Involutive Heegaard Floer homology
Using the conjugation symmetry on Heegaard Floer complexes, we define a three-manifold invariant called involutive Heegaard Floer homology, which is meant to correspond to -equivariant Seiberg-Witten Floer homology. Further, we obtain two new invariants of homology cobordism, and , and two invariants of smooth knot concordance, and . We also develop a formula for the involutive Heegaard Floer homology of large integral surgeries on knots. We give explicit calculations in the case of L-space knots and thin knots. In particular, we show that detects the non-sliceness of the figure-eight knot. Other applications include constraints on which large surgeries on alternating knots can be homology cobordant to other large surgeries on alternating knots.
In [Triangulations], the second author resolved the remaining cases of the triangulation conjecture, by showing that there are manifolds of every dimension that cannot be triangulated. The proof involves the construction of a -equivariant version of Seiberg-Witten Floer homology. ( is the group consisting of two copies of the complex unit circle with a map interchanging them such that and .) From the module structure of this homology one extracts three non-additive maps
where denotes the three-dimensional homology cobordism group. The maps are analogous to the Frøyshov-type correction terms arising from monopole or Heegaard Floer homology [FroyshovSW, KMOS, AbsGraded], but have the additional property that their reduction mod is equal to the Rokhlin invariant. Furthermore, we have for any homology sphere . This implies the non-existence of elements of order in with odd Rokhlin invariant, which in turn disproves the triangulation conjecture—in view of the previous work of Galewski-Stern and Matumoto [GS, Matumoto].
The construction of -equivariant version of Seiberg-Witten Floer homology in [Triangulations] uses finite dimensional approximation, following [Spectrum], and it is only applicable to rational homology spheres. Doing calculations with this method is rather difficult, and at the moment only accessible when one has an explicit description of the Seiberg-Witten Floer complex, e.g. for Seifert fibrations [Stoffregen]. An alternative construction was given by Lin in [Lin]; this refines the Kronheimer-Mrowka definition of monopole Floer homology from [KMBook], and works for arbitrary -manifolds. Recently, Lin established an exact triangle for his theory, which allowed him to compute it for many examples [LinExact]. However, some of Lin’s computations make use of the isomorphism between monopole and Heegaard Floer homology [KLT1, CGH2].
Indeed, among Floer theories for three-manifolds, the one most amenable to computations is Heegaard Floer homology. This was introduced by Ozsváth and Szabó in the early 2000’s [HolDisk, HolDiskTwo, HolDiskFour]. The definition starts with a pointed Heegaard diagram representing a three-manifold . One then takes the Lagrangian Floer homology of two tori in the symmetric product of the Heegaard surface . There are four flavors of this construction, denoted , , , and ; we will use to denote any of them, with . Heegaard Floer homology was shown to be isomorphic to monopole Floer homology [KLT1, CGH2]. One reason why Heegaard Floer homology is computationally tractable is because of the surgery formulas [Knots, IntSurg, RatSurg, LinkSurg] which relate it to a similar invariant for knots, knot Floer homology [Knots, RasmussenThesis]. In view of this, it would be desirable to construct a Heegaard Floer analog of -equivariant Seiberg-Witten Floer homology.
In the present paper we define a Heegaard Floer analog of -equivariant Seiberg-Witten Floer homology, which we call involutive Heegaard Floer homology. Here, is the subgroup of generated by the element . The -equivariant theory does not have the full power of a -equivariant one; in particular, one cannot use it to give another disproof of the triangulation conjecture, because the resulting homology cobordism invariants do not capture the Rokhlin invariant. Nevertheless, we will see that the information in involutive Heegaard Floer homology goes beyond that in ordinary Heegaard Floer homology. Moreover, we develop a formula for the involutive Heegaard Floer homology of large surgeries on knots, and this leads to many explicit calculations.
Both Seiberg-Witten and Heegaard Floer homology decompose as direct sums, indexed by the structures on the -manifold. In Seiberg-Witten theory, the element gives a symmetry of the equations that takes a structure to its conjugate, . In Heegaard Floer theory, there is a similar conjugation symmetry, given by switching the orientation of the Heegaard surface, as well as swapping the and the curves:
As noted in [HolDiskTwo, Theorem 2.4], this induces isomorphisms
for any structure on . We have , so is an involution on This involution was used in various arguments in the Heegaard Floer literature; see for example [LiscaStipsicz2, LiscaOwens].
We define involutive Heegaard Floer homology by making use of the construction of at the chain level. Specifically, we have a Heegaard Floer chain group and a map
that induces the map on homology. We define the flavor of involutive Heegaard Floer homology, , to be the homology of the mapping cone:
Here is just a formal variable, with , and denotes a shift in grading. If we work with coefficients in , then Heegaard Floer groups come equipped with -module structures, and we get a -module structure on .
For any flavor , the isomorphism class of the involutive Heegaard Floer homology , as a -module, is a three-manifold invariant.
The ring is the cohomology ring of the classifying space , with coefficients. If we have a space with a action, then one can obtain its -equivariant homology (as an -module) from its -equivariant homology by constructing a mapping cone and then taking homology, just as we constructed from . Since is supposed to correspond to -equivariant Seiberg-Witten Floer homology, we see that should correspond to -equivariant Seiberg-Witten Floer homology. To construct the -equivariant theory, one would have to complete the mapping cone to an infinite complex that involves not just the map , but also the chain homotopy relating to the identity, and higher homotopies. To define these higher homotopies one would need to prove that Heegaard Floer homology is natural “to infinite order,” whereas currently this is established only to first order, by the work of Juhász and Thurston [Naturality]. We refer to Section LABEL:sec:motivation below for more explanations.
The module decomposes as a direct sum indexed by the orbits of structures under the conjugation action. The most interesting case is when we have a structure that comes from a spin structure, i.e., . We then obtain a group .
Furthermore, if we have a four-dimensional spin cobordism from to , we construct maps
A priori, these depend on some additional data , which includes a choice of Heegaard diagrams for and and a handle decomposition of the cobordism . Although we expect the maps to not depend essentially on , proving this would require results about higher order naturality that are not available by current techniques.
Recall that Heegaard Floer homology (for torsion structures ) can be equipped with an absolute grading with values in ; cf. [AbsGraded]. When is a rational homology sphere, the minimal grading of the infinite -tower in gives the Ozsváth-Szabó correction term . When is spin, the involutive Heegaard Floer homology has two infinite -towers, and by imitating [AbsGraded] we obtain two new correction terms
We also have a Frøyshov-type inequality for spin four-manifolds with boundary, analogous to [AbsGraded, Theorem 9.6]:
Let be a rational homology three-sphere, and be a spin structure on . Then if is a smooth negative-definite four manifold with boundary , and is a spin structure on such that , then
If is a -homology sphere, then it admits a unique spin structure , and we can simply write and for the corresponding invariants. Recall that two three-manifolds and are called homology cobordant (resp. -homology cobordant or -homology cobordant) if there exists a smooth, compact, oriented cobordism from to such that (resp. or ) for . The homology cobordism group is generated by oriented integer homology spheres, modulo the equivalence relation given by homology cobordism. Similarly, the -homology cobordism group is generated by oriented -homology spheres, modulo -homology cobordism.
The correction terms are invariants of -homology cobordism, i.e., they descend to (non-additive) maps
Further, when is an integer homology sphere then and take even integer values, and so give maps
In some cases, for example when is an L-space (i.e., for every ), it turns out that . On the other hand, for the Brieskorn sphere we have
Thus, whereas the usual correction term cannot tell that is not homology null-cobordant, the invariant can. Of course, this can also be seen by other methods, e.g. using the Rokhlin invariant, which is for . More interesting is the following corollary:
The L-spaces that are -homology spheres generate a proper subgroup of . For example, is not -homology cobordant to any L-space.
Observe that L-spaces that are -homology spheres include, for example, all double branched covers over alternating knots in ; cf. [BrDCov]. We also remark that Corollary 1.4 can be obtained using -equivariant Seiberg-Witten Floer homology, by showing that for L-spaces; this is a consequence of the Gysin sequence that relates the - and -equivariant theories. (See [Lin, Proposition 3.10] for a version of this.)
In contrast to Corollary 1.4, the Brieskorn sphere does bound a rational homology ball, and hence is -homology cobordant to ; cf. [FSExample]. This forces the invariant to be zero.
In our theory, the calculation of and for is done using an adaptation of the large surgery formula from [Knots, RasmussenThesis] to the involutive setting. More generally, this adaptation allows us to calculate of a large integral surgery on a knot in terms of the knot Floer complex and the analogue of the map on , which we denote by .
Let us denote by the result of surgery on a knot , with framing . Recall that the usual large surgery formula (cf. [Knots, Theorem 4.4] or [IntSurg, Theorem 2.3]) identifies , for , with the homology of a quotient complex of . Here, is an integer, is its mod reduction, and we use the standard identification of structures on with the elements of . In particular, for we have a spin structure, and the map induces a similar map on . Let be the mapping cone
We now state the involutive large surgery formula.
Let be a knot, and let be its Seifert genus. Then, for all integers we have an isomorphism of relatively graded -modules
Note that, in Heegaard Floer theory, large surgeries are considered those with coefficient . If we are only interested in the spin structure , then the weaker inequality suffices. Therefore, in this paper, a “large” surgery will mean one with coefficient .
In order to compute , we need to understand the conjugation symmetry on the knot Floer complex. This can be determined explicitly for two important families of knots:
L-space knots (and their mirrors), those knots that admit a surgery that is an L-space; cf. [OSLens]. These include all torus knots, Berge knots, and pretzel knots;
Floer homologically thin knots (which we simply call thin), those for which the knot Floer homology is supported in a single diagonal; cf. [RasmussenThesis, RasSurvey, MOQuasi]. These include all alternating knots [AltKnots] and, more generally, all quasi-alternating knots [MOQuasi].
The key observation is that is equal to the map studied by Sarkar in [SarkarMoving], which corresponds to moving the basepoints around the knot. In the two cases above, knowing and the behavior of with respect to gradings suffices to determine up to chain homotopy. From knowledge of one can calculate for large surgeries on those knots (many of which are hyperbolic).
In fact, it should be noted that is in principle computable for all knots in , using grid diagrams and the maps on grid complexes [MOS, MOST]. Thus, is algorithmically computable for all large surgeries on knots. Although in this paper we limit ourselves to large surgeries, we expect that satisfies involutive analogues of the surgery exact triangle from [HolDiskTwo], of the general knot surgery formulas from [IntSurg, RatSurg], and perhaps of the link surgery formula from [LinkSurg]. Thus, it may be possible to show that is algorithmically computable for all three-manifolds, along the lines of [MOT].
Going back to correction terms, recall that, for large , the Ozsváth-Szabó correction term of in the spin structure is given by
where is an invariant of (smooth) knot concordance, coming from the knot Floer complex of ; cf. [RasmussenThesis, RasmussenGT, RatSurg, Peters, NiWu]. Similarly, using we obtain new concordance invariants and , and we have the following result.
Let be a knot of Seifert genus . Then, for each integer , we have
The calculation of and for L-space knots and mirrors of L-space knots can be found in Section LABEL:sec:Lspace, and that for thin knots in Section LABEL:sec:thin. Let us state the result for alternating knots:
Let be an alternating knot, with signature and Arf invariant . The values of the triple for are given in the following tables.
If , then
If , then
For example, the figure-eight knot (with the surgery being ) has and . Therefore,
The figure-eight knot is not slice: classically, one can prove this by checking the Fox-Milnor condition on the Alexander polynomial [FoxMilnor], or (as above) by noting that surgery on has non-trivial Rokhlin invariant. However, the non-sliceness of cannot be detected by most of the modern concordance invariants coming from Floer or Khovanov homology: from [4BallGenus, RasmussenThesis], from [RasmussenMilnor], from [MOwens], from [AltKnots, RasmussenGT, Peters], from [RatSurg], from [HomWu], from [HomEps], and from [UpsilonT] all vanish on amphichiral knots such as . By contrast, our concordance invariant does detect that is not slice.
Moreover, combining Theorems 1.6 and 1.7 we obtain various constraints on which large surgeries on an alternating knot can be homology cobordant to large surgeries on another alternating knot. For example, we have
Let and be alternating knots such that . If and are -homology cobordant for some odd , then .
The paper is organized as follows. In Section 2 we define involutive Heegaard Floer homology and prove its invariance (Theorem 1.1). In Section LABEL:sec:motivation we explain in more detail why should correspond to -equivariant Seiberg-Witten Floer homology. In Section LABEL:sec:prop we establish a few properties of the involutive Heegaard Floer groups, and define the cobordism maps. In Section LABEL:sec:ds we define the new correction terms and prove Theorems 1.2 and 1.3. Section LABEL:sec:surgery contains the proof of the involutive analog of the large surgery formula, Theorem 1.5; we also prove there Theorem 1.6 and Corollary 1.4, and show that and are not homomorphisms. In Section LABEL:sec:Lspace we apply the involutive large surgery formula to compute for large surgeries on (mirrors of) L-space knots. Large surgeries on thin knots are discussed in Section LABEL:sec:thin, where we prove Theorem 1.7 and Corollary 1.8.
Acknowledgements. We thank Jennifer Hom, András Juhász, Tye Lidman, Francesco Lin, Robert Lipshitz and Sucharit Sarkar for helpful conversations. We are also grateful to the referees for many helpful suggestions.
The goal of this section is to define involutive Heegaard Floer homology. We assume that the reader is familiar with regular Heegaard Floer homology, as in [HolDisk, HolDiskTwo, HolDiskFour]. However, we start by reviewing a few concepts in order to fix notation, and to emphasize naturality issues.
2.1. Heegaard Floer homology
Fix a closed, connected, oriented three-manifold . Denote by the space of structures on , and pick some . Heegaard Floer homology is computed from a pointed Heegaard diagram for . A pointed Heegaard diagram is a set of data where:
is an embedded, oriented surface of genus , that splits the three-manifold into two handlebodies and ;
is a set of nonintersecting simple closed curves on which bound disks in , and in fact span the kernel of ;
is a similar set of curves for instead of , such that is transverse for any ;
is a basepoint that does not lie on any of the alpha or beta curves.
The Heegaard Floer groups are variations of Lagrangian Floer cohomology for the two tori
inside the symmetric product . There is a natural map , and we will focus on those intersection points such that . To define the Floer groups, we need to impose an admissibility condition on , depending on ; cf. [HolDisk, Section 4.2.2]. We also need to choose a suitable (generic) family of almost complex structures on . We will write for the data , which we call a Heegaard pair.
Given such a pair , the Heegaard Floer chain complex is freely generated over by pairs with and , such that . The differential is given by
Here, is the space of homotopy classes of Whitney disks from to , is the Maslov index, is the moduli space of -holomorphic disks in the class (modulo the action of ), and is the algebraic intersection number of with the divisor . There is an action of on , where acts by and decreases relative grading by . The other complexes and are obtained from by considering only pairs with and . All three complexes have an induced -action, which is trivial in the case of .
We will write for any of the four flavors of the Heegaard Floer chain complex, and for the homology groups.
Theorem 2.1 (Ozsváth-Szabó [HolDisk]).
The isomorphism class of (as a -module) is an invariant of the three-manifold .
A stronger result was obtained by Juhász and Thurston, who proved naturality of the invariant:
Theorem 2.2 (Juhász-Thurston [Naturality]).
If we fix and the basepoint , then the -modules form a transitive system. That is to say, for any two Heegaard pairs and we have a distinguished isomorphism
such that for all we have:
Given a transitive system, we can get a single module as the inverse limit of this system. We can identify any with in a canonical way. We usually drop from the notation and write . We can also consider the direct sum over all structures:
Although Juhász and Thurston phrased their theorem in terms of homology, their methods actually give a result at the chain level:
If we fix , then the chain groups form a transitive system in the homotopy category of chain complexes of -modules. In other words, for every two Heegaard pairs and we have a chain homotopy equivalence
satisfying the analogs of conditions (i) and (ii) from the statement of Theorem 2.2, with equality replaced by chain homotopy. In fact, the maps are those induced on homology by .
The map is only unique up to chain homotopy. In the remainder of the paper, whenever we write , we mean a representative of this chain homotopy class of maps.
The proofs of Theorems 2.1 and 2.2 involve showing that any two Heegaard pairs are related by a sequence of the following moves: changing the almost complex structures (with the diagram being fixed); isotopies of the alpha and beta curves in ; handleslides of the curves in ; stabilizations and destabilizations of ; diffeomorphisms of induced by an ambient isotopy of in .111In fact, a diffeomorphism induced by an ambient isotopy can be obtained as the composition of some stabilizations and destabilizations. However, it is convenient to consider it as a separate move. These moves induce chain homotopy equivalences between the respective Floer chain complexes. Indeed, this was shown in [HolDisk] for most of the moves. The exceptions are handleslides, for which the argument in [HolDisk, Section 9.2] only shows that they induce quasi-isomorphisms. It is proved there that if and are the maps associated to a handleslide and its inverse, then is homotopic to the composition of the triangle map coming from a small isotopy with the nearest point map. However, one can further show that, for a small isotopy, the triangle map is chain homotopic to the nearest point map; see [LipshitzCyl, Proposition 11.3] and [OzsvathStipsicz, proof of Theorem 6.6]. This implies that is chain homotopic to the identity. The same goes for , so we can conclude that handleslides actually induce chain homotopy equivalences.
Given two Heegaard pairs and , we define the maps by choosing a sequence of moves relating and . We claim that different choices of moves yield chain homotopic maps. First, note that interpolating between two (families of) almost complex structures in two different ways produces chain homotopic maps, by the usual continuation arguments in Floer theory. (We are using here that the space of compatible almost complex structures is contractible.)
With regard to the moves on Heegaard diagrams, to show that they give rise to chain homotopic maps, in view of Theorem 2.39 in [Naturality], it suffices to prove that is a strong Heegaard invariant in the sense of [Naturality, Definition 2.33], in the homotopy category of chain complexes of -modules. There are four conditions to be checked: functoriality, commutativity, continuity, and handleswap invariance. All of these are checked in Sections 9.2 and 9.3 of [Naturality], in the context of proving the weaker statement that is a strong Heegaard invariant in the category of -modules. However, the proofs there actually work at the chain level, with the maps being considered up to chain homotopy.
Given that the maps are well-defined up to chain homotopy, conditions (i) and (ii) are almost automatic. Indeed, for (i), when , we can consider the empty sequence of moves, so that is the identity. For (ii), we consider a sequence of Heegaard moves from to , and another from to , and by composing them we get a sequence from to . ∎
We will sometimes write for . This is justified by Proposition 2.3, which says that the chain groups for different are chain homotopy equivalent (although, of course, they are not usually isomorphic).
2.2. The involution
Now let us discuss the conjugation action on Heegaard Floer homology. Given a pointed Heegaard diagram , we define the conjugate diagram by
where means with the orientation reversed. A family of almost complex structures on gives a conjugate family on . If is a Heegaard pair, we write for the conjugate pair .
Intersection points in for are in one-to-one correspondence with those for , and this correspondence takes a structure to its conjugate . Moreover, -holomorphic disks with boundaries on are in one-to-one correspondence with -holomorphic disks with boundaries on . Thus, as observed in [HolDiskTwo, Theorem 2.4], we get a canonical isomorphism between Heegaard Floer chain complexes:
Moreover, and represent the same based three-manifold . According to Proposition 2.3, we have a chain homotopy equivalence
We denote by the composition of these two maps:
The map is chain homotopic to the identity.
We have Note that , so the composition
is the conjugation of by . Recall that is the composition of maps associated to moves between the respective Heegaard pairs. When we conjugate any such map by , we get the map associated to the corresponding move between the conjugate Heegaard pairs. (This uses the identification between - and -holomorphic triangles.) In view of Proposition 2.3, the map (2) is chain homotopic to . Therefore,
when in the last step we used the properties of a transitive system. ∎
Lemma 2.5 implies that induces an involution
on Heegaard Floer homology. This was already observed in [HolDiskTwo, Theorem 2.4].
If we view Heegaard splittings as coming from self-indexing Morse functions on the three-manifold , then the equivalence is induced by moving from a Morse function to .
2.3. Involutive Heegaard Floer homology
Let denote the space of orbits of structures on , under the conjugation action. An orbit is either of the form with , or of the form with . The former case corresponds to structures that come from spin structures.
A structure with admits lifts to a spin structure; see [Lin, p. 124]. By a slight abuse of terminology, when we will refer to as a spin structure without fixing a specific lift.
Given , a Heegaard pair for , and an orbit , set
We define the involutive Heegaard Floer complex to be the mapping cone complex
Given a complex , we use to denote the same complex with the grading shifted by : Thus, as an abelian group, the cone complex above is
with the first factor being the domain of and the second the target.
To get more structure on this complex, it is helpful to introduce a formal variable of degree with , and write (3) as
Note that in the target the shift cancels out the shift due to the variable , so in fact is isomorphic to as a graded module.
We can re-write (4) as
where is the ordinary Heegaard Floer differential. We write
for the differential on . Observe that, by construction, is a complex of modules over the ring
with and decreasing the grading by and , respectively.
The quasi-isomorphism class of the complex (over ) is an invariant of the pair .
Note that in the defintion of we used the map , which was constructed from a sequence of Heegaard moves relating to . Thus, a priori, depends not only on , but also on that sequence of moves. However, Proposition 2.3 guarantees that is well-defined up to chain homotopy. Therefore, so is . Since the mapping cones of chain homotopy maps are homotopic, we conclude that changing the sequence of moves only changes by a homotopy equivalence.
Next, fix the basepoint and suppose that we have a different Heegaard pair for . Let be the corresponding map from to , which is the composition of with an involution . Consider the diagram