To a link L in the 3-sphere, we associate a spectral sequence whose E^2 page is the reduced Khovanov homology of L and which converges to a version of the monopole Floer homology of the branched double cover. The pages E^k for k > 1 depend only on the mutation equivalence class of L. We define a mod 2 grading on the spectral sequence which interpolates between the delta-grading on Khovanov homology and the mod 2 grading on Floer homology. We also derive a new formula for link signature that is well-adapted to Khovanov homology.
More generally, we construct new bigraded invariants of a framed link in a 3-manifold as the pages of a spectral sequence modeled on the surgery exact triangle. The differentials count monopoles over families of metrics parameterized by permutohedra. We utilize a connection between the topology of link surgeries and the combinatorics of graph associahedra. This also yields simple realizations of permutohedra and associahedra, as refinements of hypercubes.
Let $K$ be a rationally null-homologous knot in a $3$-manifold $Y$, equipped with a nonzero framing $λ$, and let $Y_λ(K)$ denote the result of $λ$-framed surgery on $Y$. Ozsváth and Szabó gave a formula for the Heegaard Floer homology groups of $Y_λ(K)$ in terms of the knot Floer complex of $(Y,K)$. We strengthen this formula by adding a second filtration that computes the knot Floer complex of the dual knot $K_λ$ in $Y_λ$, i.e., the core circle of the surgery solid torus. In the course of proving our refinement we derive a combinatorial formula for the Alexander grading which may be of indepe…
Given a spin$^c$ rational homology sphere $(Y,\mathfrak{s})$ with $\mathfrak{s}$ self-conjugate and for which the reduced monopole Floer homology $\mathit{HM}_{\bullet}(Y,\mathfrak{s})$ has rank one, we provide obstructions to the intersection forms of its Stein fillings which are not negative definite. The proof of this result (and of its natural generalizations we discuss) uses $\mathrm{Pin}(2)$-monopole Floer homology.
Surgery exact triangles in various 3-manifold Floer homology theories provide an important tool in studying and computing the relevant Floer homology groups. These exact triangles relate the invariants of 3-manifolds, obtained by three different Dehn surgeries on a fixed knot. In this paper, the behavior of $SU(N)$-instanton Floer homology with respect to Dehn surgery is studied. In particular, it is shown that there are surgery exact tetragons and pentagons, respectively, for $SU(3)$- and $SU(4)$-instanton Floer homologies. It is also conjectured that $SU(N)$-instanton Floer homology in gener…
For a nullhomologous Legendrian knot in a closed contact 3-manifold Y we consider a contact structure obtained by positive rational contact surgery. We prove that in this situation the Heegaard Floer contact invariant of Y is mapped by a surgery cobordism to the contact invariant of the result of contact surgery. In addition we characterize the spin-c structure on the cobordism that induces the relevant map. As a consequence we determine necessary and sufficient conditions for the nonvanishing of the contact invariant after rational surgery when Y is the standard 3-sphere, generalizing previou…
We prove the existence of an exact triangle for the Pin(2)-monopole Floer homology groups of three manifolds related by specific Dehn surgeries on a given knot. Unlike the counterpart in usual monopole Floer homology, only two of the three maps are those induced by the corresponding elementary cobordism. We use this triangle to describe the invariants associated to homology spheres obtained by (\pm1)-surgery on alternating knots.
Given an equivariant knot $K$ of order $2$, we study the induced action of the symmetry on the knot Floer homology. We relate this action with the induced action of the symmetry on the Heegaard Floer homology of large surgeries on $K$. This surgery formula can be thought of as an equivariant analog of the involutive large surgery formula proved by Hendricks and Manolescu. As a consequence, we obtain that for certain double branched covers of $S^{3}$ and corks, the induced action of the involution on Heegaard Floer homology can be identified with an action on the knot Floer homology. As an appl…
In this paper, we explore the interplay between contact structures and sutured monopole Floer homology. First, we study the behavior of contact elements, which were defined by Baldwin and Sivek, under the operation of performing Floer excisions, which was introduced to the context of sutured monopole Floer homology by Kronheimer and Mrowka. We then compute the sutured monopole Floer homology of some special balanced sutured manifolds, using tools closely related to contact geometry. For application, we obtain an exact triangle for the oriented skein relation in monopole theory and derive a con…
We study the behavior of $\mathrm{Pin}(2)$-monopole Floer homology under connected sums. After constructing a (partially defined) $\mathcal{A}_{\infty}$-module structure on the $\mathrm{Pin}(2)$-monopole Floer chain complex of a three manifold (in the spirit of Baldwin and Bloom's monopole category), we identify up to quasi-isomorphism the Floer chain complex of a connected sum with a version of the $\mathcal{A}_{\infty}$-tensor product of the modules of the summands. There is an associated Eilenberg-Moore spectral sequence converging to the Floer groups of the connected sum whose $E^2$ page i…
We write down an explicit formula for the $+$ version of the Heegaard Floer homology (as an absolutely graded vector space over an arbitrary field) of the results of Dehn surgery on a knot $K$ in $S^3$ in terms of homological data derived from $CFK^{\infty}(K)$. This allows us to prove some results about Dehn surgery on knots in $S^3$. In particular, we show that for a fixed manifold there are only finitely many alternating knots that can produce it by surgery. This is an improvement on a recent result by Lackenby and Purcell. We also derive a lower bound on the genus of knots depending on the…
In this paper, we generalize the work of the second author and prove a grading shifting property, in sutured monopole and instanton Floer theories, for general balanced sutured manifolds. This result has a few consequences. First, we offer an algorithm that computes the Floer homologies of a family of sutured handle-bodies. Second, we obtain a canonical decomposition of sutured monopole and instanton Floer homologies and build polytopes for these two theories, which was initially achieved by Juhász for sutured (Heegaard) Floer homology. Third, we establish a Thurston-norm detection result for …
We study the conjugation involution in Seiberg-Witten theory in the context of the Ozsváth-Szabó and Bloom's spectral sequence for the branched double cover of a link $L$ in $S^3$. We prove that there exists a spectral sequence of $\mathbb{F}[Q]/Q^2$-modules (where $Q$ has degree $-1$) which converges to $\widetilde{\mathit{HMI}}_*(Σ(L))$, an involutive version of the monopole Floer homology of the branched double cover, and whose $E^2$-page is a version of Bar Natan's characteristic two Khovanov homology of the mirror of $L$. We conjecture that an analogous result holds in the setting of $\ma…
对于在整同调球面中表示为框架连接（L，Λ）上的外科手术的任何三流形，Manolescu和Ozsváth构造了一个链复形的超立方体，其同调计算了Y上Λ-框架外科手术的Heegaard弗洛尔同调。这进行了一个自然的过滤，存在于任何超立方体的链复形，我们研究的E_2页的相关谱序列，称为框架弗洛尔同调。本文的目的之一是证明框架弗洛尔同调是定向框架链环的不变量，而不是surgered流形的不变量。我们将讨论如何这涉及到一个尝试在组合证明的不变性的Heegaard弗洛尔同调。我们还表明，框架弗洛尔同调满足一个外科手术确切的三角形类似于Heegaard的弗洛尔同调。这种设置导致一个完全组合的Heegaard弗洛尔同调中的外科手术精确三角形的构造。最后，我们研究了两个分量链的框架弗洛尔同调，这些链激增到S^2 x S^1 # S^2 x S^1并且具有性质2 R。
We compute the Pin(2)-equivariant monopole Floer homology for the class of plumbed 3-manifolds with at most one "bad" vertex (in the sense of Ozsvath and Szabo). We show that for these manifolds, the Pin(2)-equivariant monopole Floer homology can be calculated in terms of the Heegaard Floer/monopole Floer lattice complex defined by Nemethi. Moreover, we prove that in such cases the ranks of the usual monopole Floer homology groups suffice to determine both the Manolescu correction terms and the Pin(2)-homology as an abelian group. As an application of this, we show that $β(-Y, s) = \barμ(Y, s)…
In previous work, we introduced a natural $\mathcal{A}_{\infty}$-structure on the $\mathrm{Pin}(2)$-monopole Floer chain complex of a closed, oriented three-manifold $Y$, and showed that it is non-formal in the simplest case in which $Y$ is the three-sphere $S^3$. In this paper, we provide explicit descriptions of several Massey products induced on homology, and discuss how they can be used to compute the $\mathrm{Pin}(2)$-monopole Floer homology of connected sums in many concrete examples.
In this paper, we give a systematic study of Seiberg-Witten theory on closed oriented manifold $M$ with codimension-$3$ oriented Riemannian foliation $F$. Under a certain topological condition, we construct the basic Seiberg-Witten invariant and the monopole Floer homologies $\bar{HM}(M,F,\mfs;Γ),~\hat{HM}(M,F,\mfs;Γ), ~\widecheck{HM}(M,F,\mfs;Γ)$, for each transverse \spinc structure $\mfs$, where $Γ$ is a complete local system. We will show that these homologies are independent of the bundle-like metric and generic perturbation. The major difference between the basic monopole Floer homologie…
We study the monopole Floer homology of a SOLV rational homology sphere Y from the point of view of spectral theory. Applying ideas of Fourier analysis on solvable groups, we show that for suitable SOLV metrics on Y, small regular perturbations of the Seiberg-Witten equations do not admit irreducible solutions; in particular, this provides a geometric proof that Y is an L-space.
These lecture notes are a friendly introduction to monopole Floer homology. We discuss the relevant differential geometry and Morse theory involved in the definition. After developing the relation with the four-dimensional theory, our attention shifts to gradings and correction terms. Finally, we sketch the analogue in this setup of Manolescu's recent disproof of the long standing Triangulation Conjecture.
We compute the effect of concordance surgery, a generalization of knot surgery defined using a self-concordance of a knot, on the Ozsváth-Szabó 4-manifold invariant. The formula involves the graded Lefschetz number of the concordance map on knot Floer homology. The proof uses the sutured Floer TQFT, and a version of sutured Floer homology perturbed by a 2-form.
We define a "real" version of Kronheimer-Mrowka's monopole Floer homology for a 3-manifold equipped with an involution. As a special case, we obtain invariants for links via their double branched covers. The new input is the notion of a real spin-c structure, which consists of a spin-c structure along with a compatible anti-linear involution on the spinor bundle.
We examine surgery on a knot in $S^3$ to determine surgery obstructions to Seifert fibered integral homology spheres. We find such surgery obstructions using Heegaard Floer, Knot Floer homology and the mapping cone formula for computing Heegaard Floer homology of surgery on a knot. Here however, we take a different approach and use the number of singular fibers of a Seifert fibered integral homology sphere, which is the toroidal structure, to find obstructions. This approach allows us to show that genus one knots cannot yield Seifert fibered integral homology spheres with six or more singular …
We compute the knot Floer filtration induced by a cable of the meridian of a knot in the manifold obtained by large integer surgery along the knot. We give a formula in terms of the original knot Floer complex of the knot in the three-sphere. As an application, we show that a knot concordance invariant of Hom can equivalently be defined in terms of filtered maps on the Heegaard Floer homology groups induced by the two-handle attachment cobordism of surgery along a knot.
This is the second paper in this series. Following the setup of Meng-Taubes, we define the monopole Floer homology for any pair $(Y,ω)$, where $Y$ is a compact oriented 3-manifold with toroidal boundary and $ω$ is a suitable closed 2-form viewed as a decoration. This construction fits into a (3+1)-topological quantum field theory and generalizes the work of Kronheimer-Mrowka for closed oriented 3-manifolds. By a theorem of Meng-Taubes and Turaev, the Euler characteristic of this Floer homology recovers the Milnor-Turaev torsion invariant of the 3-manifold.
We unify two existing approaches to the tau invariants in instanton and monopole Floer theories, by identifying $τ_{\mathrm{G}}$, defined by the second author via the minus flavors $\underline{\operatorname{KHI}}^-$ and $\underline{\operatorname{KHM}}^-$ of the knot homologies, with $τ_{\mathrm{G}}^{\sharp}$, defined by Baldwin and Sivek via cobordism maps of the $3$-manifold homologies induced by knot surgeries. We exhibit several consequences, including a relationship with Heegaard Floer theory, and use our result to compute $\underline{\operatorname{KHI}}^-$ and $\underline{\operatorname{KH…
We show that the bar version of the $\mathrm{Pin}(2)$-monopole Floer homology of a three-manifold $Y$ equipped with a self-conjugate spin$^c$ structure $\mathfrak{s}$ is determined by the triple cup product of $Y$ together with the Rokhlin invariants of the spin structures inducing $\mathfrak{s}$. This is a manifestation of mod $2$ index theory, and can be interpreted as a three-dimensional counterpart of Atiyah's classic results regarding spin structures on Riemann surfaces.
We study Heegaard Floer homology and various related invariants (such as the $h$-function) for two-component L-space links with linking number zero. For such links, we explicitly describe the relationship between the $h$-function, the Sato-Levine invariant and the Casson invariant. We give a formula for the Heegaard Floer $d$-invariants of integral surgeries on two-component L-space links of linking number zero in terms of the $h$-function, generalizing a formula of Ni and Wu. As a consequence, for such links with unknotted components, we characterize L-space surgery slopes in terms of the $ν^…
This is an expansion on my talk at the Geometry and Topology conference at McMaster University, May 2004. We outline a program to relate the Heegaard Floer homologies of Ozsvath-Szabo, and Seiberg-Witten-Floer homologies as defined by Kronheimer-Mrowka. The center-piece of this program is the construction of an intermediate version of Floer theory, which exhibits characteristics of both theories.
We describe a relationship between the monopole Floer homology of three-manifolds and the geometry of Riemann surfaces. Consider an automorphism $φ$ of a compact Riemann surface $Σ$ with quotient $\mathbb{P}^1$. There is a natural correspondence between theta characteristics $L$ on $Σ$ which are invariant under $φ$ and self-conjugate spin$^c$ structures $\mathfrak{s}_L$ on the mapping torus $M_φ$ of $φ$. We show that the monopole Floer homology groups of $(M_φ,\mathfrak{s}_L)$ are explicitly determined by the eigenvalues of the (lift of the) action of $φ$ on $H^0(L)$, the space of holomorphic …
We show that the Seifert-Weber dodecahedral space $\mathsf{SW}$ is an $L$-space. The proof builds on our work relating Floer homology and spectral geometry of hyperbolic three-manifolds. A direct application of our previous techniques runs into difficulties arising from the computational complexity of the problem. We overcome this by exploiting the large symmetry group and the arithmetic and tetrahedral group structure of $\mathsf{SW}$ to prove that small eigenvalues on coexact $1$-forms must have large multiplicity.
We study which closed, connected, orientable three-manifolds $X$ containing a Klein bottle arise as integral Dehn surgery along a knot in $S^3$. Such $X$ are presentable as a gluing of the twisted $I$-bundle over the Klein bottle to a knot manifold, and we use a variety of Heegaard Floer type invariants to generate surgery obstructions. Suppose that $X$ is $8$-surgery along a genus two knot, and arises by gluing the twisted $I$-bundle over the Klein bottle to an $S^3$ knot complement. We show that $X$ is an L-space, it must be the dihedral manifold $\left(-1; \tfrac{1}{2}, \tfrac{1}{2}, \tfrac…
