Low Dimensional Topology

March 3, 2008

Group theory, as told by a Heegaard splitter

Filed under: 3-manifolds,Heegaard splittings — Jesse Johnson @ 2:08 pm

A couple of weeks ago I posted a paper to the arXiv that included a construction of infinitely many Heegaard splittings for a fixed 3-manifold that are pairwise non-isotopic but induce Nielsen equivalent generating sets for the fundamental group of the 3-manifold. Richard Weidmann recently pointed out to me another way to construct such examples if we assume a stronger version of Gordon’s conjecture claimed by David Bachman [1]. (The status Dave’s proof seems to be a somewhat divisive issue within the topology community, with opinion split between those who, for various reasons, judge his papers with an extra dose of skepticism, and those who try hard to give him the benefit of the doubt. I hope the next version of his paper is clearly written enough to settle things in his favor, but for now I am trying to remain neutral.) I don’t know if this stronger version follows from Qiu’s proof [2]. Scarlemann’s take on his proof [3] is sitting on my desk, but I haven’t had a chance to read it yet.

Given a finitely generated group we say that two sets of (the same number of) generators are Nielsen equivalent if we can turn the first set into the second by a sequence of moves in which we replace a generator from the first set with either its inverse or its product with a second generator from the same set. Equivalently, two generating sets are Nielsen equivalent if the epimorphisms they define from a free group to the original group are related by an automorphism of the free group. Given a Heegaard splitting of a 3-manifold, there is an epimorphism from the fundamental group of each handlebody (a free group) to the fundamental group of the 3-manifold. Thus every Heegaard splitting determines two Nielsen classes of generators (one for each handlebody) and if two Heegaard splittings are isotopic then the pairs of Nielsen classes must be equal. (Lustig and Moriah have used this fact to distinguish many classes of Heegaard splittings.)

The converse, as it turns out is not true. Here’s Weidmann’s idea: Montesinos has shown [4] that there are Seifert fibered spaces with irreducible genus three Heegaard splittings such that each of the two Nielsen classes they determine is redundant, i.e. has a representative consisting of two non-trivial generators and the identity. (I assume these examples are related to Boileau and Zieschang’s examples [5] of Seifert fibered spaces that have rank two fundamental groups but Heegaard genus 3.)

Consider a second 3-manifold that has two non-isotopic, irreducible Heegaard splittings that become isotopic after a single stabilization. A connect sum of these two 3-manifolds has two Heegaard splittings, each coming from connect summing the Heegaard splitting for the Seifert fibered space with one of the Heegaard splittings for the second 3-manifold. Bachman’s version of the Gordon conjecture states that a Heegaard splitting for a connect sum of 3-manifolds has a unique expression as a connect sum of Heegaard splittings. Thus the two Heegaard splittings for the connect sum are not isotopic (in fact, they’re not homeomorphic). Regardless of the status of Dave’s proof, the isotopy version of the strengthened Gordon conjecture is still open: If, given two non-isotopic Heegaard splittings of a fixed 3-manifold, we take the connect sum of each with a fixed Heegaard splitting of a second 3-manifold, can the resulting Heegaard splittings be isotopic?

To see that the Nielsen classes induced by the handlebodies are the same, note that the Nielsen classes determined by the Heegaard splittings of the second summand in the connect sum become equivalent after we add a trivial generator. (This is what happens to the generating sets when we stabilize.) The generating set determined by the connect sum is the union of the generating sets for the original Heegaard splittings, included into the free product. Thus we can use the trivial generator from the generating set for the Seifert fibered space to turn one of the generating sets into the other, then return it to a trivial generator.

The rank vs. genus problem, which I’ve mentioned previously, is known to be a difficult problem. Here’s a weaker version, suggested by the above paragraph, that might be a little easier: Are there hyperbolic 3-manifolds with Heegaard splittings in which one or both of the handlebodies induce redundant Nielsen classes (classes in which one of the generators is the identity)? This Heegaard splitting need not be minimal genus (if it were then it would have to do with the rank vs. genus question). I should also mention that Nielsen equivalent, non-isotopic Heegaard splittings of Seifert fibered spaces are probably more common than the examples in my paper. The examples I constructed (Horizontal Heegaard splittings of certain spaces with torus base space and two singular fibers) were the easiest to deal with, but I wouldn’t be surprised if many (maybe all?) horizontal Heegaard splittings had this property. Working it out would probably be a complicated but straightforward calculation.

[4] Montesinos, José M., Note on a result of Boileau-Zieschang. Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984), 241–252, London Math. Soc. Lecture Note Ser., 112, Cambridge Univ. Press, Cambridge, 1986.

[5] Boileau, M. and Zieschang, H., Heegaard genus of closed orientable Seifert $3$-manifolds. Invent. Math. 76 (1984), no. 3, 455–468.

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