irreducible.  (This is published as as appendix to a paper by Moriah and Schultens [1].)  Second, Hempel’s definition of distance, via the curve complex, suggested a new picture of strongly irreducible splittings, in which the rectangle condition does not fit neatly.  The Heegaard community moved on.

Moved on, that is, until an interesting paper appeared in the arXiv mailing last tuesday morning.  Jung Hoon Lee’s paper [2] defines a new condition, called the parity condition.  (The paper also contains a definition of the standard rectangle condition.)  The parity condition also requires only a single Heegaard diagram to check  and implies that the Heegaard splitting is irreducible (but not necessarily strongly irreducible).  One of the original applications of the rectangle condition was to find loops in a stabilized Heegaard splitting such that performing sufficiently high integral Dehn surgery along that loop produces a strongly irreducible Heegaard splitting.  Using the parity condition, one can find loops where any non-trivial integral Dehn surgery produces an unstabilized Heegaard splitting.  (Note:  An earlier version of the paper claimed a stronger result and the original version of the post described this stronger version.  As the author points out in the comments below, Eric Sedgwick and John Berge pointed out a flaw in the original.)

So, will this paper precipitate a Renaissance of rectangle condition research?  Well,… perhaps not.  I think it will depend on how easy it is to find diagrams satisfying the weak rectangle condition.  (The original rectangle condition is in some sense very rare.)  But we’ll see.  Perhaps this is a good question to add to the open problems list.  I’ll phrase it as follows:  Find an irreducible Heegaard splitting that does not have a Heegaard diagram satisfying the weak rectangle condition.

1 Comment »

  1. Hi.

    I happen to know this site today and has read some posts.
    The weak rectangle condition is wrong (because some lemma
    in the preprint is wrong).
    Eric sedgwick and independently John Berge pointed it out
    the mistake with giving counterexamples.
    (I considered lemma-first, example-second way, so have no confidence to recover the example.)
    Recently, I am working with another condition for irreducibility, called “parity condition”.

    I wonder if you have a chance to see this reply.
    (Some time has passed.)

    p.s. This is an off-topic, but
    maybe we have met in 2004 in AMS sectional meeting held in Santa Barbara. This is to express my greetings to you.

    Comment by Jung Hoon Lee — February 25, 2009 @ 7:23 am | Reply

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