(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  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 . 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.
 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.
 Follow link to arXiv.