I’m running out of open questions to write about, so I want to urge all you readers to submit your own open questions via the comments box on the open problems page. For now, though, here’s a problem about mapping class groups of Heegaard splittings that may not be that hard, but I haven’t had time to think about it myself.

The mapping class group of a Heegaard splitting is the group of automorphisms of the ambient manifold that take the Heegaard splitting onto itself, modulo isotopies that keep the Heegaard surface on itself. (This is sometimes referred to as the Goeritz group of the Heegaard splitting, though other times the Goeritz group refers specifically to Heegaard splittings of the 3-sphere.) Since every automorpism of the Heegaard splitting comes from an automorphism of the ambient manifold, there is a canonical homomorphism from the mapping class group of the Heegaard splitting to the mapping class group of the ambient manifold. In general, this homomorphism won’t be one-to-one or onto, though by stabilizing the Heegaard splitting, we can find a splitting for any manifold such that the homomorphism is onto.

Once one has a Heegaard splitting for which this homomorphism is onto, one can ask further whether there is an isomorphic copy of the mapping class group of the 3-manifold contained in the mapping class group of the Heegaard splitting. More generally, given a Heegaard splitting, consider the short exact sequence from the kernel of the canonical homomorphism to the mapping class group of the Heegaard splitting to the mapping class group of the image of the homomorphism in the mapping class group of the 3-manifold. Does this short exact sequence split? (I.e. is there an isomorphism from the mapping class group of the 3-manifold back to the mapping class group of the Heegaard splitting such that the composition of this isomorphism with the canonical homomorphism is the identity?)

If the short exact sequence splits then the mapping class group of the Heegaard splitting is a semi-direct product of the kernel of the canonical homomorphism and its image (which in many cases will be the mapping class group of the 3-manifold). We can also ask the weaker question: Given a 3-manifold, is there always a Heegaard splitting such that the short exact sequence splits? And an even weaker version: can every mapping class group of a 3-manifold be embedded in the mapping class group of a surface? I don’t know if an answer to this last question is known, though it seems like it might be easier to answer through other means. If one can find a 3-manifold whose mapping class group is not a subgroup of a surface mapping class group then no Heegaard splitting of this 3-manifold can have a split short exact sequence.

Can one easily characterize this MCG as a subgroup of the MCG of the surface in your splitting? Perhaps in terms of the action on the fundamental group?

Comment by Ben Webster — February 28, 2008 @ 11:59 pm |

As a subgroup of the surface MCG, it’s the group of automorphisms of the Heegaard surface that can be extended to automorphisms of each handlebody. If you want an algebraic interpretation, there are homomorphisms from the fundamental group of the surface to the fundamental groups of the two handlebodies (free groups) and homomorphisms from these groups to the fundamental group of the 3-manifold. This is called a splitting diagram and conversely every pair of homomorphisms from a surface group to a pair of free groups of the appropriate rank induces a Heegaard splitting. (I think this is due to Stallings.)

An automorphism of the Heegaard splitting induces an automorphism of the splitting diagram, up to conjugation (an outer automorphism of the diagram?). Since every splitting diagram induces a Heegaard splitting, I believe one can show that every automorphism of the splitting diagram corresponds to an automorphism of the Heegaard splitting. Is this along the lines of what you were looking for?

Comment by Jesse Johnson — February 29, 2008 @ 9:39 am |

That’s a neat question.

I would guess that the answer would be that there is no section in general, but I don’t know a counterexample.

If you precompose the gluing map for a given splitting with higher and higher powers of a pseudo-Anosov, will that eventually give you a surjective example? (I’m thinking of the Namazi picture of the hyperbolic metric, which has a huge collar around the heegaard surface, which must be preserved by any mapping class of the 3-manifold.) You might expect to eventually have the kernel trivial, too, as the intersection of the two handlebody groups will get pretty small as grows. This might just give you an example where both groups are trivial, though, unless you judiciously choose every invariant under some finite mapping class.

Just a thought.

(And I’m glad to have found your blog!)

Comment by Richard Kent — March 4, 2008 @ 7:46 pm |

As it turns out, for high distance splittings (like those that come from composing with a high power of a pseudo-Anosov) the homomorphism is actually an isomorphism. This is in my paper with Rubinstein and follows from Namazi’s result that for high distance Heegaard splittings, the mapping class group is finite. For hyperbolic 3-manifolds, the kernel of the homomorphism can’t contain finite order elements, so it must be trivial. So, to find examples where there’s no section one needs to look at low distance examples. But I think you’re right (and I know a couple of others who agree) that sections probably don’t exist in general.

Comment by Jesse Johnson — March 4, 2008 @ 8:37 pm |

I think (S^1)^3 would be a good candidate 3-manifold for a negative answer. Is GL_3(Z) a subgroup of a surface mapping class group? If it is, I’ve never seen an embedding of it before. The mapping class group of (S^1)^3 is GL_3(Z).

Comment by Ryan Budney — March 25, 2012 @ 7:35 pm |

GL_3(Z) is not a subgroup of a surface mapping class group. This follows from the fact that any solvable subgroup of a surface mapping class group is virtually abelian, while GL_3(Z) contains the integer Heisenberg group. The assertion about solvable subgroups of surface mapping class groups goes back to a paper of Birman, Lubotzky and McCarthy, cf http://projecteuclid.org/download/pdf_1/euclid.dmj/1077303491.

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