Low Dimensional Topology

September 2, 2010

Combinatorial Heegaard Floer Homology, Part II

Filed under: 3-manifolds,Heegaard splittings,Quantum topology — Jesse Johnson @ 12:39 pm

In my previous post about combinatorial Heegaard floer homology, I described the basic structure for how we will define the Homology groups \widehat{HF}.  We need an Abelian group C with a relative grading and a boundary map \partial that sends each graded subgroup into the subgroup one index lower, and whose square is zero.  The homology group is then defined as the kernel of \partial quotiented by the image of \partial. Last time I described the group C: Given a Heegaard diagram for a given 3-manifold, consisting of loops \{\alpha_i\}, \{\beta_j\}, then C is a direct product of copies of \mathbf{Z}_2, with one copy for each pairing of the curves.  (A pairing is a collection of points such that each \alpha_i contains exactly one of the points and each \beta_j contains exactly one point.)  In this post, I will describe the grading on this group and the boundary map \partial.

In fact, there are two relative gradings that we will need to define.  Given two pairings p_1, p_2, the first relative grading associates to (p_1, p_2) an element of the homology of the ambient 3-manifold M.  The second associates to this pair an integer.  In both cases, switching the order of the pairing will give us the negative of the original.

To get an element of the first homology of M from two pairings, we use the following construction:  Each \alpha_i contains a point in p_1 and a point in p_2, which cut \alpha_i into two arcs.  Choose one of these arcs in each \alpha_i, and a similar arc in each \beta_j.  The union of all these arcs is a collection of simple closed curves in the Heegaard surface.   Further, we can orient these curves by having each arc in each \alpha_i point towards the point in p_1 and having each arc in a \beta_j point towards the point in p_2.  (Or the other way around, as long as we choose a convention in advance.)  Since the Heegaard surface is embedded in M, this collection of oriented loops in the Heegaard surface gives us a collection of oriented loops in M, which defines an element of the first homology.  Note that if we had chosen a different arc in one of the \alpha_is, the new collection of loops would result from pushing an arc across a disk, and would thus define the same homology class.  So this homology class is well defined, independent of our choices at the beginning.

This gives us our first relative grading, which splits the generators of C into equivalence classes, where two pairings are equivalent if they define the trivial homology class.  We will define our second grading on each of these equivalence classes.  But first we need a few more definitions.

Recall that the union of the Heegaard diagram curves \{\alpha_i\}, \{\beta_i\} cut the Heegaaard diagram into a collection of regions, one of which contains a marked point.  A positive domain is a non-negative  integer-weighted sum of the complementary regions that don’t contain the marked point.  In other words, we write a non-negative integer in each complementary region, making sure to write a zero in the marked region.  Below, I’ll drop the adjective positive, and just say domain to mean a positive domain.

A region will define a (possibly negative) integer weighted sum of points in \cup (\alpha_i \cap \beta_j) as follows:  Choose an orientation on the Heegaard surface \Sigma.  The boundary a region R, is a collection of subarcs of the \alpha_i and \beta_j curves, and an orientation is induced on each of these arcs from the orientation on R that comes from \Sigma.  Define \partial(\partial(R)) to be the positive positive endpoints of the \alpha_i arcs minus the negative endpoints of the \alpha_i arcs.  (Notice that in addition to choosing an orientation on \Sigma we have also chosen the \alpha_i curves over the \beta_j curves.) Given a domain D = \sum w_i R_i, we define \partial(\partial(D)) = \sum w_i \partial(\partial(R_i)).

So \partial(\partial(D)) is a weighted sum of points in the intersection between the two sets of Heegaard diagram curves.  A generator x for C is also a sum of such points of intersection.  If we can define a domain D such that \partial(\partial(D)) = x - y (where x and y are generators for C) then we will define the Maslov index of x,y by a formula in terms of the domain D.  You can read about this formula in Lipshitz’s paper [1].  Even though it’s a very simple formula, I’m going to skip the details right now for reasons that I’ll explain below.

We want the Maslov index to be defined for any pair of generators whose homology grading is trivial.  Given such a pair x, y, the collection of loops in \Sigma they define is homology trivial in M, so they bound an immersed, orientable surface F.  Because the union of \Sigma and the disks bounded by the curves \{\alpha_i\}, \{\beta_j\} cut M into two balls,  the surface F can be homotoped into the union of \Sigma and these disks.  Moreover, we can do this in a way that avoids the marked region in \Sigma.  Then the intersection of F with \Sigma will define a domain D in \Sigma, and it is straightforward to check that \partial (\partial(D)) = x - y.

Now that we have the Maslov index, we want to define a boundary map that sends each generator x into the subspace of C spanned by the generators with Maslov index one less than x.  To do this, one defines a function called the count that assigns an integer to each domain D.  Then we define \partial(x) to be the sum of all the generators y such that there is a domain D with Malov index and count equal to one, and with \partial(\partial(D)) = x - y.

So to understand the boundary map, we only need to know which domains have Maslov index and count equal to one. This is where Sarkar-Wang and the nice Heegaard diagrams come in.  Sarkar and Wang [2] showed that if one has a nice Heegaard diagram (which I defined at the end of my last post) then a domain will have Maslov index and count equal to one if and only if each region in the domain has weight one or zero and the union of the regions with weight one forms a bigon or quadrilateral with no points of x or y in their interior.  In the case of a bigon, y will be related to x by replacing one corner of the bigon with the other corner.  For a quadrilateral, x will contain two diagonal corners, and y will result from replacing those two corners with the other two diagonal corners.  In both cases, all the remaining points of x and y will coincide.

Once all the dust has settled, there is a beautifully simple algorithm for calculating the homology:  Given a Heegaard diagram, one can enumerate all the pairings.  Deciding if two pairings define the trivial homology group of the manifold can be done via the Heegaard diagram. Then one only needs to enumerate the quadrilateral and bigon domains, in order to define the boundary map. It’s not necessarily something I would want to do by hand, but it is a whole lot simpler than trying to classify holomorphic disks.

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