Low Dimensional Topology

June 3, 2009

Frohman’s Trick

Filed under: 3-manifolds, Heegaard splittings — Jesse Johnson @ 4:34 pm

Here’s a neat trick that Charles Frohman related to me a number of years ago.  I think it’s in this paper [1], though I wasn’t able to find it in there.  The trick is beautifully simple, but there don’t seem to be that many applications of it.  In fact, as far as I know the only application is in Frohman’s paper. Here’s the Lemma:  Let \Sigma be a Heegaard surface in a 3-manifold M (other than the 3-sphere) and assume there is aball B \subset M such that the intersection of \Sigma and B is not planar.  Then \Sigma is reducible.  (Recall that a Heegaard surface is reducible if there is a sphere that intersects it in a single essential loop.)

To prove this, we first recall that a loop \ell in a handlebody H is a core if the complement in H of a neighborhood N of \ell is a compression body.  The first step of proving the Lemma is to show that having a non-planar intersection implies that the ball B contains a core \ell of one of the handlebodies of the Heegaard splitting.   This is not too difficult if you look at the intersection of the ball with each of the handlebodies, but it’s technical enough that I’ll leave the details to the interested reader.

Step two is much easier:  Because M is not the 3-sphere, the complement of B is not a ball.  If we remove a neighborhood N of \ell from M then what’s left of B will not be a ball either, so the sphere \partial B will be an essential sphere in the complement M \setminus N.  Because \ell is a core of one of the handlbodies, the surface \Sigma is also a Heegaard splitting for M \setminus N.

Haken’s Lemma states that every Heegaard splitting for a reducible 3-manifold is reducible, so \Sigma is reducible as a Heegaard splitting of M \setminus N.  But the reducing sphere for \Sigma in M \setminus N is also a reducing sphere for \Sigma in M, and that does it.

Of course, there are stronger things that one can say about the intersection of a strongly irreducible Heegaard splitting with a ball, such as what Scharlemann proved in [2], but these require a lot more work, and a much stronger assumption.  The beauty of Frohman’s trick is the simplicity of its assumptions and of its proof.  Unfortunately, this situation doesn’t seem to come up very often in practice.  Maybe this is more important as a bit of intution, something to keep in the back of your mind even if it doesn’t make it into your final proof.

[1] Frohman, Charles Minimal surfaces and Heegaard splittings of the three-torus. Pacific J. Math. 124 (1986), no. 1, 119–130.

[2] Scharlemann, Martin Local detection of strongly irreducible Heegaard splittings. Topology Appl. 90 (1998), no. 1-3, 135–147.

2 Comments »

  1. There is a paper by Kobayashi and Rieck which uses Frohman’s Lemma many times. (although I haven’t read that paper in full details…)
    “Local detection of strongly irreducible Heegaard splittings via knot exteriors” Topology Appl. 138 (2004), 239–251.
    I agree that the Lemma is interesting and useful.

    Also, if M is irreducible, the converse seems true.
    That is, if Heegaard splitting is reducible, some core is contained in a ball. So the contrapositive can possibly be a way to show irreducibility of H.S., but looks not so useful.

    Comment by JungHoon Lee — June 4, 2009 @ 12:40 am | Reply

  2. Other sources give the reference as being Frohman’s paper:
    “Topological Uniqueness of Triply Periodic Minimal Surfaces in R^3″. Frohman’s Lemma can also be stated as:
    If a spine for a compressionbody which is half of a Heegaard splitting has a cycle which is contained in a 3-ball, then the splitting is reducible.

    Here are a few other places it is used:
    – Frohman and Meeks use the trick in their classification of non-compact Heegaard splittings of R^3.

    – Scharlemann and Thompson use in their classification of splittings of (closed surface) x I. (It’s proposition 2.5 in that paper.)

    – Hayashi and Shimokawa use it in classifying Heegaard splittings of trivial arcs in compressionbodies. (Lemma 4.1).

    Comment by Scott Taylor — June 4, 2009 @ 9:13 am | Reply


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