Low Dimensional Topology

November 19, 2007

Homological Stability

Filed under: 3-manifolds,Heegaard splittings,Mapping class groups — Jesse Johnson @ 6:54 pm

(Reprinted from my old blog)

Here’s an interesting fact: The abelianizations of the mapping class groups of two surfaces are isomorphic if both surfaces have sufficiently high genus. In other words, if you take a high genus surface and add a handle to it, the abelianization of its mapping class group doesn’t change. Now, the abelianization of a group is just the first homology group of its classifying space (a space with the given group as its fundamental group and whose universal cover is contractible) and for each dimension, we can define the homology of the original group as the homology of its classifying space. Well, it turns out for these higher homology groups don’t change either, if you start with a higher genus surface.

This phenomenon is called homological stability and similar results have been proved for other sequences of groups, mostly automorphism groups of objects that become progressively more complicated. Hatcher and Wahl [1] recently proved the following: Let M and N be three manifolds and let M_k be the connect sum or boundary connect sum of M and k copies of N. Then the nth homology mapping class groups of M_k is stable for sufficiently large k.

As you might have guessed if you’ve read many of my previous posts, I’m now going to suggest a connection to Heegaard splittings. Consider a sequence of stabilizations of a fixed Heegaard splitting, i.e. the nth splitting is the result of attaching n unknotted handles to the original. The question is: are the mapping class groups of these Heegaard splittings homologically stable? Hatcher and Wahl’s result implies that mapping class groups of handlebodies are homologically stable (a handlebody is a boundary connect sum of a number of solid tori) but that isn’t enough to prove it for a Heegaard splitting.

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Heegaard splitting invariants from the symplectic group

Filed under: 3-manifolds,Heegaard splittings,Mapping class groups — Jesse Johnson @ 6:19 pm

(Reposted from my old ldt blog)

One obsession that followed me through most of graduate school and into what is now my second year as a postdoc is the quest to find invariants of Heegaard splittings, or more generally ways of determining when two Heegaard splittings are not isotopic or not homeomorphic. One of the first invariants of this sort was developed by Joan Birman [1] in 1975, who used them to show that connect sums of certain lens spaces have non-homeomorphic Heegaard splittins. Her invariant is constructed by looking at how the gluing map acts on the first homology group of the surface. The gluing map is only defined up to automorphisms of the two handlebodies in the Heegaard splitting, so to make it an invariant, Birman had to take into account the actions on the first homology group of the subgroups corresponding to automorphisms of the handlebodies.

Birman constructs a matrix such that the action of the handlebody groups corresponds to row and column operations, making its trace an invariant of the Heegaard splitting. Unfortunately, the trace must be taken modulo a number that depends on the first homology group of the 3-manifold and which in many cases is zero. This makes it difficult to find examples where Birman’s invariant is effective. After her original paper, no more papers using her invariant have been published. A number of other techniques and invariants for distinguishing Heegaard splittings have appeared in the last couple of decades, but all fail in the one case in which I am most interested: they cannot distinguish stabilized Heegaard splittings.

I am always on the lookout for new methods that might be able distinguish stabilized Heegaard splittings, so I was excited to hear Nathan Broaddus speak at last year’s AMS Joint Meetings about a project he is working on with Joan Birman and Tara Brendle. Their goal is to extend the methods behind Birman’s original invariant in order to find new Heegaard splitting invariants as follows: We can think of the first homology group of a surface as the quotient of its fundamental group by the commutator subgroup. The action of the gluing map on the first homology group is induced by its action on the fundamental group, filtered through this quotient. We can further filter this action through quotients by the lower central series of the fundamental group, i.e. the sequence of groups [G,G], then [G,[G,G]], [G,[G,[G,G]]], etc. (This is called the Johnson-Morita representation, where “Johnson” refers to Dennis Johnson, no relation.)

The first step in their program recently appeared on the arXiv [2]. They look at the second Johnson-Morita representation, the one defined by the quotient by [G,[G,G]], and calculate the image of the whole mapping class group and of the handlebody group in the automorphism group of the quotient. The next step is to turn this understanding into invariants. In conversations, Broaddus and Birman suggested that the main problem with finding invariants is that the information they can extract from the representations appears in many cases to be invariant for the 3-manifold and thus would not distinguish Heegaard splittings. (I’ve run into the same problem often enough when trying to concoct my own invariants.) Still, I’m keeping my eye on the arXiv in anticipation of their next paper.

[1] On the equivalence of Heegaard splittings of closed, orientable $3$-manifolds. Knots, groups, and $3$-manifolds (Papers dedicated to the memory of R. H. Fox), pp. 137–164. Ann. of Math. Studies, No. 84, Princeton Univ. Press, Princeton, N.J., 1975.
[2] Follow link to arXiv.

Mapping class actions on handlebodies

Filed under: 3-manifolds,Heegaard splittings,Mapping class groups — Jesse Johnson @ 5:48 pm

(Reposted from my old ldt blog)

Now, those of you who have been unfortunate enough to see me give a talk in the last year know that I have been thinking a lot about mapping class groups of Heegaard splittings. These are groups of automorphisms of a 3-manifold that take a Heegaard surface onto itself, modulo isotopies of the 3-manifold that preserve the Heegaard surface setwise. When I was visiting Rutgers, New Brunswick a few months ago, spreading the gospel of the mapping class group, Saul Schleimer suggested an interesting idea and I hope he will forgive me for mentioning it here.Any automorphism of a Heegaard splitting that preserves each handlebody induces an automorphism the fundamental group of each handlebody (which is a free group). Saul suggested trying to understand this pair of actions of the mapping class group on these two free groups. If you Dehn twist the boundary of a handlebody along the boundary of a properly embedded disk, the action on the fundamental group is trivial. Luft [1] showed, moreover, that any automorphism of a handlebody that acts trivially on the fundamental group is a composition of Dehn twists along properly embedded disks.

If a Heegaard splitting is reducible, i.e. there is an embedded sphere that intersects the Heegaard surface in a single essential loop, then Dehn twisting along this loop extends into each handlebody as a Dehn twist along a properly embedded disk. Thus such an automorphism acts trivially on the fundamental groups of both handlebodies. The question is whether there is a converse analogous to Luft’s result. In other words, given an automorphism of a Heegaard splitting that fixes the fundamental groups of both handlebodies, is this automorphism always a composition of twists along reducing spheres? A slightly weaker question is: Is there an irreducible Heegaard splitting that admits an automorphism that fixes both fundamental groups?

In the case when the the automorphism is a composition of disjoint Dehn twists, a positive answer to the first question follows from a result proved by Oertel [2] and and generalized by McCullough [3]. Oertel’s theorem states that if a composition of Dehn twists along disjoint loops extends into a handlebody then these loops cobound properly embedded disks and annuli in the handlebody. If an automorphism of a Heegaard surface is a composition of Dehn twists along disjoint loops then the disks and annuli in the two handlebodies form spheres and tori in the ambient manifold. If the automorphism fixes the fundamental groups of both handlebodies then these surfaces must in fact be reducing spheres.

Note also that if one can find a sphere that intersects one handlebody in a pair of non-parallel essential disks and the other handlebody in an annulus then Dehn twisting the Heegaard surface along the loops on intersection induces an automorphism that fixes the fundamental group of one handlebody but not the other. It should be possible to construct an irreducible (but weakly reducible) Heegaard splitting that has such a sphere.

[1] Actions of the homeotopy group of an orientable $3$-dimensional handlebody. Math. Ann. 234 (1978), no. 3, 279–292.
[2] U. Oertel. Automorphisms of 3-dimensional handlebodies. Topology, 41:363–410, 2002.
[3] Follow link to arXiv.

Casson’s invariant and the Torelli group

Filed under: 3-manifolds,Heegaard splittings,Mapping class groups — Jesse Johnson @ 5:40 pm

(Reposted from my old ldt blog)

Here’s an interesting paper, which appeared on the ArXiv a couple of weeks ago: “The Casson invariant and the word metric on the Torelli group” by Broaddus, Farb and Putman [1]. This paper continues a line of inquiry started by Morita [2] in 1991. Given a Heegaard splitting for a homology 3-sphere, we can think of the manifold as being the result of cutting the 3-sphere open along a Heegaard surface of the same genus, then re-gluing along a new map. Morita showed that you can get any Heegaard splitting for any homology sphere by composing the original gluing map for the 3-sphere with an element of the Torelli group. (A surface homomorphism is in the Torelli group if the induced automorphism of the first homology group of the surface is the identity.) Morita then showed that one can compute the Casson invariant from this regluing map.

Broaddus, Farb and Putman have shown that for any fixed generating set for the Torelli group, the Casson invariant of a manifold determined by a given re-gluing map is bounded above by a constant times the square of the distance from the origin in the word metric. (The word metric for a group is the path metric for a Cayley graph defined by the generating set.) They also demonstrate that there is sequence of elements of the Torelli group determining manifolds whose Casson invariants grow asymptotically quadratically with the distance from the origin. Thus a quadratic bound is the best that one can hope for.

Consider the map from the Torelli group to the (non-negative) integers that sends each element to the Casson invariant of the homology 3-sphere it defines. Broaddus, Farb and Putman’s result implies that as you follow a path through a Cayley graph for the Torelli group, the numbers at each element grow at most quadratically. However, there are many paths for which the numbers will not grow at all. For example, if the re-gluing map extends to an automorphism of either handlebody then the resulting manifold is the 3-sphere, so the Casson invariant is zero. There are also many homology spheres for which the Casson invariant is zero (see for example [3]). It would be interesting to classify the set of all automorphism that map to zero. This along with a classification of the Torelli automorphisms that extend into a given handlebody (I don’t know if this is known.) might lead to a nice characterization of the set of homology 3-spheres with trivial Casson invariants.

[1] and [3], follow links
[2] S. Morita, On the structure of the Torelli group and the Casson invariant, Topology 30 (1991), no. 4, 603–621.

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