Involutive Heegaard Floer homology and plumbed three-manifolds
We compute the involutive Heegaard Floer homology of the family of three-manifolds obtained by plumbings along almost-rational graphs. (This includes all Seifert fibered homology spheres.) We also study the involutive Heegaard Floer homology of connected sums of such three-manifolds, and explicitly determine the involutive correction terms in the case that all of the summands have the same orientation. Using these calculations, we give a new proof of the existence of an infinite-rank subgroup in the three-dimensional homology cobordism group.
In [HMinvolutive], Hendricks and the second author defined an invariant of three-manifolds called involutive Heegaard Floer homology. This is a variation of the well-known Heegaard Floer homology of Ozsváth and Szabó [HolDisk, HolDiskTwo], taking into account the conjugation action on the Heegaard Floer complex. Specifically, let denote any of the four flavors of the Heegaard Floer complex (with , , , or ), for a three-manifold with a self-conjugate structure . The Heegaard Floer complex is constructed starting from a Heegaard diagram for . The conjugation is induced by reversing the orientation of the Heegaard surface and swapping the alpha and beta curves, and observing that the result is also a Heegaard diagram for , which can be related to the original one by a sequence of Heegaard moves. The corresponding involutive Heegaard Floer homology is the homology of the mapping cone
This is a module over the ring , where denotes the field of two elements.
The construction of involutive Heegaard Floer homology was inspired by developments in gauge theory [Triangulations, Lin, Stoffregen]. Indeed, in the case when is a rational homology sphere, is conjectured to be isomorphic to the -equivariant Seiberg-Witten Floer homology of the pair . Just as Heegaard Floer homology is more amenable to computations than Seiberg-Witten theory, involutive Heegaard Floer homology should be easier to calculate than its gauge-theoretic counterpart. So far, was computed in [HMinvolutive] for large surgeries on -space and thin knots (including many hyperbolic examples); see also [BorodzikHom] for related applications. Moreover, a formula for of connected sums was established in [HMZ].
The purpose of this paper is to compute for the class of rational homology spheres introduced by Némethi in [NemethiOS], namely those associated to plumbings along AR (almost-rational) graphs. For the sake of the exposition, we will focus on the minus version, .
If is a weighted graph, let us denote by the boundary of the corresponding plumbing of two-spheres. In [Plumbed], Ozsváth and Szabó calculated the Heegaard Floer homology of when is a negative definite graph with at most one bad vertex. (In particular, if a rational homology sphere is Seifert fibered over a base orbifold with underlying space , then can be expressed as for such a graph .) In [NemethiOS], Némethi defined AR graphs as those obtained from a rational graph by increasing the decoration of at most one vertex. We will call the resulting manifolds AR plumbed three-manifolds, for simplicity. They include those considered by Ozsváth and Szabó in [Plumbed], as well as the links of rational and weakly elliptic singularities. Further, extending the work of [Plumbed], Némethi computed the Heegaard Floer homology of AR plumbed three-manifolds. Specifically, to each plumbing graph and characteristic element one can associate a decorated infinite tree , called a graded root. To one can associate, in a combinatorial fashion, a lattice homology group . Némethi’s result, phrased in terms of the minus rather than the plus flavor, is that
Here, denotes the stucture on associated to , and denotes a grading shift,111In this paper, by we will denote a grading shift by , so that an element in degree in a module becomes an element in degree in the shifted module This is the standard convention, but opposite to the one in [NemethiOS]. where .
To understand the involutive Heegaard Floer homology, we focus on self-conjugate . In that case, there is a natural involution on (and hence on its grading-shifted counterpart), coming from reflecting the graded root along its vertical axis.
Let be the plumbed three-manifold associated to an AR graph , and oriented as the boundary of the plumbing. Let be a self-conjugate structure on . Let also be the graded root associated to , and let be the reflection involution on the lattice homology .
Then, we have an isomorphism of graded -modules
Under this isomorphism, the action of on is given by the quotient map
The main sources of applications for involutive Heegaard Floer homology are the involutive correction terms and , the analogues of the Ozsváth-Szabó correction term defined in [AbsGraded]. In [HMinvolutive], it is shown that and can be used to constrain the intersection forms of spin four-manifolds with boundary, and the existence of homology cobordisms between three-manifolds.
For the class of plumbed three-manifolds studied in this paper, we have the following result.
Let and be as in Theorem 1.1. Then, the involutive Heegaard Floer correction terms are given by
where is the Ozsváth-Szabó -invariant and is the Neumann-Siebenmann invariant from [Neu], [Sieb].
Theorems 1.1 and 1.2 should be compared with the corresponding results for -monopole Floer homology, obtained by the first author in [Dai]. For the smaller class of rational homology spheres Seifert fibered over a base orbifold with underlying space , the -equivariant Seiberg-Witten Floer homology was computed by Stoffregen in [Stoffregen].
The proof of Theorem 1.1 is in two steps. First, we identify the action of on (or, more precisely, with the conjugation involution on , using an argument from [Dai]. Second, we prove that, in the case at hand, the action of on determines (up to chain homotopy) the underlying action at the chain level, on any free chain complex that computes . This allows us not only to compute the involutive Heegaard Floer homology, but also to characterize the pair up to the notion of equivalence considered in [HMZ, Definition 8.3].
The equivalence class of is the input needed for the connected sum formula in involutive Heegaard Floer homology proved in [HMZ]. Thus, we can use that formula to calculate for connected sums of AR plumbed three-manifolds. In this paper we will focus on explicitly calculating the involutive correction terms for such manifolds. To compute the correction terms we only need to understand the pair up to a weaker equivalence relation, called local equivalence; cf. [HMZ, Definition 8.5].
where is the Ozsváth-Szabó correction term and is the Neumann-Siebenmann invariant. The values of can be read from the graded root associated to . Specifically, these values describe a simplified graded root, called monotone, which is obtained from by deleting some branches according to a certain algorithm.
We refer to the integer above (the number of quantities ) as the local complexity of . In particular, having local complexity one is the same as the projective type condition introduced by Stoffregen in [Stoffregen, Section 5.2].
It will be convenient to also define
Let be AR plumbed three-manifolds, oriented as the boundaries of those plumbings, and equipped with self-conjugate structures . Suppose that has local complexity .
For each tuple of integers with , define
where the maximum is taken over all tuples with .
Part (b) of Theorem 1.3 has the following consequence.
Let be as in Theorem 1.3. Without loss of generality, assume that
Moreover, if are all of projective type, then we have equality in (2).
In [Stoffregen2, Theorem 1.4], Stoffregen calculated the homology cobordism invariants (coming from -equivariant Seiberg-Witten Floer homology, cf. [Triangulations]) in the case of connected sums of Seifert fibered integral homology spheres of projective type. Corollary 1.4 implies that or for those manifolds.
In a different direction, we can specialize Theorem 1.3 to the case of self-connected sums . In that case, the expression for simplifies significantly. Indeed, the following corollary shows that satisfies a curious “stabilization” property for self-connected sums of AR manifolds. More precisely, it turns out that for sufficiently large, is a linear function of :
Let be an AR plumbed three-manifold and let be a self-conjugate structure on . Then
Moreover, for all sufficiently large, this is equal to .
Corollary 1.5 should be compared with Theorem 1.3 of [Stoffregen2], which shows that for , , and , the difference (for example) is a bounded function of . Here, in the case of AR manifolds, we have the stronger result that for and , the corresponding expressions are eventually constant (namely, identically zero for , and for ).
We can also study for connected sums of AR plumbed three-manifolds, when some of these manifolds are equipped with the opposite orientation. Recall that in [HMZ] such a connected sum (of large surgeries on torus knots) was used to give an example of a -homology sphere with and all different. Using the methods of this paper, we can now find an integral homology sphere with the same property.
Consider the integral homology -sphere
Then, we have
Consequently, is not homology cobordant to any AR plumbed three-manifold.
Finally, we give a new proof of the following result:
Theorem 1.7 (cf. Fintushel-Stern [FSinstanton], Furuta [FurutaHom]).
The homology cobordism group is infinitely generated.
The original proofs of Theorem 1.7 used Yang-Mills theory. Recently, Stoffregen [Stoffregen2] gave another proof, which is based on -equivariant Seiberg-Witten Floer homology. Our proof is modelled on that of Stoffregen, but uses involutive Heegaard Floer homology—thus, replacing gauge theory with symplectic geometry. Specifically, we show that the Brieskorn spheres
are linearly independent in , forming a subgroup. (The same manifolds were considered by Stoffregen in [Stoffregen2].)
Organization of the paper. Section 2 reviews some background material on involutive Heegaard Floer homology, plumbings, graded roots, and lattice homology. In Section 3, we identify the action of the conjugation involution on the Heegaard Floer homology of AR plumbed three-manifolds. In Section 4 we describe a free complex that computes the lattice homology , which we call the standard complex associated to the graded root; moreover, we prove that a complex of this form (with its involution) is uniquely determined by up to equivalence. This key fact is used in Section 5 to prove Theorems 1.1 and 1.2. In Section 6 we describe the local equivalence class of a graded root, in terms of an associated monotone root. In Section 7 we study the tensor products of standard complexes associated to monotone graded roots. These results are then put to use in Section 8, where we prove Theorems 1.3, 1.6 and 1.7, as well as Corollaries 1.4 and 1.5.
Acknowledgements. We thank Tye Lidman, Matthew Stoffregen, Zoltán Szabó and Ian Zemke for helpful conversations and suggestions.
2.1. Involutive Heegaard Floer homology
We assume that the reader is familiar with Heegaard Floer homology, as in [HolDisk, HolDiskTwo, HolDiskFour, AbsGraded]. Throughout the paper we will work with coefficients in . Furthermore, we will only consider rational homology spheres , equipped with self-conjugate structures . In this situation, the Heegaard Floer groups
have absolute -gradings, and are modules over the ring . We focus on the minus version, which is the homology of a complex , consisting of free -modules. The other three Floer complexes can be obtained from :
The construction of relies on fixing a particular Heegaard pair for , consisting of a pointed Heegaard diagram , with embedded in , together with a family of almost-complex structures on . We use the notation to emphasize the dependence of on . If and are two Heegaard pairs for with the same basepoint , then by work of Juhász and Thurston [Naturality], we obtain a preferred isomorphism
More precisely, choosing any sequence of moves relating and yields a chain homotopy equivalence
inducing the isomorphism above. The map is itself unique up to chain homotopy in the sense that choosing any other sequence of moves yields a map chain-homotopic to it; cf. [HMinvolutive, Proposition 2.3]. This justifies the use of the notation , rather than .
We now define a grading-preserving homotopy involution on . Given a Heegaard pair for , define the conjugate Heegaard pair to be the conjugate Heegaard diagram
together with the conjugate family of almost-complex structures on . Then intersection points in for are in one-to-one correspondence with those for , and -holomorphic disks with boundary on are in one-to-one correspondence with -holomorphic disks with boundary on . Thus, as observed in [HolDiskTwo, Theorem 2.4], we obtain a canonical isomorphism
where we have used the fact that in our case . Since and are Heegaard pairs for the same three-manifold, we may further define the composition
In [HMinvolutive, Section 2.2] it is shown that is a well-defined map on (up to the notion of equivalence considered below) and that is chain-homotopic to the identity.
The involutive Heegaard Floer complex is then in [HMinvolutive] defined to be the mapping cone of on , with a new variable that marks the target (as opposed to the domain) of the cone; cf. the formula (1). By taking homology we obtain the invariant . The other three flavors of involutive Heegaard Floer homology are constructed similarly.
We can formalize the algebra underlying as in [HMZ, Section 8]. The following is Definition 8.1 in [HMZ], slightly modified to allow for -gradings:
An -complex is a pair , consisting of
a -graded, finitely generated, free chain complex over the ring , where . Moreover, we ask that there is some such that the complex is supported in degrees differing from by integers. We also require that there is a relatively graded isomorphism
and that is supported in degrees differing from by even integers;
a grading-preserving chain homomorphism , such that is chain homotopic to the identity.
An example of an -complex is , equipped with the conjugation involution. Further, when is an integral homology sphere, this is an -complex where we can take .
Let us also recall Definition 8.3 in [HMZ]:
Two -complexes and are called equivalent if there exist chain homotopy equivalences
that are homotopy inverses to each other, and such that
where denotes -equivariant chain homotopy.
To each -complex , we can associate an involutive complex
For example, . We refer to the homology of the mapping cone as the involutive homology of and denote it by .
The proof of the following lemma is based on a simple filtration argument; compare [HMinvolutive, proof of Proposition 2.8].
An equivalence of -complexes induces a quasi-isomorphism between the respective involutive complexes (5).
In this paper, we will describe the equivalence class of the pair for certain three-manifolds. By the above lemma, the equivalence class determines the involutive Heegaard Floer homology , up to isomorphism. Furthermore, from we can derive the other Heegaard Floer complexes using (3), and the map descends to similar maps on those complexes. From here, we can get the corresponding flavors of involutive Heegaard Floer homology, again up to isomorphism.
The equivalence class of also appears in the following connected sum formula, which follows from [HMZ, Theorem 1.1].
Theorem 2.4 ([Hmz]).
Suppose and are rational homology spheres equipped with self-conjugate structures and . Let , and denote the conjugation involutions on the Floer complexes , and . Then, the equivalence class of the -complex is the same as that of
where denotes a grading shift.
The grading shift is due to the fact that, with the usual conventions in Heegaard Floer theory, is supported in degree .
We now review the definition of the correction terms. In [AbsGraded], Ozsváth and Szabó defined the -invariant as the minimal degree of a nonzero element in the infinite tower of ; or, equivalently, as the maximal degree of a nonzero element in the infinite tail of , plus two:
In involutive Heegaard Floer homology, there are two analogous invariants
See [HMinvolutive, Section 5.1] and [HMZ, Lemma 2.9].
(The shifts by and in these definitions are chosen so that for .)
The above invariants satisfy
Moreover, they descend to maps
where denotes the homology cobordism group (consisting of integral homology -spheres, up to the equivalence given by homology cobordisms). Of the three maps above, only is a homomorphism.
For the purpose of computing the involutive Floer correction terms, it will be useful for us to consider a weaker notion of equivalence than discussed above. Recall Definition 8.5 of [HMZ]:
Two -complexes and are called locally equivalent if there exist (grading-preserving) homomorphisms
and and induce isomorphisms on homology after inverting the action of .
Definition 2.6 is modelled on the relation between Floer chain complexes induced by a homology cobordism. As observed in [HMZ, Lemma 8.7], the relation of local equivalence respects taking tensor products, in the sense that if or is changed by a local equivalence, then their tensor product also changes by a local equivalence. One can thus define a group structure on the set of -complexes modulo local equivalence, with multiplication given by the tensor product; cf. [HMZ, Section 8.3]. In [HMZ] only -complexes with were considered, and the resulting group was denoted . One can also form such a group from all -complexes, which we denote by . Of course, is just the direct sum of infinitely many copies of , one for each .
Given an -complex , we define and analogously to the involutive Floer correction terms and above. The same argument that shows and are homology cobordism invariants (cf. [HMinvolutive, Proposition 5.4]) proves that these depend only on the local equivalence class of . Hence we have (non-homomorphisms)
If we have an actual Heegaard Floer complex , we can identify it with its equivalence class in , in which case and . Theorem 2.4 states that under this identification, the Floer -complex corresponding to the connected sum of and is mapped to the grading-shifted tensor product of their images in .
2.2. Plumbing graphs
We review here some facts about plumbings, with an emphasis on the almost-rational graphs introduced in [NemethiOS]. We refer to [Neu], [Plumbed] and [NemethiOS] for more details.
Let be a weighted graph, i.e., a graph equipped with a function , where denotes the set of vertices of . Let be the four-manifold with boundary obtained by attaching two-handles to (or, equivalently, plumbing together disk bundles over ) according to the graph . We denote by the boundary of , with the induced orientation.
Each vertex gives rise to a generator of represented by the core of the corresponding two-handle. The intersection form on the integer lattice is given by
From now on we will assume that the graph is a tree and the intersection form on is negative definite; this is equivalent to being a rational homology sphere that is realized as the link of a normal surface singularity.
For , we write if is a linear combination of the generators with non-negative coefficients. Further, we write if and .
The dual lattice may be identified with , and the exact sequence
shows that we can view as a sublattice of . The intersection form on extends to a -valued intersection form of .
Let be the set of characteristic vectors for , i.e., vectors such that
for all . There is a natural action of on given by , and we denote the orbit of any under this action by . Characteristic vectors may be identified with structures on , and equivalence classes may be identified with structures on . Self-conjugate structures on correspond to the orbits lying entirely in the sublattice , that is, those which satisfy .
Fix , and define by
There is a canonical characteristic element characterized by for all . If we view as the link of a normal surface singularity with resolution , then is the first Chern class of the canonical bundle of the complex structure on .
A normal surface singularity is called rational if its geometric genus is zero. By the work of Artin [Artin1, Artin2], the manifold is the boundary of a rational normal surface singularity if and only if
If satisfies this property, we will say that is a rational plumbing graph.
Definition 2.7 (Némethi [NemethiOS]).
Let be a weighted tree with a negative definite intersection form on , as above. We say that is AR (almost-rational) if there exists a vertex of such that by replacing the value with some we obtain a rational plumbing graph .
In this situation, we refer to as an AR plumbed three-manifold.
The class of AR graphs includes:
elliptic graphs (corresponding to normal surface singularities of geometric genus equal to one);
negative-definite, star-shaped graphs;
negative-definite plumbing trees with at most one bad vertex, that is, a vertex such that , where is the degree of . This is the class of plumbings considered by Ozsváth and Szabó in [Plumbed].
In particular, if is a rational homology sphere that is Seifert fibered over an orbifold with underlying space , then is an AR plumbed three-manifold coming from a star-shaped graph . Indeed, suppose that is the Seifert bundle with negative orbifold Euler number , and Seifert invariants
with , . Write as a continued fraction
Then, is the star-shaped graph with arms, such that the decoration of the central vertex is
and along the arm we see the decorations , in that order starting from the central vertex.
then is an integral homology sphere, denoted . The values of are uniquely determined by the values of , the fact that , and the condition (7). Moreover, any Seifert fibered integral homology sphere is of the form for some .
All Seifert fibered rational homology spheres have base orbifold with underlying space or . (Further, if they are integer homology spheres, this has to be .) When the underlying space is , then the Seifert fibered rational homology sphere is an L-space, by [BoyerGordonWatson, Proposition 18]. We do not study this case in our paper, since the involutive Heegaard Floer homology of L-spaces is easily determined by the ordinary Heegaard Floer homology; cf. [HMinvolutive, Corollary 4.8].
2.3. Graded roots and lattice homology
The Heegaard Floer homology of an AR plumbed three-manifold can be described in terms of a combinatorial object, called a graded root, which is associated to the weighted graph . Graded roots were introduced in [NemethiOS], where the focus was on describing the groups . In this paper we will concentrate on the minus version, . By the duality isomorphism from [HolDiskFour, Proposition 7.11], we have
Thus, we can consider the same graded roots as in [NemethiOS], except here we find it more convenient to reverse the grading, i.e., reflect them vertically. While the graded roots in [NemethiOS] have an infinite upward stem and finite roots below, ours will have an infinite downward stem and a finite tree opening upward.
Specifically, with our conventions, a graded root is an infinite tree equipped with a grading function such that
for any edge ,
for any edges and with ,
is bounded above,
is finite for any , and
Note that, since is connected, the difference in the values of at any two vertices is an integer.
To every graded root we can associate an -module , with one generator for each , and relations
We define the degree of to be , so that is a map of degree . It will usually be convenient for us to think of in terms of and view the vertices of being graded by their degree, rather than the function . Thus (for example), when we speak of a graded root shifted by , we mean that the grading is shifted by , and when we construct graded roots, we will sometimes use the terms “degree” and “grading” interchangeably to mean . This unfortunate overlap of terminology is due to the fact that , rather than , is more naturally related to the Maslov grading in Floer homology.
Let us also mention that by reflecting a graded root across the horizontal line of grading yields a downwards opening graded root , of the kind considered in [NemethiOS]. There is also an associated -module . Concretely, has one generator for each , in grading , and has relations
Now let be a plumbing graph as in Section 2.2, and pick . Following Némethi in [NemethiOS, Nem] (cf. also [Dai, Section 1.2]), we can associate to a graded root as follows. We consider the lattice , and the function from (6). Let us identify with using the basis coming from the vertices of . Let be the set of -dimensional side-length-one lattice cubes in . We define a weight function
by letting be the maximum of over all vertices of . Set
Note that there is an inclusion .
We let the vertices of in grading be the connected components of . Further, for and , we let be an edge of if and only if .
We define the lattice homology to be .
Building on the work of Ozsváth and Szabó [Plumbed], Némethi proved:
Theorem 2.9 (Némethi [NemethiOS]).
Let be an AR plumbing graph and be the corresponding three-manifold. For , let
where denotes the number of vertices of .
Then, the Heegaard Floer homology of is given by
In particular, the -invariants of are given by
Note that of an AR plumbed three-manifold is always supported in degrees congruent to modulo .
We will sometimes abuse notation slightly and denote the shifted root simply by when no confusion is possible. We refer to as the graded root associated to . Note that this depends only on , not on itself, as indeed we have
Since the highest degree of a nonzero element in is , the -invariant is the maximum degree of a vertex in , plus two.
With regard to the plus versions, in view of (8), we have isomorphisms
Consider the Brieskorn sphere , which is surgery on the torus knot . The manifold can be represented as the boundary of a plumbing on the AR graph
The graded root associated to was computed in [NemethiGRS, Tweedy], and is shown in Figure 1. The -invariant is zero.
We now introduce the following concept.
A symmetric graded root is a graded root together with an involution such that
for any vertex ,
is an edge in if and only if is an edge on ,
for every , there is at most one -invariant vertex with .
We can represent every symmetric graded root by a planar tree that is symmetric about the vertical axis (the infinite downwards stem). The involution is the reflection about this axis.
Note that, for a symmetric graded root, the involution induces an involution (still denoted ) on the corresponding lattice homologies , . The action of there commutes with that of the variable .
Symmetric graded roots appear naturally when we consider self-conjugate structures on an AR plumbed three-manifold . Let be such a structure. As before, let be the integer lattice spanned by the vertices of . There is an obvious involution on given by
where is viewed as lying in the lattice . Since is self-conjugate, maps into itself, and it is easily checked that the weight function is invariant under . Thus, induces an involution on the set of connected components of each set , and hence an involution on the graded root . This makes into a symmetric graded root. See [Dai, Section 2.1] for further discussion.
3. The involution on Heegaard Floer homology
In Section 2.1 we introduced the homotopy involution on the Heegaard Floer complexes of three-manifolds equipped with self-conjugate structures. The goal of this section will be to identify the action induced by at the level of Heegaard Floer homology, for AR plumbed three-manifolds.
We begin by reviewing the construction of the isomorphism between Heegaard Floer homology and lattice homology. For convenience, we work with the plus versions, as in the original picture of [Plumbed, NemethiOS]; that is, we will focus on the isomorphisms (10).
The plumbing graph gives an oriented cobordism between and . Let denote the set of characteristic vectors on limiting to on . We denote a typical element of by , with being fixed.
Let be the graded -module . For any element , define a map by
where is the Heegaard Floer cobordism map associated to the structure on with . In [Plumbed], it is shown that for any , the map satisfies a set of “adjunction relations” relating the values of on different characteristic vectors in . Specifically, for every , if we set we have
The set of all maps from to satisfying these relations forms a graded -module, which can be identified with the lattice homology ; cf. Proposition 4.7 in [NemethiOS]. Under this identification, the reflection involution on corresponds to precomposing a map with the reflection on .
Theorem 8.3 in [NemethiOS] says that the correspondence effects a graded -module isomorphism between and the lattice homology , as in (10).
Unwinding the definitions, to check that on corresponds to under these isomorphisms is equivalent to proving that
for all and . In other words, we must establish the equality