Ronald Fintushel, Ronald J. Stern
For 5 <= k <= 8 we show that the infinite family of exotic smooth structures on CP^2# k(-CP^2) can be achieved by 1/n - surgeries on a single embedded nullhomologous torus in a manifold R_k which is homeomorphic to CP^2# k(-CP^2).
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By studying the example of smooth structures on CP^2#3(-CP^2) we illustrate how surgery on a single embedded nullhomologous torus can be utilized to change the symplectic structure, the Seiberg-Witten invariant, and hence the smooth structure on a 4-manifold.
We present a method for finding embedded nullhomologous tori in standard 4-manifolds which can be utilized to change their smooth structure. As an application, we show how to obtain infinite families of simply connected smooth 4-manifolds with b^+ = 1 and b^- = 2,...,7, via surgery on nullhomologous tori embedded in the standard manifolds CP^2 # k (-CP^2), k=2,...,7.
We define an simple invariant of an embedded nullhomologous Lagrangian torus and use this invariant to show that many symplectic 4-manifolds have infinitely many pairwise symplectically inequivalent nullhomologous Lagrangian tori. We further show that for a large class of examples that lambda(T) is actually a C-infinity invariant. In addition, this invariant is used to show that many symplectic 4-manifolds have nontrivial homology classes which are represented by infinitely many pairwise inequivalent Lagrangian tori, a result first proved by S Vidussi for the homotopy K3-surface obtained from …
We give an explicit formula to compute the rotation number of a nullhomologous Legendrian knot in contact (1/n)-surgery diagrams along Legendrian links and obtain a corresponding result for the self-linking number of transverse knots. Moreover, we extend the formula by Ding-Geiges-Stipsicz for computing the d3-invariant to (1/n)-surgeries.
Smoothly knotted and topologically unknotted nullhomologous surfaces in 4-manifolds
本文利用不等价光滑结构，在4-流形中构造无穷多光滑互异但拓扑同伦的零同调曲面，并指出其拓扑边界为柄体。
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 study finite type invariants of nullhomologous knots in a closed 3-manifold $M$ defined in terms of certain descending filtration $\{\mathscr{K}_n(M)\}_{n\geq 0}$ of the vector space $\mathscr{K}(M)$ spanned by isotopy classes of nullhomologous knots in $M$. The filtration $\{\mathscr{K}_n(M)\}_{n \geq 0}$ is defined by surgeries on special kinds of claspers in $M$ having one special leaf. More precisely, when $M$ is fibered over $S^1$ and $H_1(M)=\mathbb{Z}$, we study how far the natural surgery map from the space of $\mathbb{Q}[t^{\pm 1}]$-colored Jacobi diagrams on $S^1$ of degree $n$ to…
In this article we use the technique of Luttinger surgery to produce small examples of simply connected and non-simply connected minimal symplectic 4-manifolds. In particular, we construct: (1) An example of a minimal symplectic 4-manifold that is homeomorphic but not diffeomorphic to CP^2#3(-CP^2) which contains a symplectic surface of genus 2, trivial normal bundle, and simply connected complement and a disjoint nullhomologous Lagrangian torus with the fundamental group of the complement generated by one of the loops on the torus. (2) A minimal symplectic 4-manifold that is homeomorphic but …
We discuss the properties of a certain type of Dehn surgery along a Lagrangian torus in a symplectic 4-manifold, known as Luttinger's surgery, and use this construction to provide a purely topological interpretation of a non-isotopy result for symplectic plane curves with cusp and node singularities due to Moishezon.
Mixed tori in contact surgery diagrams
本研究提出将辛JSJ分解应用于三维切触流形切触手术图的图示框架，可推导原流形的恰当/弱辛填充，既重现已知分类结果，又给出大类配边三维流形的填充分类算法。
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 $S^4$.
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.
This short note presents a simple construction of nonisotopic symplectic tori representing the same primitive homology class in the symplectic 4-manifold E(1)_K, obtained by knot surgery on the rational elliptic surface E(1) with the left-handed trefoil knot K. E(1)_K has the simplest homotopy type among simply-connected symplectic 4-manifolds known to exhibit such property.
Bohr-Sommerfeld profile surgeries and Disk Potentials
本文提出一种通过切换两个精确填充Legendrians的手术操作（BSP手术），证明了在特定情况下该手术能保持Lagrangian的单调性，并通过盘势变化的墙穿越公式，构造了非平凡单调Lagrangian tori。
Let E(1)_K denote the closed 4-manifold that is homotopy equivalent (hence homeomorphic) to the rational elliptic surface E(1) and is obtained by performing Fintushel-Stern knot surgery on E(1) using a knot K in S^3. We construct an infinite family of homologous non-isotopic symplectic tori representing a primitive homology class in E(1)_K when K is any nontrivial fibred knot in S^3. We also show how these tori can be non-isotopically embedded as homologous symplectic submanifolds in other symplectic 4-manifolds.
Hyperbolic knots with a large toroidal surgery
本文构造了无限族双曲纽结，其例外手术产生含五个互不相交非平行不可压缩环面的图流形。
Transverse tori in Engel manifolds
研究Engel流形中环面的横结构，揭示其与接触3流形中结的类比关系，分类形式不变量并探讨其在同伦中的唯一性作用。
Let K \subset Y be a knot in a three manifold which admits a longitude-framed surgery such that the surgered manifold has first Betti number greater than that of Y. We find a formula which computes the twisted Floer homology of the surgered manifold, in terms of twisted knot Floer homology. Using this, we compute the twisted Heegaard Floer homology \underline{HF}^+ of the mapping torus of a diffeomorphism of a closed Riemann surface whose mapping class is periodic, giving an almost complete description of the structure of these groups. When the mapping class is nontrivial, we find in particula…
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.
We use surgery along 2-tori embedded in a union of two copies of a product of punctured 2-tori to produce a new collection of homotopy 4-spheres (4-manifolds homotopy equivalent to $S^4$ and hence homeomorphic to $S^4$ but possibly not diffeomorphic to $S^4$). It is still unknown if these new examples are in fact exotic.
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…
$\mathbb{Z}$-Torus Exteriors and Small Knot-Surgery Four-Manifolds
本文通过切割-粘贴操作替换环面邻域，构造了与$S^2\times S^2$和$\\#_3(S^2\times S^2)$同胚的单连通四流形及光滑同伦$4$-球面，并证明相关Seiberg-Witten不变量消失，所得流形非辛。
Let $M_λ$ be the $λ$-component Milnor link. For $λ\ge 3$, we determine completely when a finite slope surgery along $M_λ$ yields a lens space including $S^3$ and $S^1\times S^2$, where {\it finite slope surgery} implies that a surgery coefficient of every component is not $\infty$. For $λ=3$ (i.e.\ the Borromean rings), there are three infinite sequences of finite slope surgeries yielding lens spaces. For $λ\ge 4$, any finite slope surgery does not yield a lens space. As a corollary, $M_λ$ for $λ\ge 3$ does not yield both $S^3$ and $S^1\times S^2$ by any finite slope surgery. We generalize the…
In this short article we investigate the topology of the moduli space of two-convex embedded tori $S^{n-1}\times S^1\subset \mathbb{R}^{n+1}$. We prove that for $n \geq 3$ this moduli space is path-connected, and that for $n = 2$ the connected components of the moduli space are in bijective correspondence with the knot classes associated to the embeddings. Our proof uses a variant of mean curvature flow with surgery developed in our earlier article (arXiv:1607.05604) where neck regions are deformed to tiny strings instead of being cut out completely, an approach which preserves the global topo…
Taut foliations, left-orders, and pseudo-Anosov mapping tori
本文通过构造基本群在实数上的保向同胚作用来刻画紧叶理3-流形的横向几何，并证明八字结所有非平凡手术的基本群左可序，同时将该方法应用于 Hodgson-Weeks 普查中 44.7% 的非 L-空间有理同调球。
Surgery and Excision for Furuta-Ohta invariants on Homology $S^1 \times S^3$
本文为同调 $S^1 \times S^3$ 上的 Furuta-Ohta 不变量建立了手术公式与切除公式，并将其应用于计算映射环面族的不变量，同时完整刻画了相关柱形末端 $4$-流形上的零次 ASD 瞬子模空间。
Bordered manifolds with torus boundary and the link surgery formula
本文利用链环手术公式发展带边$\text{HF}^-$理论，通过结合代数$\text{K}$上的$D$型模解释链环手术复形，证明连通和公式并计算多类例子，可计算特定3流形的Heegaard Floer同调，后续将验证格同调猜想。
A knot K is called n-adjacent to another knot K', if K admits a projection containing n generalized crossings such that changing any 0 < m \leq n of them yields a projection of K'. We apply techniques from the theory of sutured 3-manifolds, Dehn surgery and the theory of geometric structures of 3-manifolds to answer the question of the extent to which non-isotopic knots can be adjacent to each other. A consequence of our main result is that if K is n-adjacent to K' for all n, then K and K' are isotopic. This provides a partial verification of the conjecture of V. Vassiliev that the finite type…
Tight surgeries on torus knots
本文对沿任意环面上纽结手术所得流形上的紧接触结构完成分类，还分类了相关曲面丛的接触结构，研究其辛可填充性并反驳了一项相关猜想。
We prove that for any V>0, there exist a hyperbolic manifold M_V, so that Vol(M_V) < 2.03 and LinVol(M_V) > V. The proof requires study of cosmetic surgery on links (equivalently, fillings of manifolds with boundary tori). There is no bound on the number of components of the link (or boundary components). For statements, see the second part of the introduction. Here are two examples of the results we obtain: 1) Let K be a component of a link L in S^3. Then "most" slopes on K cannot be completed to a cosmetic surgery on L, unless K becomes a component of a Hopf link. 2) Let X be a manifold and …
We develop a calculus of surgery data, called bridged links, which involves besides links also pairs of balls that describe one-handle attachements. As opposed to the usual link calculi of Kirby and others this description uses only elementary, local moves(namely modifications and isolated cancellations), and it is valid also on non-simply connected and disconnected manifolds. In particular, it allows us to give a presentation of a 3-manifold by doing surgery on any other 3-manifold with the same boundary. Bridged link presentations on unions of handlebodies are used to give a Cerf-theoretical…
Braided Surfaces and Small Exotic 4-Manifolds
本文证明纤维化椭圆对合给出零手术结与环面乘积的二重分支覆盖，计算编织词，并推广至亏格二纤维化结，同时给出扩展障碍。
On nonorientable $4$--manifolds
本文研究了非定向4维流形的结构性结果，包括简化断裂Lefschetz纤维和三重分割的分类，以及通过链环手术构造4维流形的方法。
Round handles are affiliated with smooth 4-manifolds in two major ways: 5-dimensional round handles appear extensively as the building blocks in cobordisms between 4-manifolds, whereas 4-dimensional round handles are the building blocks of broken Lefschetz fibrations on them. The purpose of this article is to shed more light on these interactions. We prove that if X and X' are cobordant closed smooth 4-manifolds with the same euler characteristics, and if one of them is simply-connected, then there is a cobordism between them which is composed of round 2-handles only, and therefore one can pas…
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