A survey of 4-manifolds through the eyes of surgery

On dissolving knot surgery $4$-manifolds under a $\mathbb{CP}^2$-connected sum

In this article we prove that, if $X$ is a smooth $4$-manifold containing an embedded double node neighborhood, all knot surgery $4$-manifolds $X_K$ are mutually diffeomorphic to each other after a connected sum with $\mathbb{CP}^2$. Hence, by applying to the simply connected elliptic surface $E(n)$, we also show that every knot surgery $4$-manifold $E(n)_K$ is almost completely decomposable.

Surgery on closed 4-manifolds with free fundamental group

Even though the disk embedding theorem is not available in dimension 4 for free fundamental groups, some surgery problems may be shown to have topological solutions. We prove that surgery problems may be solved if one considers closed 4-manifolds and the intersection pairing is extended from the integers, and prove a related splitting result.

Surgery Description of Orientation-Preserving $D_{2p}$-Actions on Compact Orientable 3-Manifolds

This paper has been withdrawn because its contents have become subsumed in Section 5.2 of arXiv:0805.2307.

Reidemeister torsion of a 3-manifold obtained by a Dehn-surgery along the figure-eight knot

Let M be a closed 3-manifold obtained by Dehn surgery along the figure-eight knot. We give a formula of the Reisdemeisiter torsion of M for any irreducible SL(2;C)-representation. It is described as a rational expression of the trace of the image of the meridian.

Complexity of surgery manifolds and Cheeger-Gromov invariants

We present new lower bounds on the complexity of Dehn surgery manifolds of knots, using our recent result on the Cheeger-Gromov rho invariants and triangulations. As an application, we give explicit examples of closed hyperbolic 3-manifolds with fixed first homology for which the gap between the Gromov norm and the complexity is arbitrarily large.

Lefschetz fibration structures on knot surgery 4-manifolds

In this article we study Lefschetz fibration structures on knot surgery 4-manifolds obtained from an elliptic surface E(2) using Kanenobu knots $K$. As a result, we get an infinite family of simply connected mutually diffeomorphic 4-manifolds coming from a pair of inequivalent Kanenobu knots. We also obtain an infinite family of simply connected symplectic 4-manifolds, each of which admits more than one inequivalent Lefschetz fibration structures of the same generic fiber.

通过手术探讨3流形的并行性

Parallelisability of 3-manifolds via surgery

本文通过手术方法证明所有闭合可定向3流形均可平行化，并结合接触几何与Kaplan的改进补充了相关研究。

关于三维流形的框架链环上的圆手术及其图

On Round Surgery on Framed Links and Their Diagrams for 3-manifolds

本文提出三维流形中框架链环上的圆手术方法，建立圆手术图与Dehn手术图的对应，证明Lickorish-Wallace定理的圆手术类比，并建立圆版本Kirby演算，应用于构造标准Kirby图及证明切触叶状结构的存在。

Lefschetz fibrations on knot surgery $4$-manifolds via Stallings twist

In this article we construct a family of knot surgery $4$-manifolds admitting arbitrarily many nonisomorphic Lefschetz fibration structures with the same genus fiber. We obtain such families by performing knot surgery on an elliptic surface $E(2)$ using connected sums of fibered knots obtained by Stallings twist from a slice knot $3_1 \sharp 3^*_1$. By comparing their monodromy groups induced from the corresponding monodromy factorizations, we show that they admit mutually nonisomorphic Lefschetz fibration structures.

Round surgery and contact structures on 3-manifolds

Contact round surgery of contact 3-manifolds is introduced in this paper. By using this method, an alternative proof of the existence of a contact structure on any closed orientable 3-manifold is given. It is also proved that any contact structure on any closed orientable 3-manifold is constructed from the standard contact structure on the 3-dimensional sphere by contact round surgeries. For the proof, important operations in contact topology, the Lutz twist and the Giroux torsion, are described in terms of contact round surgeries.

Surgery and involutions on 4-manifolds

We prove that the canonical 4-dimensional surgery problems can be solved after passing to a double cover. This contrasts the long-standing conjecture about the validity of the topological surgery theorem for arbitrary fundamental groups (without passing to a cover). As a corollary, the surgery conjecture is reformulated in terms of the existence of free involutions on a certain class of 4-manifolds. We consider this question and analyze its relation to the A,B-slice problem.

Concordance surgery and the Ozsváth-Szabó 4-manifold invariant

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.

Irreducible 3-manifolds that cannot be obtained by 0-surgery on a knot

We give two infinite families of examples of closed, orientable, irreducible 3-manifolds $M$ such that $b_1(M)=1$ and $π_1(M)$ has weight 1, but $M$ is not the result of Dehn surgery along a knot in the 3-sphere. This answers a question of Aschenbrenner, Friedl and Wilton, and provides the first examples of irreducible manifolds with $b_1=1$ that are known not to be surgery on a knot in the 3-sphere. One family consists of Seifert fibered 3-manifolds, while each member of the other family is not even homology cobordant to any Seifert fibered 3-manifold. None of our examples are homology cobord…

Only finitely many alternating knots can yield a given manifold by surgery

We show that given a 3-manifold $Y$ there is only a finite number of alternating knots $K \subset S^3$ such that $Y$ can be obtained by surgery on $K$. A very similar but somewhat not complete statement has been obtained in a recent preprint of Lackenby and Purcell.

3-manifolds Modulo Surgery Triangles

Surgery triangles are an important computational tool in Floer homology. Given a connected oriented surface $Σ$, we consider the abelian group $K(Σ)$ generated by bordered 3-manifolds with boundary $Σ$, modulo the relation that the three manifolds involved in any surgery triangle sum to zero. We show that $K(Σ)$ is a finitely generated free abelian group and compute its rank. We also construct an explicit basis and show that it generates all bordered 3-manifolds in a certain stronger sense. Our basis is strictly contained in another finite generating set which was constructed previously by Bal…

关于手术三维流形的纤维性

On the fiberedness of surgery 3-manifolds

研究闭可定向三维流形中沿纽结的Dehn手术，证明除稀疏斜率外，手术所得流形非曲面纤维化。

通过手术表示法在3流形中构造Seifert表面的算法

An algorithm for Seifert surfaces in 3-manifolds via surgery presentations

本文提出了一种通过手术表示法在任意3流形中构造无界链接Seifert表面的算法，扩展了经典的Seifert算法。

Stable classification of 4-manifolds obtained by the surgery on loops

This paper is concerned with the problem of stable diffeomorphism classification of 4-manifolds obtained using the surgery on loops. The main theorem states that under the assumption that the normal 1-type of two 4-manifolds in question is the same, the only classifying invariant is the signature. In particular, in some cases, any two closed smooth 4-manifolds with a given fundamental group, obtained by the standard construction, are stably diffeomorphic.

Surgery on links with unknotted components and three-manifolds

It is shown that any closed three-manifold M obtained by integral surgery on a knot in the three-sphere can always be constructed from integral surgeries on a 3-component link L with each component being an unknot in the three-sphere. It is also interesting to notice that infinitely many different integral surgeries on the same link L could give the same three-manifold M.

沿亏格为一的经典扭结进行0-手术所得3-流形的JSJ分解

The JSJ-decomposition of the 3-manifold obtained by 0-surgery along a classical pretzel knot of genus one

本文研究沿亏格一经典扭结作0-手术所得3-流形的JSJ分解，并借助最小扭曲五链环异常填充的分类结果进行分析。

On surgery along Brunnian links in 3-manifolds

We consider surgery moves along (n+1)-component Brunnian links in compact connected oriented 3-manifolds, where the framing of the each component is 1/k for k in Z. We show that no finite type invariant of degree < 2n-2 can detect such a surgery move. The case of two link-homotopic Brunnian links is also considered. We relate finite type invariants of integral homology spheres obtained by such operations to Goussarov-Vassiliev invariants of Brunnian links.

Maps on 3-manifolds given by surgery

Suppose that the 3-manifold M is given by integral surgery along a link L in S^3. In the following we construct a stable map from M to the plane, whose singular set is canonically oriented. We obtain upper bounds for the minimal numbers of crossings and non-simple singularities and of connected components of fibers of stable maps from M to the plane in terms of properties of L.

Dehn Surgery Equivalence Relations on Three-Manifolds

When can one 3-manifold be transformed to another by a finite sequence of Dehn surgeries which are restricted to preserve the first homology of the manifolds ? What is the resulting equivalence relation on 3-manifolds ? What if the surgery circle is further restricted to lie more deeply in the lower central series of the fundamental group ? We answer these questions. It is shown that many of these questions have answers in terms of classical toplogical invariants. Relations with Heegard splittings and the Torelli group are discussed. This is also related to whether or not one 3-manifold may be…

3-manifolds and 4-dimensional surgery

Let $X$ be a connected compact 3-manifold with non-empty boundary. Consider the boundary $M$ of $X\times D^2$. $M$ is a 4-dimensional closed manifold and has the same fundamental group as $X$. Various examples of $X$ are known for which a certain assembly map $A:H_4(X;L)\to L_4(π_1(X))$ is injective. For such an $X$ and any CW-spine $B$ of $X$, there is a $UV^1$-map $p:M\to B$. For any $ε>0$, if the surgery obstruction for a TOP normal map $(f,b):N\to M$ vanishes, we can perform surgery on $f$ to change it into a $p^{-1}(ε)$-controlled homotopy equivalence.

Obstructing finite surgery

For a fixed p, there are only finitely many elliptic 3-manifolds given by p/q-surgery on a knot in S^3. We prove this result by using the Heegaard Floer correction terms (d-invariants) to obstruct elliptic manifolds from arising as knot surgery.

Combinatorial Dehn surgery on cubed and Haken 3-manifolds

A combinatorial condition is obtained for when immersed or embedded incompressible surfaces in compact 3-manifolds with tori boundary components remain incompressible after Dehn surgery. A combinatorial characterisation of hierarchies is described. A new proof is given of the topological rigidity theorem of Hass and Scott for 3-manifolds containing immersed incompressible surfaces, as found in cubings of non-positive curvature.

通过德恩手术得到双曲三维流形的一般性

Genericity of hyperbolic 3-manifolds via Dehn surgery

研究如何描述链环和德恩手术性质，引入计数模型，证明在该模型下随机链环和随机三维流形是双曲的。

Surfaces in 4-manifolds and the surgery conjecture

We give a survey of geometric approaches to the topological 4-dimensional surgery and 5-dimensional s-cobordism conjectures, with a focus on the study of surfaces in 4-manifolds. The geometric lemma underlying these conjectures is a statement about smooth immersions of disks and of certain 2-complexes, capped gropes, in a 4-manifold. We also mention a reformulation in terms of the A,B-slice problem, and the relation of this question to recent developments in the study of the classical knot concordance group.

Exceptional surgery curves in triangulated 3-manifolds

For the purposes of this paper, Dehn surgery along a curve K in a 3-manifold M with slope r is `exceptional' if the resulting 3-manifold M_K(r) is reducible or a solid torus, or the core of the surgery solid torus has finite order in the fundamental group of M_K(r). We show that, providing the exterior of K is irreducible and atoroidal, and the distance between r and the meridian slope is more than one, and a homology condition is satisfied, then there is (up to ambient isotopy) only a finite number of such exceptional surgery curves in a given compact orientable 3-manifold M, with the boundar…

Non-concordant links with homology cobordant zero framed surgery manifolds

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…

Knotted surfaces in 4-manifolds by Knot surgery and Stabilization

Given a simply-connected closed 4-manifold $X$ and a smoothly embedded oriented surface $Σ$, various constructions based on Fintushel-Stern knot surgery have produced new surfaces in $X$ that are pairwise homeomorphic to $Σ$, but not diffeomorphic. We prove that for all known examples of surface knots constructed from knot surgery operations that preserve the fundamental group of the complement of surface knots, they become pairwise diffeomorphic after stabilizing by connected summing with one $S^2\tilde{\times}S^2$. When $X$ is spin, we show in addition that any surfaces obtained by a knot su…

Degree-one maps, surgery and four-manifolds

We give a description of degree-one maps between closed, oriented 3-manifolds in terms of surgery. Namely, we show that there is a degree-one map from a closed, oriented 3-manifold $M$ to a closed, oriented 3-manifold $N$ if and only if $M$ can be obtained from $N$ by surgery about a link in $N$ each of whose components is an unknot. We use this to interpret the existence of degree-one maps between closed 3-manifolds in terms of smooth 4-manifolds. More precisely, we show that there is a degree-one map from $M$ to $N$ if and only if there is a smooth embedding of $M$ in $W=(N\times I)#_n \bar{…

Dehn surgery and negatively curved 3-manifolds

We show that, for any given 3-manifold M, there are at most finitely many hyperbolic knots K in the 3-sphere and fractions p/q (with q > 22), such that M is obtained by p/q surgery along K. This is a corollary of the following result. If M is obtained by Dehn filling the cusps of a hyperbolic 3-manifold X, where each filling slope has length more than 2 π+ ε, then, for any given M and ε> 0, there are only finitely many possibilities for X and for the filling slopes. In this paper, we also investigate the length of boundary slopes, and sequences of negatively curved metrics on a given 3-manifol…

Exceptional Dehn surgery on large arborescent knots

A Dehn surgery on a knot $K$ in $S^3$ is exceptional if it produces a reducible, toroidal or Seifert fibred manifold. It is known that a large arborescent knot admits no such surgery unless it is a type II arborescent knot. The main theorem of this paper shows that up to isotopy there are exactly three large arborescent knots admitting exceptional surgery, each of which admits exactly one exceptional surgery, producing a toroidal manifold.

Complexities of 3-manifolds from triangulations, Heegaard splittings, and surgery presentations

We study complexities of 3-manifolds defined from triangulations, Heegaard splittings, and surgery presentations. We show that these complexities are related by linear inequalities, by presenting explicit geometric constructions. We also show that our linear inequalities are asymptotically optimal. Our results are used in [arXiv:1405.1805] to estimate Cheeger-Gromov $L^2$ $ρ$-invariants in terms of geometric group theoretic and knot theoretic data.

结手术4-流形E(n)_K不包含1-和3-柄

Knot surgery $4$-manifolds $E(n)_K$ without $1$- and $3$-handles

本文研究了通过结手术构造的4-流形E(n)_K，并证明其可分解为不包含1-和3-柄的结构，适用于特定类型的结。

Seiberg-Witten不变量的手术公式

A surgery formula for Seiberg-Witten invariants

本文证明了$b_1=1$的光滑$4$-流形普通Seiberg-Witten不变量的手术公式，并展示了其在寻找奇异光滑结构和嵌入曲面亏格下界中的应用。

Surgery on tori in the 4-sphere

We investigate the operation of torus surgery on tori embedded in $S^4$. Key questions include which 4-manifolds can be obtained in this way, and the uniqueness of such descriptions. As an application we construct embeddings of 3-manifolds into 4-manifolds by viewing Dehn surgery as a cross section of a surgery on a surface. In particular, we give new embeddings of homology spheres into $S^4$.

Coisotropic Luttinger surgery and some new symplectic 6-manifolds with vanishing canonical class

We introduce a surgery operation on symplectic manifolds called coisotropic Luttinger surgery, which generalizes Luttinger surgery on Lagrangian tori in symplectic 4-manifolds. We use it to produce infinitely many distinct symplectic non-Kahler 6-manifolds $X$ with $c_1(X)=0$ which are not of the form $M\times F$ for $M$ a symplectic 4-manifold and $F$ a closed surface.

A note on knot surgery

In this paper we clarify an issue in the knot surgery construction of Fintushel and Stern. Using knot surgery, they construct an infinite number of smooth structures on 4-manifolds satisfying certain conditions, but they do not explicitly work out the circumstances under which two manifolds that arise from their construction will fail to be diffeomorphic on the grounds of Seiberg-Witten theory. This paper fills in that gap.

Braid equivalence in 3-manifolds with rational surgery description

In this paper we describe braid equivalence for knots and links in a 3-manifold $M$ obtained by rational surgery along a framed link in $S^3$. We first prove a sharpened version of the Reidemeister theorem for links in $M$. We then give geometric formulations of the braid equivalence via mixed braids in $S^3$ using the $L$-moves and the braid band moves. We finally give algebraic formulations in terms of the mixed braid groups $B_{m,n}$ using cabling and the techniques of parting and combing for mixed braids. We also provide concrete formuli of the braid equivalence in the case where $M$ is a …

稳定微分同胚流形与修正手术障碍的实现

Stably diffeomorphic manifolds and the realisation of modified surgery obstructions

本文构造了无限多个稳定微分同胚但不同伦等价的 $4k$ 维流形，并通过证明 Kreck 修正手术幺半群子集的实现结果克服了修正手术障碍。

A Surgery Theory for Manifolds of Bounded Geometry

A surgery classification theory is introduced for manifolds of bounded geometry up to quasi-isometry. The Borel conjecture for this theory is proven for flat Euclidean space.

Heegaard diagrams and surgery descriptions for twisted face-pairing 3-manifolds

The twisted face-pairing construction of our earlier papers gives an efficient way of generating, mechanically and with little effort, myriads of relatively simple face-pairing descriptions of interesting closed 3-manifolds. The corresponding description in terms of surgery, or Dehn-filling, reveals the twist construction as a carefully organized surgery on a link. In this paper, we work out the relationship between the twisted face-pairing description of closed 3-manifolds and the more common descriptions by surgery and Heegaard diagrams. We show that all Heegaard diagrams have a natural deco…

3维流形的手术等价关系

Surgery equivalence relations for 3-manifolds

本文综述通过限制手术类型在3维流形上定义的多种等价关系，阐明曲面映射类群滤过在其定义与比较中的作用，并介绍有限型等不变量的刻画方法。

Can Dehn surgery yield three connected summands?

A consequence of the Cabling Conjecture of Gonzalez-Acuña and Short is that Dehn surgery on a knot in $S^3$ cannot produce a manifold with more than two connected summands. In the event that some Dehn surgery produces a manifold with three or more connected summands, then the surgery parameter is bounded in terms of the bridge number by a result of Sayari. Here this bound is sharpened, providing further evidence in favour of the Cabling Conjecture.

Seiberg-Witten不变量与族Seiberg-Witten不变量的配边公式

Surgery formulas for Seiberg-Witten invariants and family Seiberg-Witten invariants

本文证明了普通及族Seiberg-Witten不变量的配边公式，用配置空间上同调类切割模空间表达配边后流形的不变量，并借此研究配边如何保持或产生奇异现象。

接触手术距离

Contact surgery distance

本文研究了接触3流形的接触手术距离，证明其不超过底层光滑流形拓扑手术距离加5，并分类了特定的有理同调3球面。

Surgery formula for torsions and Seiberg-Witten invariants of 3-manifolds

We give a surgery formula for the torsions and Seiberg-Witten invariants associated with $Spin^c$-structures on 3-manifolds. We use the technique of Reidemeister-type torsions and their refinements.

沿纽结的唯一手术描述

Unique Surgery Descriptions along Knots

本文证明非平凡纽结存在无穷多r-手术具唯一纽结手术描述，双曲L-空间纽结的无穷多整数斜率手术流形亦然，还构造了具多个不同手术描述的流形族，推广了特征斜率概念。

Hurwitz-type bound, knot surgery, and smooth $\s^1$-four-manifolds

In this paper we prove several related results concerning smooth $\Z_p$ or $\s^1$ actions on 4-manifolds. We show that there exists an infinite sequence of smooth 4-manifolds $X_n$, $n\geq 2$, which have the same integral homology and intersection form and the same Seiberg-Witten invariant, such that each $X_n$ supports no smooth $\s^1$-actions but admits a smooth $\Z_n$-action. In order to construct such manifolds, we devise a method for annihilating smooth $\s^1$-actions on 4-manifolds using Fintushel-Stern knot surgery, and apply it to the Kodaira-Thurston manifold in an equivariant setting…

Systoles and Dehn surgery for hyperbolic 3-manifolds

Given a closed hyperbolic 3-manifold M of volume V, and a link L in M such that the complement M \ L is hyperbolic, we establish a bound for the systole length of M \ L in terms of V. This extends a result of Adams and Reid, who showed that in the case that M is not hyperbolic, there is a universal bound of 7.35534... . As part of the proof, we establish a bound for the systole length of a non-compact finite volume hyperbolic manifold which grows asymptotically like (4/3)log(V).

Contact surgery and supporting open books

Let $(M,ξ)$ be a contact 3-manifold. We present two new algorithms, the first of which converts an open book $(Σ,Φ)$ supporting $(M,ξ)$ with connected binding into a contact surgery diagram. The second turns a contact surgery diagram for $(M,ξ)$ into a supporting open book decomposition. These constructions lead to a refinement of a result of Ding-Geiges, which states that every such $(M,ξ)$ may be obtained by contact surgery from $(S^{3},ξ_{std})$, as well as bounds on the support norm and genus of contact manifolds obtained by surgery in terms of classical link data. We then introduce Kirby …

Knot surgery and Scharlemann manifolds

We discuss the relation between Fintushel-Stern knot surgery operation on 4-manifolds and Scharlemann manifolds, and as a corollary show that they all are standard. Along the way we show that the fishtail can exotically knot in the 4-sphere infinitely many ways.