Frank Quinn
Withdrawn May 2005. There is an error in the even-dimensional case of the proof in the April 2005 version. The hoped-for 4-dimensional applications are unlikely to survive the repairs.
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We establish surgery formulas for filtration of the Heegaard Floer homology associated with p/q surgery on a null-homologous knot K in a three-manifold Y, induced by K_{p/q}. Here K_{p/q} is the core of the attached solid torus (which produces the surgery). This would generalize a result of Ozsvath and Szabo. We will re-prove that surgery on non-trivial knots can not produce S^3, as a corollary of a non-vanishing result for the knot Floer homology of K_{p/q}.
The aim of this paper is to show the importance of the Steenrod construction of homology theories for the disassembly process in surgery on a generalized $n$-manifold $X^n$, in order to produce an element of generalized homology theory, which is basic for calculations. In particular, we show how to construct an element of the $n$-th Steenrod homology group $H^{st}_{n} (X^{n}, \mathbb{L}^+),$ where $\mathbb{L}^+$ is the connected covering spectrum of the periodic surgery spectrum $\mathbb{L}$, avoiding the use of the geometric splitting procedure, which is standardly used in surgery on topologi…
Using Taubes' periodic ends theorem, Auckly gave examples of toroidal and hyperbolic irreducible integer homology spheres which are not surgery on a knot in the three-sphere. We give an obstruction to a homology sphere being surgery on a knot coming from Heegaard Floer homology. This is used to construct infinitely many small Seifert fibered examples.
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 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.
The results become part of an upcoming paper.
Surgery exact triangles in involutive Heegaard Floer homology
本文通过对合加倍模型建立对合Heegaard Floer同调的手术正合三角形，并由此得到Ozsváth-Szabó映射锥公式的对合版本，进而构造出不同调配边于Seifert纤维空间线性组合的整数同调球。
We show that if a positive integral surgery on a knot K inside a homology sphere X with Seifert genus g(K) results in an induced knot K_n in X_n(K)=Y which has simple Floer homology, we should have n>=2g(K). Moreover, if X is the standard sphere, the three-manifold Y is a L-space and the Heegaard Floer homology groups of K are determined by its Alexander polynomial.
Auckly gave two examples of irreducible integer homology spheres (one toroidal and one hyperbolic) which are not surgery on a knot in the three-sphere. Using Heegaard Floer homology, the authors and Karakurt provided infinitely many small Seifert fibered examples. In this note, we extend those results to give infinitely many hyperbolic examples, as well as infinitely many examples with arbitrary JSJ decomposition.
This paper presents an alternative approach to controlled surgery obstructions. The obstruction for a degree one normal map $(f,b): M^n \rightarrow X^n$ with control map $q: X^n \rightarrow B$ to complete controlled surgery is an element $σ^c (f, b) \in H_n (B, \mathbb{L})$, where $M^n, X^n$ are topological manifolds of dimension $n \geq 5$. Our proof uses essentially the geometrically defined $\mathbb{L}$-spectrum as described by Nicas (going back to Quinn) and some well known homotopy theory. We also outline the construction of the algebraically defined obstruction, and we explicitly describ…
Using the combinatorial approach to Heegaard Floer homology we obtain a relatively easy formula for computation of hat Heegaard Floer homology for the three-manifold obtained by rational surgery on a knot K inside a homology sphere Y.
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.
We prove a surgery formula of the Casson-Seiberg-Witten invariant of integral homology $S^1 \times S^3$ along an embedded torus, which could either be regarded as an extension of the product formula for Seiberg-Witten invariants or a manifestation of the surgery exact triangle in $4$-dimensional Seiberg-Witten theory of homology $S^1 \times S^3$. As an application, we compute this invariant for mapping tori of $3$-manifolds under diffeomorphisms of finite order and fixed-point set being a simple closed curve. This computation generalizes the result of Lin-Ruberman-Saveliev.
We provide related Dehn surgery descriptions for rational homology spheres and a class of their regular finite cyclic covering spaces. As an application, we use the surgery descriptions to relate the Casson invariants of the covering spaces to that of the base space. Finally, we show that this places restrictions on the number of finite and cyclic Dehn fillings of the knot complements in the covering spaces beyone those imposed by Culler-Gordon-Luecke-Shalen and Boyer-Zhang.
The cosmetic surgery conjecture is a longstanding conjecture in 3-manifold theory. We present a theorem about exceptional cosmetic surgery for homology spheres. Along the way we prove that if the surgery is not a small seifert $\mathbb{Z}/2\mathbb{Z}$-homology sphere or a toroidal irreducible non-Seifert surgery then there is at most one pair of exceptional truly cosmetic slope. We also prove that toroidal truly cosmetic surgeries on integer homology spheres must be integer homology spheres.
Surgery obstructions for knots in integer homology spheres
将Hom-Karakurt-Lidman关于S^3中纽结手术的Heegaard Floer同调障碍推广到所有整数同调球面Y的正负1/m手术，用于检验无穷多小Seifert纤维化和双曲例子，并推导光滑配边下b_2(W)的下界。
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…
We use topological surgery in dimension four to give sufficient conditions for the zero framed surgery manifold of a 3-component link to be homology cobordant to the 3-torus, which arises from zero framed surgery on the Borromean rings, via a topological homology cobordism preserving the homotopy classes of the meridians. This enables us to give new examples of 3-component links with unknotted components and vanishing pairwise linking numbers, such that any two of these links have homology cobordant zero surgeries in the above sense, but the zero surgery manifolds are not homeomorphic. Moreove…
We define a twisted version of Manolescu and Woodward's Symplectic Instanton homology, prove that this invariant fits into the framework of Wehrheim and Woodward's Floer Field theory, and describe its behaviour for connected sum and Dehn surgery.
Let $M_n$ be a homology 3-sphere obtained by $\frac1n$-Dehn surgery along a $(p,q)$-torus knot. We consider a polynomial $σ_{(p,q,n)}(t)$ whose zeros are the inverses of the Reideimeister torsion of $M_n$ for $\mathit{SL}(2;\mathbb{C})$-irreducible representations. We give an explicit formula of this polynomial by using Tchebychev polynomials of the first kind. Further we also give a 3-term relations of these polynomials.
When can surgery on a null-homologous knot K in a rational homology sphere produce a non-separating sphere? We use Heegaard Floer homology to give sufficient conditions for K to be unknotted. We also discuss some applications to homology cobordism, concordance, and Mazur manifolds.
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…
Filtered instanton homology and cosmetic surgery
该研究利用带Chern-Simons滤过的瞬子Floer同调排除了美容手术猜想中±1/n手术的可能，将猜想简化为亏格2平凡Alexander多项式纽结的±2手术情形，同时定义了新的三维流形不变量与瞬子同调手术关系，并推广到S²×S¹中的纽结。
Instantons, indefinite 4-manifolds, and Dehn surgery
该研究利用基于模2瞬子同调的Froyshov不变量$q_3$，证明存在Dehn手术数可任意大的双曲整数同调球，并将相关不等式扩展至有理同调球。
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…
Let K be a (2p,q)-torus knot and M_n is a 3-manifold obtained by 1/n-Dehn surgery along K. We consider a polynomial whose zeros are the inverses of the Reideimeister torsion of M_n for SL(2;C)-irreducible representations. Johnson gave a formula for the case of the (2,3)-torus knot under some modification and normalization. We generalize this formula by using Tchebychev polynomials.
Contact cosmetic surgery on Legendrian knots in integer homology sphere $L$-spaces
研究整数同调球L-空间中Legendrian纽结的接触整容手术，证明除可能的拉格朗日切片纽结外，所有非平凡Legendrian纽结均满足接触整容手术猜想。
Knot surgery formulae for instanton Floer homology I: the main theorem
本文针对3-流形Y中满足[K]=0∈H₁(Y;ℚ)且m≠0的纽结K，证明了带边瞬子同调Iᵛ(Yₘ(K))的整手术公式，其证明基于缝合瞬子同调SHI和导出范畴的八面体引理，还得到相关正合三角形及独立价值的新结果。
In this paper, we use Heegaard Floer homology to study reducible surgeries. In particular, suppose K is a non-cable knot in the three-sphere with an L-space surgery. If p-surgery on K is reducible, we show that p equals 2g(K)-1. This implies that any knot with an L-space surgery has at most one reducible surgery, a fact that we show additionally for any knot of genus at most two.
Let $K$ be a non-trivial knot in $S^3$, and let $r$ and $r'$ be two distinct rational numbers of same sign, allowing $r$ to be infinite; we prove that there is no orientation-preserving homeomorphism between the manifolds $S^3_r(K)$ and $S^3_{r'}(K)$. We further generalize this uniqueness result to knots in arbitrary integral homology L-spaces.
For any three-manifold presented as surgery on a framed link (L,Λ) in an integral homology sphere, Manolescu and Ozsváth construct a hypercube of chain complexes whose homology calculates the Heegaard Floer homology of Λ-framed surgery on Y. This carries a natural filtration that exists on any hypercube of chain complexes; we study the E_2 page of the associated spectral sequence, called Framed Floer homology. One purpose of this paper is to show that Framed Floer homology is an invariant of oriented framed links, but not an invariant of the surgered manifold. We discuss how this relates to an…
Rational surgery exact triangles in Heegaard Floer homology
本文提出了一种在Heegaard Floer同调中适用于所有正有理斜率的手术精确三角形构造方法，通过解决组合问题并利用Spin^c共轭对称性实现了扩展。
Surgery Exact Triangles in Instanton Theory
本文证明了联系纽结instanton Floer同调与手术instanton同调的正合三角形，这是整数系数下的首个此类结果，并据此导出Poincaré同调球在$\mathbb{Z}/2$-系数下非instanton $L$-空间。
For a 3-manifold with torus boundary admitting an appropriate involution, we show that Khovanov homology provides obstructions to certain exceptional Dehn fillings. For example, given a strongly invertible knot in S^3, we give obstructions to lens space surgeries, as well as obstructions to surgeries with finite fundamental group. These obstructions are based on homological width in Khovanov homology, and in the case of finite fundamental group depend on a calculation of the homological width for a family of Montesinos links.
By recent results of Baker--Etnyre--Van Horn-Morris, a rational open book decomposition defines a compatible contact structure. We show that the Heegaard Floer contact invariant of such a contact structure can be computed in terms of the knot Floer homology of its (rationally null-homologous) binding. We then use this description of contact invariants, together with a formula for the knot Floer homology of the core of a surgery solid torus, to show that certain manifolds obtained by surgeries on bindings of open books carry tight contact structures. Possible applications to lens space surgerie…
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…
In the paper \cite{wall_1}, C.T.C. Wall proved that two smooth closed simply connected 4-manifolds which are homeomorphic are in fact stably diffeomorphic. We prove a similar result which states that two smooth closed 4-manifolds satisfying certain properties are stably diffeomorphic if and only if their signatures agree. The manifolds in question are obtained by surgery on loops. The methods we use are modified surgery of Kreck \cite{kreck} and Kirby calculus.
A regular fiber of the Seifert fibering of the Poincaré homology sphere admits a Dehn surgery to $L(2,1)\#L(3,2)\#L(5,4)$. We prove that this is the only knot in the Poincaré homology sphere with a surgery to a connected sum of more than two lens spaces.
Matveev introduced Borromean surgery on 3-manifolds, and proved that the equivalence relation on closed, oriented 3-manifolds generated by Borromean surgeries is characterized by the first homology group and the torsion linking pairing. Massuyeau generalized this result to closed, spin 3-manifolds, and the second author to compact, oriented 3-manifolds with boundary. In this paper we give a partial generalization of these results to compact, spin 3-manifolds with boundary.
The Wall surgery obstruction groups have two interesting geometrically defined subgroups, consisting of the surgery obstructions between closed manifolds, and the inertial elements. We show that the inertia group $I_{n+1}(π,w)$ and the closed manifold subgroup $C_{n+1}(π,w)$ are equal in dimensions $n+1\geq 6$, for any finitely-presented group $π$ and any orientation character $w\colon π\to \cy 2$.
We introduce a homology surgery problem in dimension 3 which has the property that the vanishing of its algebraic obstruction leads to a canonical class of π-algebraically-split links in 3-manifolds with fundamental group π. Using this class of links, we define a theory of finite type invariants of 3-manifolds in such a way that invariants of degree 0 are precisely those of conventional algebraic topology and surgery theory. When finite type invariants are reformulated in terms of clovers, we deduce upper bounds for the number of invariants in terms of π-decorated trivalent graphs. We also con…
The Bryant-Ferry-Mio-Weinberger surgery exact sequence for high-dimensional compact ANR homology manifolds is used to obtain transversality, splitting and bordism results for homology manifolds, generalizing previous work of Johnston.
Insensitivity of Khovanov homology under rim surgery
研究\(B^4\)中光滑定向恰当嵌入曲面边缘手术对考夫曼同调的影响，通过局部环面替换操作，结合已有工作得出奇异曲面，利用考夫曼缠结千层饼模给出两个证明。
Under certain homological hypotheses on a compact 4-manifold, we prove exactness of the topological surgery sequence at the stably smoothable normal invariants. The main examples are the class of finite connected sums of 4-manifolds with certain product geometries. Most of these compact manifolds have non-vanishing second mod 2 homology and have fundamental groups of exponential growth, which are not known to be tractable by Freedman-Quinn topological surgery. Necessarily, the *-construction of certain non-smoothable homotopy equivalences requires surgery on topologically embedded 2-spheres an…
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 l…
An iterable surgery formula on involutive knot lattice homotopy
本文提出可迭代的纽结格点同调手术公式，生成对偶纽结的双重过滤空间及对合数据，并在无穷范畴中实现函子性，用于计算具体例子。
For a knot $K$ in a homology $3$-sphere $Σ$, let $M$ be the result of $2/q$-surgery on $K$, and let $X$ be the universal abelian covering of $M$. Our first theorem is that if the first homology of $X$ is finite cyclic and $M$ is a Seifert fibered space with $N\ge 3$ singular fibers, then $N\ge 4$ if and only if the first homology of the universal abelian covering of $X$ is infinite. Our second theorem is that under an appropriate assumption on the Alexander polynomial of $K$, if $M$ is a Seifert fibered space, then $q=\pm 1$ (i.e.\ integral surgery).
Knot surgery formulae for instanton Floer homology II: applications
本文通过整数手术公式研究了3流形的瞬子同调，揭示了几乎L空间结的纤维化和强准正性质，并确定了特定类型的结结构。
We first present three graphic surgery formulae for the degree $n$ part $Z_n$ of the Kontsevich-Kuperberg-Thurston universal finite type invariant of rational homology spheres. Each of these three formulae determines an alternate sum of the form $$\sum_{I \subset N} (-1)^{\sharp I}Z_n(M_I)$$ where $N$ is the set of components of a framed algebraically split link $L$ in a rational homology sphere $M$, and $M_I$ denotes the manifold resulting from the Dehn surgeries on the components of $I$. The first formula treats the case when $L$ is a boundary link with $n$ components, while the second one i…
Closed manifold surgery obstructions and the Oozing Conjecture
本文通过新例子反驳了渗出猜想，揭示了闭流形在高维中的非平凡手术障碍特性。
We give a Dehn surgery characterization of the trefoil and the figure eight knots. These results are gotten by combining surgery formulas in Heegaard Floer homology from an earlier paper with the characterization of these knots in terms of their knot Floer homology given in a recent paper of Ghiggini.
Kawauchi defined a group structure on the set of homology $S^1$$\times$$S^2$'s under an equivalence relation called $\widetilde{H}$-cobordism. This group receives a homomorphism from the knot concordance group, given by the operation of zero-surgery. It is natural to ask whether the zero-surgery homomorphism is injective. We show that this question has a negative answer in the smooth category. Indeed, using knot concordance invariants derived from knot Floer homology we show that the kernel of the zero-surgery homomorphism contains a $\mathbb{Z}^\infty$-subgroup.
Homology manifolds and euclidean bundles
构造一个庞加莱复形，其周期性全手术障碍消失但斯皮瓦克法丛不能约化为稳定欧几里得丛，以此反驳文献中关于相关庞加莱复形同伦等价于同调流形及同调流形斯皮瓦克法丛可约化的两个断言。
If a closed, orientable hyperbolic 3--manifold M has volume at most 1.22 then H_1(M;Z_p) has dimension at most 2 for every prime p not 2 or 7, and H_1(M;Z_2) and H_1(M;Z_7) have dimension at most 3. The proof combines several deep results about hyperbolic 3--manifolds. The strategy is to compare the volume of a tube about a shortest closed geodesic C in M with the volumes of tubes about short closed geodesics in a sequence of hyperbolic manifolds obtained from M by Dehn surgeries on C.
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.
On lens space surgeries from the Poincaré homology sphere
本文基于Greene的changemaker格发展格嵌入障碍，确定庞加莱同调球面上纽结手术产生透镜空间的条件，证明半整数手术仅有两个特例，并得出双曲纽结的透镜空间手术必为整数的结论。
We provide a proof of the controlled surgery sequence, including stability, in the special case that the local fundamental groups are trivial. Stability is a key ingredient in the construction of exotic homology manifolds by Bryant, Ferry, Mio and Weinberger, but no proof has been available. The development given here is based on work of M. Yamasaki.
Cosmetic surgeries on knots in homology spheres and the Casson--Walker invariant
针对同调球中纽结的美容手术问题，本文利用三维球内2分支链环的 Casson–Walker 不变量有理手术公式，给出纽结与手术斜率存在此类手术时的若干约束。
In this paper we define and investigate Z/2-homology cobordism invariants of Z/2-homology 3-spheres which turn out to be related to classical invariants of knots. As an application we show that many lens spaces have infinite order in the Z/2-homology cobordism group and we prove a lower bound for the slice genus of a knot on which integral surgery yields a given Z/2-homology sphere. We also give some new examples of 3-manifolds which cannot be obtained by integral surgery on a knot.
An explicit (-1)^n-quadratic form over Z[Z^{2n}] representing the surgery problem E_8 x T^{2n} is obtained, for use in the Bryant-Ferry-Mio-Weinberger construction of 2n-dimensional exotic homology manifolds.
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