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.
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…
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 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.
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.
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 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 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.
This is the third paper of this series. In \cite{Wang20}, we defined 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. In this paper, we establish a gluing theorem for this Floer homology when two such 3-manifolds are glued suitably along their common boundary, assuming that $\partial Y$ is disconnected, and $ω$ is small and yet non-vanishing on $\partial Y$. As applications, we construct a monopole Floer 2-functor and the generalized cobordism maps. Using results of Kr…
The contact invariant is an element in the monopole Floer homology groups of an oriented closed three manifold canonically associated to a given contact structure. A non-vanishing contact invariant implies that the original contact structure is tight, so understanding its behavior under symplectic cobordisms is of interest if one wants to further exploit this property. By extending the gluing argument of Mrowka and Rollin to the case of a manifold with a cylindrical end, we will show that the contact invariant behaves naturally under a strong symplectic cobordism. As quick applications of the …
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.
We prove a Smith-type inequality for regular covering spaces in monopole Floer homology. Using the monopole Floer / Heegaard Floer correspondence, we deduce that if a 3-manifold Y admits a p^n-sheeted regular cover that is a Z/pZ-L-space (for p prime), then Y is a Z/pZ-L-space. Further, we obtain constraints on surgeries on a knot being regular covers over other surgeries on the same knot, and over surgeries on other knots.
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.
In the present work we generalize the construction of monopole Floer homology due to Kronheimer and Mrowka to the case of a gradient flow with Morse-Bott singularities. Focusing then on the special case of a three-manifold equipped with a spin$^c$ structure isomorphic to its conjugate, we define the counterpart in this context of Manolescu's recent $\mathrm{Pin}(2)$-equivariant Seiberg-Witten-Floer homology. In particular, we provide an alternative approach to his disproof of the celebrated Triangulation conjecture.
We show that monopole Floer homology (as defined by Kronheimer and Mrowka) is isomorphic to the S^1-equivariant homology of the Seiberg-Witten Floer spectrum constructed by the second author.
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…
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 study the general properties of the moduli spaces of SO(3) vortices over orbifold Riemann surfaces and use these to characterize the solutions of the SO(3) monopole equations on Seifert manifolds following in the footsteps of Mrowka, Ozsváth and Yu. We also study the solutions to the SO(3) monopole equations on S1 x (Riemann Surface) in order to motivate the construction of a version of monopole Floer homology, which we call framed monopole Floer homology, in analogy with the construction given by Kronheimer and Mrowka for the case of instanton Floer homology. Finally, the SO(3) vortex modu…
In a recent paper, Lin, Ruberman and Saveliev proved a splitting formula expressing the Seiberg-Witten invariant $λ_{SW}(X)$ of a smooth $4$-manifold with rational homology of $S^1\times S^3$ in terms of the Frøyshov invariant $h(X)$ and a Lefschetz number in reduced monopole Floer homology. In this note we observe that a similar splitting formula holds in reduced instanton Floer homology.
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.
Given an involution on a rational homology 3-sphere $Y$ with quotient the $3$-sphere, we prove a formula for the Lefschetz number of the map induced by this involution in the reduced monopole Floer homology. This formula is motivated by a variant of Witten's conjecture relating the Donaldson and Seiberg--Witten invariants of 4-manifolds. A key ingredient is a skein-theoretic argument, making use of an exact triangle in monopole Floer homology, that computes the Lefschetz number in terms of the Murasugi signature of the branch set and the sum of Frøyshov invariants associated to spin structures…
In this remark, we show how the monopole Frøyshov invariant, as well as the analogues of the Involutive Heegaard Floer correction terms $\underline{d},\overline{d}$, are related to the $\mathrm{Pin}(2)$-equivariant Floer homology $\mathit{SWFH}^G_*$. We show that the only interesting correction terms of a $\mathrm{Pin}(2)$-space are those coming from the subgroups $\mathbb{Z}/4$, $S^1$, and $\mathrm{Pin}(2)$ itself.
We refine some classical estimates in Seiberg-Witten theory, and discuss an application to the spectral geometry of three-manifolds. In particular, we show that on a rational homology three-sphere $Y$, for any Riemannian metric the first eigenvalue of the laplacian on coexact one-forms is bounded above explicitly in terms of the Ricci curvature, provided that $Y$ is not an $L$-space (in the sense of Floer homology). The latter is a purely topological condition, and holds in a variety of examples. Performing the analogous refinement in the case of manifolds with $b_1>0$, we obtain a gauge-theor…
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.
In this paper we construct possible candidates for the minus versions of monopole and instanton knot Floer homologies. For a null-homologous knot $K\subset Y$ and a base point $p\in K$, we can associate the minus versions, $\underline{\rm KHM}^-(Y,K,p)$ and $\underline{\rm KHI}^-(Y,K,p)$, to the triple $(Y,K,p)$. We prove that a Seifert surface of $K$ induces a $\mathbb{Z}$-grading, and there is an $U$-map on the minus versions, which is of degree $-1$. We also prove other basic properties of them. If $K\subset Y$ is not null-homologous but represents a torsion class, then we can also construc…
Inspired by Kronheimer and Mrowka's approach to monopole Floer homology, we develop a model for $\mathbb{Z}/2$-equivariant symplectic Floer theory using equivariant almost complex structures, which admits a localization map to a twisted version of Floer cohomology in the invariant set. We then present applications to Steenrod operations on Lagrangian Floer cohomology, and to the Heegaard Floer homology of double covers of three-manifolds.
We recently defined an invariant of contact manifolds with convex boundary in Kronheimer and Mrowka's sutured monopole Floer homology theory. Here, we prove that there is an isomorphism between sutured monopole Floer homology and sutured Heegaard Floer homology which identifies our invariant with the contact class defined by Honda, Kazez and Matić in the latter theory. One consequence is that the Legendrian invariants in knot Floer homology behave functorially with respect to Lagrangian concordance. In particular, these invariants provide computable and effective obstructions to the existence …
The endpoint of this series of papers is to construct 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. In the first paper, we exploit the framework of the gauged Landau-Ginzburg models to address two model problems for the (perturbed) Seiberg-Witten moduli spaces on either $\mathbb{C}\timesΣ$ or $\mathbb{H}^2_+\timesΣ$, where $Σ$ is any compact Riemann surface of genus $\geq 1$. Our first result states that finite energy solutions to the perturbed equations on $\mathbb{C}\timesΣ$ are necessa…
This paper establishes a new technique that enables us to access some fundamental structural properties of instanton Floer homology. As an application, we establish, for the first time, a relation between the instanton Floer homology of a $3$-manifold or a null-homologous knot inside a $3$-manifold and the Heegaard diagram of that $3$-manifold or knot. We further use this relation to compute the instanton knot homology of some families of $(1,1)$-knots, including all torus knots in $S^3$, which were mostly unknown before. As a second application, we also study the relation between the instanto…
Sutured instanton Floer homology was introduced by Kronheimer and Mrowka. In this paper, we prove that for a taut balanced sutured manifold with vanishing second homology, the dimension of the sutured instanton Floer homology provides a bound on the minimal depth of all possible taut foliations on that balanced sutured manifold. The same argument can be adapted to the monopole and even the Heegaard Floer settings, which gives a partial answer to one of Juhasz's conjectures. Using the nature of instanton Floer homology, on knot complements, we can construct a taut foliation with bounded depth, …
