Selman Akbulut, Kouichi Yasui
We discuss corks, and introduce new objects which we call plugs. Though plugs are fundamentally different objects, they also detect exotic smooth structures in 4-manifolds like corks. We discuss relation between corks, plugs and rational blow-downs. We show how to detect corks and plugs inside of some exotic manifolds. Furthermore, we construct knotted corks and plugs.
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By using corks we construct diffeomorphic ribbon disks $D\subset B^{4}$, which are non-isotopic rel boundary to each other.
We construct an infinite family $\{ C_{n,k}\}_{k=1}^{\infty}$ of corks of Mazur type satisfying $2n\leq \mathrm{sc}^{\mathrm{sp}}(C_{n,k})\leq O(n^{3/2})$ for any positive integer $n$. Furthermore, using these corks, we construct an infinite family $\{(W_{n,k},W'_{n,k})\}_{k=1}^{\infty}$ of exotic pairs of $4$-manifolds with boundary whose special shadow-complexities satisfy the above inequalities. We also discuss exotic pairs with small shadow-complexity.
Exotic Surfaces in 4-manifolds and Surface Corks
本文引入曲面软木塞的概念，通过切除并重新粘合一个紧致可缩的零余维子流形来改变光滑嵌入曲面的微分同胚类型，并证明了其存在性及构造了首个横截曲面软木塞的例子。
We provide the first information on diffeotopy groups of exotic smoothings of R^4: For each of uncountably many smoothings, there are uncountably many isotopy classes of self-diffeomorphisms. We realize these by various explicit group actions. There are also actions at infinity by nonfinitely generated groups, for which no nontrivial element extends over the whole manifold. In contrast, every diffeomorphism of the end of the universal R^4 extends. Our techniques apply to many other open 4-manifolds, and are related to cork theory. We show that under broad hypotheses, cork twisting is equivalen…
Suppose that $X,X'$ are simply-connected closed exotic 4-manifolds. It is well-known that $X'$ is obtained by an order 2 cork twist of $X$. We give an infinite exotic family of 4-manifolds not generated by any infinite order cork. This is the first example admitting such a condition. We prove a necessary condition of 4-dimensional OS-invariants for a family to be generated by an infinite order cork and give non-contractible relatively exotic 4-manifolds that are never induced by any cork. Furthermore, we prove an estimate of the number of OS-invariants of 4-manifolds generated by a cork.
It is known that every exotic smooth structure on a simply connected closed 4-manifold is determined by a codimention zero compact contractible Stein submanifold and an involution on its boundary. Such a pair is called a cork. In this paper, we construct infinitely many knotted imbeddings of corks in 4-manifolds such that they induce infinitely many different exotic smooth structures. We also show that we can imbed an arbitrary finite number of corks disjointly into 4-manifolds, so that the corresponding involutions on the boundary of the contractible 4-manifolds give mutually different exotic…
Every exotic pair in 4-dimension is obtained each other by twisting a {\it cork} or {\it plug} which are codimension 0 submanifolds embedded in the 4-manifolds. The twist was an involution on the boundary of the submanifold. We define cork (or plug) with order $p\in {\Bbb N}\cup \{\infty\}$ and show there exists a plug with infinite order. Furthermore we show twisting $(P,φ^2)$ gives to enlargements of $P$ compact exotic manifolds with boundary.
The Akbulut cork is not universal
本文通过构造无法由 Akbulut-Yasui cork 族 W_n 联系的奇异单连通闭 4-流形对，证明 Akbulut cork 不具通用性，并说明带边界情形不存在 ∂-通用 cork。
We show that, for each integer n, there exist infinitely many pairs of n-framed knots representing homeomorphic but non-diffeomorphic (Stein) 4-manifolds, which are the simplest possible exotic 4-manifolds regarding handlebody structures. To produce these examples, we introduce a new description of cork twists and utilize satellite maps. As an application, we produce knots with the same 0-surgery which are not concordant for any orientations, disproving the Akbulut-Kirby conjecture given in 1978.
Corks for exotic diffeomorphisms
该研究证明了某些 exotic diffeomorphisms 可以通过 Corks 技术被同伦到可收缩子流形上，并探讨了其在 exotic diffeomorphisms 中的应用。
A cork is a smooth, contractible, oriented, compact 4-manifold $W$ together with a self-diffeomorphism $f$ of the boundary 3-manifold that cannot extend to a self-diffeomorphism of $W$; the cork is said to be strong if $f$ cannot extend to a self-diffeomorphism of any smooth integer homology ball bounded by $\partial W$. Surprising recent work of Dai, Hedden, and Mallick showed that most of the well-known corks in the literature are strong. We construct the first non-strong corks, which also give rise to new examples of absolutely exotic Mazur manifolds. Additionally we give the first examples…
From any 4-dimensional oriented handlebody X without 3- and 4-handles and with b_2>0, we construct arbitrary many compact Stein 4-manifolds which are mutually homeomorphic but not diffeomorphic to each other, so that their topological invariants (their fundamental groups, homology groups, boundary homology groups, and intersection forms) coincide with those of X. We also discuss the induced contact structures on their boundaries. Furthermore, for any smooth 4-manifold pair (Z,Y) such that the complement Z-intY is a handlebody without 3- and 4-handles and with b_2>0, we construct arbitrary many…
It is known that every compact Stein 4-manifolds can be embedded into a simply connected, minimal, closed, symplectic 4-manifold. By using this property, we discuss a new method of constructing corks. This method generates a large class of new corks including all the previously known ones. We prove that every one of these corks can knot infinitely many different ways in a closed smooth manifold, by showing that cork twisting along them gives different exotic smooth structures. We also give an example of infinitely many disjoint embeddings of a fixed cork into a non-compact 4-manifold which pro…
Minimal genus relative trisections of corks
本文证明了Akbulut软木塞的三截面亏格为3，构造了无穷多个此类软木塞，并给出了带边界4-流形三截面亏格的下界，以及异种对的低亏格相对三截面图。
Corks, exotic 4-manifolds and genus functions
本文证明了4维流形可通过修改获得无限多个奇异光滑结构，并展示了亏格函数的代数实现与稳定性，同时提出了亏格函数类型的概念及其应用。
Exotically knotted closed surfaces from Donaldson's diagonalization for families
本研究利用族版本Donaldson对角化定理，提出无需显式用4-流形不变量的奇异曲面检测法，构造出首个补空间微分同胚的不可定向奇异纽结曲面例子，并得到多类奇异嵌入的相关结果。
In the previous paper the author defined an infinite order plug $(P,φ)$ which gives rise to infinite Fintushel-Stern's knot-surgeries. Here, we give two 4-dimensional infinitely many exotic families $Y_n$, $Z_n$ of exotic enlargements of the plug. The families $Y_n$, $Z_n$ have $b_2=3$, $4$ and the boundaries are 3-manifolds with $b_1=1$, $0$ respectively. We give a plug (or g-cork) twist $(P,φ_{p,q})$ producing the 2-bridge knot or link surgery by combining the plug $(P,φ)$. As a further example, we describe a 4-dimensional twist $(M,μ)$ between knot-surgeries for two mutant knots. The twiste…
It is known that the only Stein filling of the standard contact structure on S^3 is B^4. In this paper, we construct simply connected exotic compact Stein 4-manifold pairs for any Betti number $b_2 \geq 1$; we do this by enlarging corks and plugs.
One stabilization is not enough for contractible 4-manifolds
该研究构造了与$S^2 \ imes S^2$取连通和后仍具异质性的cork例子，结合已有工作证明存在取该连通和后仍保持绝对异质性的可缩4维流形异质对，推进了4维流形稳定化相关研究。
From diffeomorphisms to exotic phenomena in small 4-manifolds
本文利用族Seiberg-Witten理论对4-流形微分同胚的约束，统一研究小型4-流形中的多种奇异现象，并给出检测非光滑同痕同胚的新方法和纽结相对亏格下界。
A short survey of exotic smooth structutes on 4-manifolds is given with a special emphasis on the corresponding cork structures. Along the way we discuss some of the more recent results in this direction, obtained jointly with R.Matveyev, B.Ozbagci, C.Karakurt and K.Yasui.
We show how to construct absolutely exotic smooth structures on compact 4-manifolds with boundary, including contractible manifolds. In particular, we prove that any compact smooth 4-manifold W with boundary that admits a relatively exotic structure contains a pair of codimension-zero submanifolds homotopy equivalent to W that are absolutely exotic copies of each other. In this context, {\em absolute} means that the exotic structure is not relative to a particular parameterization of the boundary. Our examples are constructed by modifying a relatively exotic manifold by adding an invertible ho…
We introduce a new generalization of Gompf nuclei and give applications. We construct infinitely many exotic smooth structures for a large class of compact 4-manifolds with boundary, regarding topological invariants. We prove that a large class of closed 3-manifolds (including disjoint unions of Stein fillable 3-manifolds) bound compact connected oriented 4-manifolds which admit infinitely many smooth structures. To detect smooth structures, we introduce a relative genus function. As a side result, we show that log transform and knot surgery of 4-manifolds do not admit any Stein structure, und…
On localizing groups of exotic diffeomorphisms of 4-manifolds
该研究围绕4-流形外徽微分同胚群的无限生成能否局部化到更小嵌入子流形的问题展开，通过Seiberg-Witten理论等得到多项相关结论，并指出4-流形映射类群也有类似结果。
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。
Practical Software for Triangulating and Simplifying 4-Manifolds
本文开发了从Kirby图构建4-流形三角剖分的软件工具和新的简化启发式方法，并据此给出了若干有界奇异对、cork、plug及K3曲面的更小三角剖分，揭示了4-流形三角剖分的精细结构特征。
We prove that the Dolgachev surface E(1)_{2,3} (which is an exotic copy of the elliptic surface E(1)=CP^2 # 9(-CP^2)) can be obtained from E(1) by twisting along a simple "plug", in particular it can be obtained from E(1) by twisting along an RP^2.
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