Grid diagrams and Heegaard Floer invariants
We give combinatorial descriptions of the Heegaard Floer homology groups for arbitrary three-manifolds (with coefficients in ). The descriptions are based on presenting the three-manifold as an integer surgery on a link in the three-sphere, and then using a grid diagram for the link. We also give combinatorial descriptions of the mod 2 Ozsváth-Szabó mixed invariants of closed four-manifolds, also in terms of grid diagrams.
Starting with the seminal work of Donaldson [Donaldson], gauge theory has found numerous applications in low-dimensional topology. Its role is most important in dimension four, where the Donaldson invariants [DonaldsonPolynomials], and later the Seiberg-Witten invariants [SW1, SW2, Witten], were used to distinguish between homeomorphic four-manifolds that are not diffeomorphic. More recently, Ozsváth and Szabó introduced Heegaard Floer theory [HolDisk, HolDiskTwo], an invariant for low-dimensional manifolds inspired by gauge theory, but defined using methods from symplectic geometry. Heegaard Floer invariants in dimension three are known to detect Thurston norm [GenusBounds] and fiberedness [Ni3]. Heegaard Floer homology can be used to construct various four-dimensional invariants as well [HolDiskFour, Zemke]. Notable are the so-called mixed invariants, which are conjecturally the same as (the mod two reduction of) the Seiberg-Witten invariants and, further, are known to share many of their properties: in particular, they are able to distinguish homeomorphic four-manifolds with different smooth structures. In a different direction, there also exist Heegaard Floer invariants for null-homologous knots and links in three-manifolds (see [Knots, RasmussenThesis, Links]); these have applications to knot theory.
One feature shared by the Donaldson, Seiberg-Witten, and Heegaard Floer invariants is that their original definitions are based on counting solutions to some nonlinear partial differential equations. This makes it difficult to exhibit algorithms which, given a combinatorial presentation of the topological object (for example, a triangulation of the manifold, or a diagram of the knot), calculate the respective invariant. The first such general algorithms appeared in 2006, in the setting of Heegaard Floer theory. Sarkar and Wang [SarkarWang] gave an algorithm for calculating the hat version of Heegaard Floer homology of three-manifolds. The corresponding maps induced by simply connected cobordisms were calculated in [LMW]. In a different direction, all versions of the Heegaard Floer homology for links in were found to be algorithmically computable using grid diagrams, see [MOS]. See also [CubeResolutions, OSS, LOThf] for other developments. This progress notwithstanding, a combinatorial description of the smooth, closed four-manifold invariants remained elusive.
Our aim in this paper is to present combinatorial descriptions of all (completed) versions of Heegaard Floer homology for three-manifolds, as well as of the mixed invariants of closed four-manifolds. It should be noted that we only work with invariants defined over the field . However, we expect a sign refinement of our descriptions to be possible.
Our strategy is to use (toroidal) grid diagrams to represent links, and to represent three- and four-manifolds in terms of surgeries on those links. Grid diagrams were previously used in [MOS] to give a combinatorial description of link Floer homology. From here one automatically obtains a combinatorial description of the Heegaard Floer homology of Dehn surgeries on knots in , since these are known to be determined by the knot Floer complex, cf. [IntSurg, RatSurg]. Since every three-manifold can be expressed as surgery on a link in , it is natural to pursue a similar approach for links.
In [LinkSurg], the Heegaard Floer homology of an integral surgery on a link is described in terms of some data associated to Heegaard diagrams for the link and its sublinks. In particular, we can consider this description in the case where the Heegaard diagrams come from a grid diagram for the link. For technical reasons, we need to consider slightly different grid diagrams than the ones used in [MOS] and [MOST]. Whereas the grids in [MOS] and [MOST] had one marking and one marking in each row and column, here we will use grids with at least one extra free marking, that is, a marking such that there are no other markings in the same row or column. (An example of a grid diagram with four free markings is shown on the right hand side of Figure 1.)
In the setting of grids with at least one free marking, the problem of computing the Heegaard Floer homology of surgeries boils down to counting isolated pseudo-holomorphic polygons in the symmetric product of the grid torus. Pseudo-holomorphic bigons of index one are easy to count, as they correspond bijectively to empty rectangles on the grid, cf. [MOS]. In general, one needs to count -gons of index that relate the Floer complex of the grid to those of its destabilizations at points, where .
Just as in the case of bigons, one can associate to each polygon a certain object on the grid, which we call an enhanced domain. Roughly, an enhanced domain consists of an ordinary domain on the grid plus some numerical data at each destabilization point. (See Definition 3.1 below.) Further, when the enhanced domain is associated to a pseudo-holomorphic polygon, it satisfies certain positivity conditions. (See Definition 3.3 below.) Thus, the problem of counting pseudo-holomorphic polygons reduces to finding the positive enhanced domains of the appropriate index, and counting the number of their pseudo-holomorphic representatives.
When , this problem is almost as simple as in the case . Indeed, the only positive enhanced domains for are the “snail-like” ones used to construct the destabilization map in [MOST, Section 3.2], see Figure 6 below. It is not hard to check that each such domain has exactly one pseudo-holomorphic representative, modulo .
The key fact that underlies the calculation is that in that case there exist no positive enhanced domains of index corresponding to destabilization at points. Hence, the positive domains of index are what is called indecomposable: in particular, their counts of pseudo-holomorphic representatives depend only on the topology of the enhanced domain (i.e., they are independent of its conformal structure). Unfortunately, this fails to be true for : there exist positive enhanced domains of index , so the counts in index depend on the almost complex structure on the symmetric product of the grid.
Nevertheless, we know that different almost complex structures give rise to chain homotopy equivalent Floer complexes for the surgeries on the link, though they may give different counts for particular domains. Let us think of an almost complex structure as a way of assigning a count (zero or one, modulo ) to each positive enhanced domain on the grid, in a compatible way. Of course, there may exist other such assignments: we refer to an assignment that satisfies certain compatibility conditions as a formal complex structure. (See Definition 4.5 below.) Each formal complex structure produces a model for the Floer complex of the link surgery. If two formal complex structures are homotopic (in a suitable sense, see Definition 4.7), the resulting Floer complexes are chain homotopy equivalent. Suppose now we could show that all formal complex structures on a grid are homotopic. Then, we would easily arrive at a combinatorial formulation for the Floer homology of surgeries. Indeed, one could pick an arbitrary formal complex structure on the grid, which is a combinatorial object; and the homology of the resulting complex would be the right answer, whether or not the formal structure came from an actual almost complex structure on the symmetric product.
The question of whether or not any two formal complex structures on a grid are homotopic basically reduces to whether or not a certain cohomology group vanishes in degrees . Here, is the cohomology of a complex generated by positive enhanced domains, modulo periodic domains. (See Definition 4.5.) Computer experimentation suggests the following:
Let be a toroidal grid diagram (with at least one free marking) representing a link . Then for any .
We can prove only a weaker version of this conjecture, but one that is sufficient for our purposes. This version applies only to grid diagrams that are sparse, in the sense that if a row contains an marking, then the adjacent rows do not contain markings (i.e., they each contain a free marking); also if a column contains an marking, then the adjacent columms do not contain markings. One way to construct a sparse diagram is to start with any grid, and then double its size by interspersing free markings along a diagonal. The result is called the sparse double of the original grid: see Figure 1.
We can show that if is a sparse grid diagram, then for . As a consequence, we obtain:
Let be an oriented link with a framing . Let be the manifold obtained by surgery on along , and let be a structure on . Then, given as input a sparse grid diagram for , the Heegaard Floer homology groups , , with coefficients in are algorithmically computable.
The algorithm alluded to in the statement of Theorem 1.2 is given in the proof. In the above statement, and are the completed versions of the usual Heegaard Floer homology groups and , respectively. The groups and are constructed from complexes defined over the power series rings and , respectively, cf. [LinkSurg, Section 2], compare also [KMBook]. More precisely, the respective chain complexes are
One could also define a completed version of in a similar way; but in that case, since multiplication by is nilpotent on , the completed version is the same as the original one.
The three completed variants of Heegaard Floer homology are related by a long exact sequence:
Moving to four dimensions, in [LinkSurg] it is shown that every closed four-manifold with admits a description as a three-colored link with a framing. This description is called a link presentation, see Definition 2.8 below.
Let be a closed four-manifold with , and a structure on . Given as input a cut link presentation for and a sparse grid diagram for , one can algorithmically compute the mixed invariant with coefficients in .
Although Theorems 1.2 and 1.3 give combinatorial procedures for calculating the respective invariants, in practice our algorithms are very far from being effective. This is especially true since we need to double the size of an ordinary grid in order to arrive at a sparse one. However, one could just assume the truth of Conjecture 1.1, and try to do calculations using any grid with at least one free marking. This is still not effective, and it remains an interesting challenge to modify the algorithm in such a way as to make it more suitable for actual calculations. We remark that, in the case of the hat version of knot Floer homology, the algorithm from [MOS] has been implemented on computers, see [BaldwinGillam, Beliakova, Droz]; at present, one can calculate these groups for links with grid number up to .
Another application of the methods in this paper is to describe combinatorially the pages of the link surgeries spectral sequence from [BrDCov, Theorem 4.1]; see Section 2.14 and Theorem 4.10 below for the relevant discussion. Recall that this spectral sequence can be used to give a relationship between Khovanov homology to the Heegaard Floer homology of branched double covers. In the setting of , the spectral sequence can also be described combinatorially using bordered Floer homology; see [LOTbrcov], [LOTbrcov2].
This paper is organized as follows. In Section 2 we state the results from [LinkSurg] about how of an integral surgery on a link (and all the related invariants) can be expressed in terms of counts of holomorphic polygon counts on the symmetric product of a grid (with at least one free marking). In Section 3 we define enhanced domains, and associate one to each homotopy class of polygons on the symmetric product. We also introduce the positivity condition on enhanced domains, which is necessary in order for the domain to have holomorphic representatives. In Section 4 we associate to a grid the complex whose generators are positive enhanced domains modulo periodic domains. We show that, if Conjecture 1.1 is true, then all possible ways of assigning holomorphic polygon counts to enhanced domains are homotopic. We explain how this would lead to a combinatorial description of the Heegaard Floer invariants. In Section 5 we prove the weaker (but sufficient for our purposes) version of the conjecture, which is applicable to sparse grid diagrams.
Throughout the paper we work over the field .
Acknowledgement. The first author would like to thank the mathematics department at the University of Cambridge for its hospitality during his stay there in the Spring of 2009. We would also like to thank Robert Lipshitz and Zoltán Szabó for interesting conversations.
2. The surgery theorem applied to grid diagrams
Our main goal here is to present the statement of Theorem 2.5 below, which expresses the Heegaard Floer homology of an integral surgery on a link in terms of a grid diagram for the link (or, more precisely, in terms of holomorphic polygon counts on a symmetric product of the grid). We also state similar results for the cobordism maps on induced by two-handle additions (Theorem 2.6), for the other completed versions of Heegaard Floer homology (Theorem 2.7), and for the mixed invariants of four-manifolds (Proposition 2.10).
The proofs of all the results from this section are given in [LinkSurg].
2.1. Hyperboxes of chain complexes
We start by summarizing some homological algebra from [LinkSurg, Section 5].
When is a function, we denote its iterate by , i.e., .
For a collection of nonnegative integers, we set
In particular, is the set of vertices of the -dimensional unit hypercube.
For , set
We can view the elements of as vectors in . There is a partial ordering on , given by . We write if and . We say that two multi-indices with are neighbors if , i.e., none of their coordinates differ by more than one.
We define an -dimensional hyperbox of chain complexes of size to consist of a collection of -graded vector spaces
together with a collection of linear maps
defined for all and such that (i.e., the multi-indices of the domain and the target are neighbors). The maps are required to satisfy the relations
for all . If , we say that is a hypercube of chain complexes.
Note that in principle also depends on , but we omit that from notation for simplicity. Further, if we consider the total complex
we can think of as a map from to itself, by extending it to be zero when is not defined. Observe that is a chain map.
Let be an -dimensional hyperbox of chain complexes. Let us imagine the hyperbox to be split into unit hypercubes. At each vertex we see a chain complex . Associated to each edge of one of the unit hypercubes is a chain map. Along the two-dimensional faces we have chain homotopies between the two possible ways of composing the edge maps, and along higher-dimensional faces we have higher homotopies.
There is a natural way of turning the hyperbox into an -dimensional hypercube , which we called the compressed hypercube of . The compressed hypercube has the property that along its edge we see the composition of all the edge maps on the axis of the hyperbox.
In particular, if , then is a string of chain complexes and chain maps:
and the compressed hypercube is
For general and , the compressed hypercube has at its vertices the same complexes as those at the vertices of the original hyperbox :
If along the coordinate axis in we have the edge maps , then along the respective axis in we see . Given a two-dimensional face of with edge maps and and chain homotopies between and , to the respective compressed face in we assign the map
which is a chain homotopy between and . The formulas for what we assign to the higher-dimensional faces in are more complicated, but they always involve sums of compositions of maps in .
Let and be two hyperboxes of chain complexes, of the same size . A chain map is defined to be a collection of linear maps
for all such that . In particular, gives an ordinary chain map between the total complexes and .
Starting with from here, we can define chain homotopies and chain homotopy equivalences between hyperboxes by analogy with the usual notions between chain complexes.
The construction of from is natural in the following sense. Given a chain map , there is an induced chain map between the respective compressed hypercubes. Moreover, if is a chain homotopy equivalence, then so is .
2.2. Chain complexes from grid diagrams
Consider an oriented, -component link . We denote the components of by . Let
where denotes linking number. Further, let
Let be a toroidal grid diagram representing the link and having at least one free marking, as in [LinkSurg, Section 15.1]. Precisely, consists of a torus , viewed as a square in the plane with the opposite sides identified, and split into annuli (called rows) by horizontal circles , and into other annuli (called columns) by vertical circles . Further, we are given some markings on the torus, of two types: and , such that:
each row and each column contains exactly one marking;
each row and each column contains at most one marking;
if the row of an marking contains no markings, then the column of that marking contains no markings either. An marking of this kind is called a free marking. We assume that the number of free markings is at least .
Observe that contains exactly markings and exactly markings. A marking that is not free is called linked. The number is called the grid number or the size of .
We draw horizontal arcs between the linked markings in the same row (oriented to go from the to the ), and vertical arcs between the linked markings in the same column (oriented to go from the to the ). Letting the vertical arcs be overpasses whenever they intersect the horizontal arcs, we obtain a planar diagram for a link in , which we ask to be the given link .
We let be the set of matchings between the horizontal and vertical circles in . Any admits a Maslov grading and an Alexander multi-grading
(For the precise formulas for and , see [MOST], where they were written in the context of grid diagrams without free markings. However, the same formulas also apply to the present setting.)
For , we let be the space of empty rectangles between and , as in [MOST]. Specifically, a rectangle from to is an embedded rectangle in the torus, whose lower left and upper right corners are coordinates of , and whose lower right and upper left corners are coordinates of ; and moreover, if all other coordinates of coincide with all other coordinates of . We say that the rectangle is empty if its interior contains none of the coordinates of (or ).
For , we denote by and the number of times (resp. ) appears in the interior of . Let and be the set of ’s (resp. ’s) belonging to the component of the link. The coordinate of the Alexander multi-grading is characterized uniquely up to an overall additive constant by the property that
where here is any rectangle from to .
The Floer complex associated to the grid is the free module over generated by the elements of , and endowed with the differential:
The Maslov grading gives the homological grading on , such that each variable decreases it by . The Alexander functions give filtrations on , such that decreases the filtration level by one, and preserves all filtrations for .
Note that we can view as a Lagrangian Floer chain complex in a symplectic manifold, the symmetric product . The two Lagrangian submanifolds are the tori , the product of horizontal circles, and , the product of vertical circles. Empty rectangles are the same as holomorphic strips of index one in , with boundaries on and ; compare [MOS].
Given , we denote by
the subcomplex given by , .
When is a knot, the complex is a multi-basepoint, completed version of the subcomplex of the knot Floer complex , in the notation of [Knots]. A similar complex appeared in the integer surgery formula in [IntSurg], stated there in terms of rather than .
Next, we construct a resolution of , following [LinkSurg, Section 13]. First, we introduce some more notation. For each , we re-label the elements of and as
in the cyclic order in which they appear on . We denote by the multiplicities of a rectangle at the given marking, and we will sometimes use the alternate name for the variable associated to . Further, we arrange so that the free markings are labeled .
Consider the differential graded algebra
Here, the prime signifies that the elements of the algebra are infinite sums of monomials in the , and variables, with the property that, for each ,
where denotes the exponent of a variable in the monomial .
We define an intermediate complex , which is the dg module over generated by , with differential
This complex admits filtrations , for , such that
and takes all the other , and variables to zero (i.e, the action of those variables preserves ).
For , we define to be the filtered part of given by , The complex is a resolution of , via the -linear projection
Finally, we mention the curved link Floer complex that appeared in [ZemkeHFL]. Recall that a curved complex (or a matrix factorization) over a ring is a -module with a endomorphism such that for some in the base ring . In our case, we take
We define the curved link Floer complex to be the -module freely generated by , with the endomorphism
See for example [ZemkeQuasi, Lemma 2.1]; compare also [FOOO1], [KR1]. The contributions to come from the index two periodic domains, that is, the rows and columns of the grid.
There is a homological (Maslov) grading on , just as in the case of , and preserved by the action of the variables.
2.3. Summary of the construction
For the benefit of the reader, before moving further we include here a short summary of Sections 2.4–2.10 below. The aim of these sections is to be able to state the Surgery Theorem 2.5, which expresses of an integral surgery on a link (with framing ) as the homology of a certain chain complex associated to a grid diagram for (with at least one free marking). Given a sublink , we let be the grid diagram for obtained from by deleting all the rows and columns that support components of . Roughly, the complex is built as an -dimensional hypercube of complexes. Each vertex corresponds to a sublink , and the chain complex at that vertex is the direct product of generalized Floer complexes , over all possible values ; the reader is encouraged to peek ahead at the expression (29).
The differential on is a sum of some maps denoted , which are associated to oriented sublinks : given a sublink , we need to consider all its possible orientations, not just the one induced from the orientation of . When has only one component, the maps are chain maps going from a generalized Floer complex associated to (for containing ) to one associated to . When has two components, are chain homotopies between different ways of composing the respective chain maps removing one component at a time; for more components of we get higher homotopies. The maps fit together in a hypercube of chain complexes, as in Section 2.1.
Each map will be a composition of three kinds of maps, cf. Equation (28) below: a “projection-inclusion map” , a “descent map” , and an isomorphism . (Here, refers to a natural projection from the set to itself, and to a projection from to ; see Section 2.4.) The maps are defined in Section 2.5, and go between different generalized Floer complexes associated to the same grid. The descent maps are defined in Section 2.8, and go from a generalized Floer complex for a grid to one associated to another diagram, which is obtained from the grid by handlesliding some beta curves over others. Finally, the isomorphisms relate these latter complexes to generalized Floer complexes for the smaller grid ; these isomorphisms are defined in Section 2.9.
The main difficulty lies in correctly defining the descent maps. When the link has one component, they are constructed by composing some transition maps (involving just counts of empty rectangles on ) with some maps that count pseudo-holomorphic triangles. For example, there is a triangle map that corresponds to a single handleslide, done so that the new beta curve encircles a marking . The type ( or ) of the marking is determined by the chosen orientation for . More generally, by counting pseudo-holomorphic polygons with more sides, we define maps corresponding to a whole set of markings; see Section 2.7 below. When has two markings, the respective maps count quadrilaterals, and are chain homotopies between triangle maps. In general, the polygon maps fit into a hypercube of chain complexes. To construct the descent maps for a sublink (which we do in Section 2.8), we build a hyperbox out of both transition maps and polygon maps (as well as hybrids of these), and apply the compression procedure mentioned in Section 2.1.
2.4. Sublinks and reduction
Suppose that is a sublink. We choose an orientation on (possibly different from the one induced from ), and denote the corresponding oriented link by . We let (resp. ) to be the set of indices such that the component is in and its orientation induced from is the same as (resp. opposite to) the one induced from . We let
so that is the disjoint union of and . We also denote by
the set of all indices with .
For , we define a map by
Then, for , we set
Let be the complement of the sublink in . We define the reduction of the grid diagram with respect to the sublink to be obtained from by deleting the basepoints in for , deleting the basepoints in for , and viewing the basepoints in as free basepoints, for . Note that is a grid diagram for the link .
Thus, if we let be the subset consisting of the basepoints on , this is exactly the set of remaining basepoints on . The set of basepoints on is
We define a reduction map
as follows. The map depends only on the summands of corresponding to . Each of these ’s appears in with a (possibly different) index , so there is a corresponding summand of . We then set
where is considered with the orientation induced from , while is with its own orientation. We then define to be the direct sum of the maps , pre-composed with projection to the relevant factors. Note that .
The Alexander filtrations behave nicely with respect to reduction maps. If a generator has filtration level
then the same generator has multi-grading when viewed as a generator for the reduced grid . Hence, for example, we have an identification of generalized Floer complexes
2.5. Projection-inclusion maps
Let be an oriented sublink of , as in the previous subsection. We refine the construction of the resolution from Section 2.2 by introducing a relative version, denoted . This version looks like
the Floer complex with respect to , and
the intermediate complex with respect to , and
the complex with respect to .
Precisely, we first consider , the complex similar to that denoted in Section 2.2, but using the diagram . Thus, compared to , in the definition of we do not introduce variables and for ; instead, for such , we use as exponents for in the formula for the differential. The complex has filtrations for , where we use the convention that the Alexander filtration levels of are the ones from the diagram (rather than those from , where becomes ).
We let be the subcomplex of in filtration degrees
We now define a projection-inclusion map
This will be the first ingredient in the construction of the maps advertised in Section 2.3. The map is constructed as the composition of a projection similar to from (4), but taken only with respect to the indices , and the natural inclusion into . Specifically, for , we set
We then extend to be equivariant with respect to the action of the variables and for , as well as to that of all the variables.
2.6. Handleslides over a set of markings
Consider a subset consisting only of linked markings. We say that is consistent if, for any , at most one of the sets and is nonempty. From now on we shall assume that is consistent.
We let be the sublink consisting of those components such that at least one of the markings on is in . We orient as , such that a component is given the orientation coming from when , and is given the opposite orientation when .
Let us define a new set of curves on the torus . Let be the index corresponding to the vertical circle just to the left of a marking . We let be a circle encircling and intersecting , as well as the curve just below , in two points each; in other words, is obtained from by handlesliding it over the vertical curve just to the right of . For those that are not for any , we let be a curve isotopic to and intersecting it in two points.
Our assumption on the existence of a free marking ensures that contains at least one vertical beta curve.
For any consistent collection , we denote
is a multi-pointed Heegaard diagram representing the link . More generally, let be any sublink of containing , such that the restriction to of the orientation on coincides with . We can then consider a diagram111The notation is specific to this paper. In [LinkSurg, Section 15.3], this diagram was part of a “hyperbox of Heegaard diagrams,” denoted .