Alan L. Carey, Bai-Ling Wang
This paper gives a detailed construction of Seiberg-Witten-Floer homology for a closed oriented 3-manifold with a non-torsion $\spinc$ structure. Gluing formulae for certain 4-dimensional manifolds splitting along an embedded 3-manifold are obtained.
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For a rational homology 3-sphere $Y$ with a $\spinc$ structure $\s$, we show that simple algebraic manipulations of our construction of equivariant Seiberg-Witten Floer homology lead to a collection of variants which are topological invariants. We establish exact sequences relating them, we show that they satisfy a duality under orientation reversal, and we explain their relation to ou previous construction of equivariant Seiberg-Witten Floer (co)homologies. We conjecture the equivalence of these versions of equivariant Seiberg-Witten Floer homology with the Heegaard Floer invariants introduce…
This is the second part of the proof of the exact traiangles in Seiberg-Witten Floer theory. We analyse the splitting and gluing of flow lines of the Chern-Simons-Dirac functional when the underlying three-manifold splits along a torus. (two corrections added)
Given a three-manifold with b_1=1 and a nontorsion spin^c structure, we use finite dimensional approximation to construct from the Seiberg-Witten equations two invariants in the form of a periodic pro-spectra. Various functors applied to these invariants give different flavors of Seiberg-Witten Floer homology. We also construct stable homotopy versions of the relative Seiberg-Witten invariants for certain four-manifolds with boundary.
We prove a gluing formula for the families Seiberg-Witten invariants of families of $4$-manifolds obtained by fibrewise connected sum. Our formula expresses the families Seiberg-Witten invariants of such a connected sum family in terms of the ordinary Seiberg-Witten invariants of one of the summands, under certain assumptions on the families. We construct some variants of the families Seiberg-Witten invariants and prove the gluing formula also for these variants. One variant incorporates a twist of the families moduli space using the charge conjugation symmetry of the Seiberg-Witten equations.…
Various Seiberg-Witten Floer cohomologies are defined for a closed, oriented 3-manifold; and if it is the mapping torus of an area-preserving surface automorphism, it has an associated periodic Floer homology as defined by Michael Hutchings. We construct an isomorphism between a certain version of Seiberg-Witten Floer cohomology and the corresponding periodic Floer homology, and describe some immediate consequences.
We show that the contact gluing map of Honda, Kazez, and Matic has a natural algebraic description. In particular, we establish a conjecture of Zarev, that his gluing map on sutured Floer homology is equivalent to the contact gluing map.
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 construct Bott-type and equivariant Seiberg-Witten Floer homology and cohomology for 3-manifolds, in particular rational homology spheres, and prove their diffeomorphism invariance. This paper is a revised version of math.DG/9701010. Some typos are removed.
In this paper we construct gluing maps and cobordism maps for sutured monopole Floer homology.
Some aspects of the construction of SW Floer homology for manifolds with non-trivial rational homology are analyzed. In particular, the case of manifolds that are obtained as zero-surgery on a knot in a homology sphere, and for torsion spinc structures. We discuss relative invariants in the case of torsion spinc structures.
Let $Y$ be a closed and oriented $3$-manifold. We define different versions of unfolded Seiberg-Witten Floer spectra for $Y$. These invariants generalize Manolescu's Seiberg-Witten Floer spectrum for rational homology $3$-spheres. We also compute some examples when $Y$ is a Seifert space.
This is the last of five papers that construct an isomorphism between the Seiberg-Witten Floer homology and the Heegaard Floer homology of a given compact, oriented 3-manifold.
Let M be a closed, connected and oriented 3-manifold. This article is the first of a five part series that constructs an isomorphism between the Heegaard Floer homology groups of M and the corresponding Seiberg-Witten Floer homology groups of M.
Seiberg-Witten Floer spectra for $b_1>0$
本文将 Seiberg-Witten Floer 谱推广到 b_1>0 的三维流形，在配边上定义 Bauer-Furuta 映射并计算若干例子。
This note corrects an erroneous statement in Lemma 3.8 of the author's paper Embedded Contact Homology and Seiberg-Witten Floer Homology IV which was published in Volume 14 of Geometry and Topology in 2009.
Using Furuta's idea of finite dimensional approximation in Seiberg-Witten theory, we refine Seiberg-Witten Floer homology to obtain an invariant of homology 3-spheres which lives in the S^1-equivariant graded suspension category. In particular, this gives a construction of Seiberg-Witten Floer homology that avoids the delicate transversality problems in the standard approach. We also define a relative invariant of four-manifolds with boundary which generalizes the Bauer-Furuta stable homotopy invariant of closed four-manifolds.
We introduce an extra filtration of $\CFK(Y,K)$ and use it in order to obtain formulas for Floer homology of $(Y,K)$, which is obtained from $(Y_i,K_i), i=1,2$ by gluing the knot complements on the framed torus boundaries.
This is the fourth of five papers that construct an isomorphism between the Seiberg-Witten Floer homology and the Heegaard Floer homology of a given compact, oriented 3-manifold. The isomorphism is given as a composition of three isomorphisms; the first of these relates a version of embedded contact homology on an an auxillary manifold to the Heegaard Floer homology on the original. The second isomorphism relates the relevant version of the embedded contact homology on the auxilliary manifold with a version of the Seiberg-Witten Floer homology on this same manifold. The third isomorphism relat…
We construct equivariant and Bott-type Seiberg-Witten Floer homology and cohomology for 3-manifolds, in particular rational homology spheres, and prove their diffeomorphism invariance. We present several versions of the equivariant theory: the singular version, the de Rham version and the Cartan version, with the first playing the most important role. These versions are shown to be equivalent to each other. A few typos are removed.
This is the first of five papers that construct an isomorphism between the embedded contact homology and Seiberg-Witten Floer cohomology of a compact 3-manifold with a given contact 1-form. This paper describes what is involved in the construction.
A satellite formula for real Seiberg-Witten Floer homotopy types
本文建立了具有奇模式的纽结的实Seiberg-Witten Floer同伦类型的卫星公式，基于实Floer同伦类型的切除定理，导出了纽结配边理论的应用，并证明配边不变量仅依赖于零框架手术。
We give a partial characterization of bordered Floer homology in terms of sutured Floer homology. The bordered algebra and modules are direct sums of certain sutured Floer complexes. The algebra multiplication and algebra action correspond to a new gluing map on SFH. It is defined algebraically, and is a special case of a more general "join" map. In a follow-up paper we show that this gluing map can be identified with the contact cobordism map of Honda-Kazez-Matic. The join map is conjecturally equivalent to the cobordism maps on SFH defined by Juhasz.
A previous result about the decategorified bordered (sutured) Heegaard Floer invariants of surfaces glued together along intervals, generalizing the decategorified content of Rouquier and the author's higher-tensor-product-based gluing theorem in cornered Heegaard Floer homology, was proved only over $\mathbb{F}_2$ and without gradings. In this paper we add signs and prove a graded version of the interval gluing theorem over $\mathbb{Z}$, enabling a more detailed comparison of these aspects of decategorified Heegaard Floer theory with modern work on non-semisimple 3d TQFTs in mathematics and p…
We introduce a Floer homotopy version of the contact invariant introduced by Kronheimer-Mrowka-Ozváth-Szabó. Moreover, we prove a gluing formula relating our invariant with the first author's Bauer-Furuta type invariant, which refines Kronheimer-Mrowka's invariant for 4-manifolds with contact boundary. As an application, we give a constraint for a certain class of symplectic fillings using equivariant KO-cohomology.
We will define a version of Seiberg-Witten-Floer stable homotopy types for a closed, oriented 3-manifold $Y$ with $b_1(Y) > 0$ and a spin-c structure $\mathfrak{c}$ on $Y$ with $c_1(\mathfrak{c})$ torsion under an assumption on $Y$. Using the Seiberg-Witten-Floer stable homotopy type, we will construct a gluing formula for the stable cohomotopy version of Seiberg-Witten invariants of a closed 4-manifold $X$ which has a decomposition $X = X_1 \cup_Y X_2$ along $Y$.
Unfolded Seiberg-Witten Floer spectra, II: Relative invariants and the gluing theorem
本文将相对Bauer-Furuta不变量推广到一般带边四维流形，并详细证明了相对不变量的粘合定理。
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 compute the $\mathrm{Pin}(2)$-equivariant Seiberg-Witten Floer homology of Seifert rational homology three-spheres in terms of their Heegaard Floer homology. As a result of this computation, we prove Manolescu's conjecture that $β=-\barμ$ for Seifert integral homology three-spheres. We show that the Manolescu invariants $α, β,$ and $γ$ give new obstructions to homology cobordisms between Seifert fiber spaces, and that many Seifert homology spheres $Σ(a_1,...,a_n)$ are not homology cobordant to any $-Σ(b_1,...,b_n)$. We then use the same invariants to give an example of an integral homology …
Equivariant Seiberg-Witten-Floer cohomology
本文为有理同调3-球面上的有限群作用发展了等变Seiberg-Witten-Floer上同调，基于Manolescu的等变稳定同伦型构造，定义了由群上同调索引的$d$-不变量，并展示了其在纽结配边、群作用扩展障碍及流形嵌入等方面的应用。
Let M be a closed, connected, orientable 3-manifold. The purpose of this paper is to study the Seiberg-Witten Floer homology of M given that S^1 X M admits a symplectic form. In particular, we prove that M fibers over the circle if M has first Betti number 1 and S^1 X M admits a symplectic form with non-torsion canonical class.
The equivalence between two real Seiberg-Witten Floer homologies
本文证明了对配备实spin^c结构的实有理同调球，Li定义的实单极Floer同调与Konno、Miyazawa和Taniguchi定义的实Seiberg-Witten Floer同调同构，并由此识别了某些Froyshov型不变量并证明了两个Smith型不等式。
We define Pin(2)-equivariant Seiberg-Witten Floer homology for rational homology 3-spheres equipped with a spin structure. The analogue of Froyshov's correction term in this setting is an integer-valued invariant of homology cobordism whose mod 2 reduction is the Rokhlin invariant. As an application, we show that there are no homology 3-spheres Y of Rokhlin invariant one such that Y # Y bounds an acyclic smooth 4-manifold. By previous work of Galewski-Stern and Matumoto, this implies the existence of non-triangulable high-dimensional manifolds.
The equivalence of two Pin(2)-equivariant Seiberg-Witten Floer homologies
本文证明了对于配备自共轭 spin$^c$-结构的有理同调 3-球，Lin 定义的 Pin(2)-等变单极 Floer 同调与 Manolescu 定义的 Pin(2)-等变 Seiberg-Witten Floer 同调是同构的。
Lattice homology and Seiberg-Witten Floer spectra
利用格同调，对几乎有理的plumbed同调球面Y给出Seiberg-Witten-Floer谱的组合描述，并计算了某些连通和的Manolescu κ-不变量。
Seiberg-Witten Floer Spectra and Contact Structures
本文利用Seiberg-Witten Floer谱的上同伦集为带接触结构的有理同调3-球面定义新不变量，证明其自然恢复已知接触元素，并引入等变版本以处理有限覆盖情形。
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…
An estimate on energy of min-max Seiberg-Witten Floer generators
本文改进极小极大 Seiberg-Witten Floer 生成元的能量估计，从而无需辛 4 维流形中的嵌入球即可直接证明 ECH 容量恢复体积，并进一步估计收敛速度。
New invariants of involutions from Seiberg-Witten Floer theory
本文利用对合与电荷共轭对称性的耦合发展等变 Seiberg-Witten Floer 理论，构造三类 δ 不变量，证明其 Frøyshov 型不等式，并在 plumbing、分支覆盖和 Dehn 手术中计算及应用。
We describe the natural gluing map on sutured Floer homology which is induced by the inclusion of one sutured manifold (M',Γ') into a larger sutured manifold (M,Γ), together with a contact structure on M-M'. As an application of this gluing map, we produce a (1+1)-dimensional TQFT by dimensional reduction and study its properties.
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…
On the slice-torus invariant $q_M$ from $\mathbb{Z}_2$-equivariant Seiberg--Witten Floer cohomology
该数学研究证明Iida–Taniguchi提出的$\u2124$-值切片环面不变量$q_M$无法由多种已知的纽结配边不变量线性组合表示，明确了该不变量的独立性。
Motivated by various possible generalizations of Taubes's \(SW=Gr\) theorem [T] to Floer-theoretic setting, we prove certain variants of Taubes's convergence theorem in \cite{T} (the first part of his proof of \(SW=Gr\)). In place of the closed symplectic 4-manifold considered in [T], this article considers non-compact manifolds with cylindrical ends, equipped with a self-dual harmonic 2-form with non-degenerate zeroes. This extends and simplifies some central technical ingredients of the author's prior work in [LT] and [KLT5]. Other expected applications include: extending the \(HM=PFH\) theo…
We prove a gluing formula for Seiberg--Witten invariants which describes in particular the behaviour of the invariant under blow-up and rational blow-down.
Given a self-diffeomorphism h of a closed, orientable surface S and an embedding f of S into a three-manifold M, we construct a mutant manifold N by cutting M along f(S) and regluing by h. We will consider whether there are any gluings such that for any embedding, the manifold and its mutant have isomorphic Heegaard Floer homology. In particular, we will demonstrate that if the gluing is not isotopic to the identity, then there exists an embedding of S into a three-manifold M such that the rank of the non-torsion summands of the Heegaard Floer homology of M differs from that of its mutant. We …
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.
Floer homotopy type and eta invariants of Seifert $3$-manifolds fibering over $\mathbb{RP}^2$
研究Seifert有理同调3球面的Floer同调类型及eta不变量，证明其为L空间，且Floer同调类型为S^0的悬垂，同时计算Ozsváth-Szabó d-不变量及Seiberg-Witten δ-不变量。
In a previous paper we have constructed an invariant of four-dimensional manifolds with boundary in the form of an element in the stable homotopy group of the Seiberg-Witten Floer spectrum of the boundary. Here we prove that when one glues two four-manifolds along their boundaries, the Bauer-Furuta invariant of the resulting manifold is obtained by applying a natural pairing to the invariants of the pieces. As an application, we show that the connected sum of three copies of the K3 surface contains no exotic nuclei. In the process we also compute the Floer spectrum for several Seifert fibratio…
We survey the different versions of Floer homology that can be associated to three-manifolds. We also discuss their applications, particularly to questions about surgery, homology cobordism, and four-manifolds with boundary. We then describe Floer stable homotopy types, the related Pin(2)-equivariant Seiberg-Witten Floer homology, and its application to the triangulation conjecture.
We prove a surgery formula for the renormalized Euler characteristic of Ozsvath and Szabo. Equality between this Euler cahracteristic and the Seiberg-Witten invariant follows for rational homology three-spheres.
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.
Protocorks and monopole Floer homology
本文引入 protocorks 类流形，证明其扭转与 Seiberg-Witten 不变量变化及 Floer 同调分裂定理，改进 Morgan-Szabó 结果并推广约化 Floer 同调贡献的结论。
We define larger variants of the vector spaces one obtains by decategorifying bordered (sutured) Heegaard Floer invariants of surfaces. We also define bimodule structures on these larger spaces that are similar to, but more elaborate than, the bimodule structures that arise from decategorifying the higher actions in bordered Heegaard Floer theory introduced by Rouquier and the author. In particular, these new bimodule structures involve actions of both odd generators $E$ and $F$ of $\mathfrak{gl}(1|1)$, whereas the previous ones only involved actions of $E$. Over $\mathbb{F}_2$, we show that t…
We prove Witten's formula relating the Donaldson and Seiberg-Witten series modulo powers of degree $c+2$, with $c=-{1/4}(7χ+11σ)$, for four-manifolds obeying some mild conditions, where $χ$ and $σ$ are their Euler characteristic and signature. We use the moduli space of SO(3) monopoles as a cobordism between a link of the Donaldson moduli space of anti-self-dual SO(3) connections and links of the moduli spaces of Seiberg-Witten monopoles. Gluing techniques allow us to compute contributions from Seiberg-Witten moduli spaces lying in the first (or `one-bubble') level of the Uhlenbeck compactific…
It has been a central open problem in Heegaard Floer theory whether cobordisms of links induce homomorphisms on the associated link Floer homology groups. We provide an affirmative answer by introducing a natural notion of cobordism between sutured manifolds, and showing that such a cobordism induces a map on sutured Floer homology. This map is a common generalization of the hat version of the closed 3-manifold cobordism map in Heegaard Floer theory, and the contact gluing map defined by Honda, Kazez, and Matić. We show that sutured Floer homology, together with the above cobordism maps, forms…
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 prove that, like the Seiberg-Witten monopole homology, the Heegaard Floer homology for a three-manifold determines its Thurston norm. As a consequence, we show that knot Floer homology detects the genus of a knot. This leads to new proofs of certain results previously obtained using Seiberg-Witten monopole Floer homology (in collaboration with Kronheimer and Mrowka). It also leads to a purely Morse-theoretic interpretation of the genus of a knot. The method of proof shows that the canonical element of Heegaard Floer homology associated to a weakly symplectically fillable contact structure i…
Involutions, knots, and Floer K-theory
本文提出了一种针对 knots 和具有 involution 的 3-流形的 Seiberg--Witten Floer K-理论，并建立了 10/8 型不等式，用于研究 knots 的稳定数和相对 genus，同时检测了非光滑可延展的 involution。
We construct a variant of Floer homology groups and prove a gluing formula for a variant of Donaldson invariants. As a corollary, the variant of Donaldson invariants is non-trivial for connected sums of 4-manifolds which satisfy a condition for Donaldson invariants. We also show a non-existence result of compact, spin 4-manifolds with boundary some homology 3-spheres and with certain intersection forms.
We outline Hutchings's prescription that produces an ECH analog of Latschev and Wendl's algebraic $k$-torsion in the context of $ech$, a variant of ECH used in a proof of the isomorphism between Heegaard Floer and Seiberg-Witten Floer homologies; and we explain how it translates into Heegaard Floer homology.
Knot concordance invariants from Seiberg-Witten theory and slice genus bounds in 4-manifolds
本文利用等变Seiberg-Witten-Floer上同调构造了一族新的纽结同痕不变量 $θ^{(q)}(K)$，并证明它们能给出定4-流形中同调平凡光滑曲面的亏格下界，推广了 $ν^+(K)$ 的相关结果。
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