Anar Akhmedov
In this article, we construct the first example of a simply connected minimal symplectic 4-manifold homeomorphic but not diffeomorphic to 3CP^2#7CP^2b. We also construct the first exotic symplectic structure on CP^2#5CP^2b.
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In a recent paper, Park constructs certain exotic simply-connected four-manifolds with small Euler characteristics. Our aim here is to prove that the four-manifolds in his constructions are minimal.
On absolutely exotic diffeomorphisms of 4-manifolds
本文证明存在无穷多个可缩紧致光滑4-流形具有无限阶绝对奇异微分同胚，并通过从相对奇异微分同胚构造绝对奇异微分同胚的定理实现，类比Akbulut-Ruberman定理。
New constructions and invariants of closed exotic 4-manifolds
该研究借助Heegaard Floer同调提出可在覆盖中保持区分性的新型闭四维流形不变量，并构造多类闭异质4-流形。
We produce infinite families of exotic actions of finite cyclic groups on simply connected smooth 4-manifolds with nontrivial Seiberg-Witten invariants.
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 study the possibility of realizing exotic smooth structures on punctured simply connected $4$-manifolds as leaves of a codimension one foliation on a compact manifold. In particular, we show the existence of uncountably many smooth open $4$-manifolds which are not diffeomorphic to any leaf of a codimension one transversely $C^{2}$ foliation on a compact manifold. These examples include some exotic ${\mathbb R}^4$'s and exotic cylinders $S^3\times{\mathbb R}$.
We construct an invariant of open four-manifolds using the Heegaard Floer theory of Ozsvath and Szabo. We show that there is a manifold X homeomorphic to R^4 for which the invariant is non-trivial, showing that X is an exotic R^4.
Exotic structures on 4-manifolds with infinite dihedral fundamental group
本文研究了具有无限二面体基群的4维闭定向流形上的无限exotic结构，假设b2+和b2-至少为12。
This article intends to provide an introduction to the construction of small exotic 4-manifolds. Some of the necessary background is covered. An exposition is given of J. Park's construction in arXiv:math.GT/0311395 of an exotic CP^2#7(-CP^2). This article does not intend to present any new results. It was originally a Master's thesis, and its aim is merely to provide a leisurely introduction to exotic 4-manifolds that might be of use to interested graduate students.
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.
Exotic 4-manifolds with small trisection genus
本研究证明了存在三分亏格为4的带边4-流形的异乎寻常对，并为Akbulut-Kirby无限族可缩4-流形构造了亏格3的相对三分，推进了低维拓扑中4-流形三分结构的研究。
Irreducible 4-manifolds can admit exotic diffeomorphisms
本文证明了多种极小复曲面容许奇异微分同胚，提供了不可约4-流形奇异微分同胚的首批已知实例，并给出了自旋4-流形上边界Dehn扭转非平凡的充分条件。
Exotic definite four-manifolds with non-trivial fundamental group
本文受Levine-Lidman-Piccirillo结果启发，在基本群为Z/2Z且具有定交形式的闭四维流形上构造了无穷多个奇异光滑结构，并推广到更多含偶数b_2^+的情形。
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。
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.
In a small simply-connected closed 4-manifold, we construct infinitely many pairs of exotic codimension-$1$ submanifolds with diffeomorphic complements that remain exotic after any number of stabilizations by $ S^2 \times S^2$. We also give new constructions of exotic embeddings of 3-spheres in 4-manifolds with diffeomorphic complements.
Exotic definite four-manifolds with non-cyclic fundamental group
本文在基本群为 $\mathbb{Z}/2\times \mathbb{Z}/2$ 的定四维流形上构造了无穷多个两两非微分同胚的光滑结构，揭示了该流形上光滑结构的丰富性。
We show examples of pairs of smooth, compact, homeomorphic 4-manifolds, whose diffeomorphism types are distinguished by the topology of the singular sets of smooth stable maps defined on them. In this distinction we rely on results from Seiberg-Witten theory.
Some examples of small irreducible exotic 4-manifolds with free abelian fundamental group
本研究构造了若干秩小于3、带非平凡自由阿贝尔基本群且欧拉示性数小的两两非同胚闭不可约奇异4-流形，提出了计算环面手术所得4-流形等变相交形式的显式机制，并给出了任意基本群奇异光滑结构的新稳定化结果。
Links of Mazur manifolds and exotica
本文在简单4-流形中研究Mazur流形的链环，构造了S^4中的非平凡2分支链环，并利用它们得到#^n CP^2中拓扑可分解但光滑不可分解的链环，进而导出带边界的单连通确定4-流形的奇异对以及S^4中多种Mazur流形的奇异嵌入。
Exotic diffeomorphisms on closed aspherical $4$-manifolds
该研究通过Davis反射技巧构造闭非球面4-流形的微分同胚，得到拓扑同痕于恒等但光滑不同痕于恒等的首批此类流形例子，丰富了微分拓扑相关研究。
We construct potentially new manifolds homeomorphic but not diffeomorphic to $\mathbb{CP}^{2} \# 8 \overline{\mathbb{CP}^{2}}$ and $\mathbb{CP}^{2} \# 9 \overline{\mathbb{CP}^{2}}$ via rational blowdown surgery along certain $4$-valent plumbing graphs. This way all the graph classes from \cite{weighted} have a representative which admits a rational blowdown leading to an exotic manifold. We emphasize the simplicity of the constructions which boils down to finding a good configuration of complex lines and quadrics in $\mathbb{CP}^{2}$, and deciding which intersections to blow up.
Exotic diffeomorphisms on $4$-manifolds with $b_2^+ = 2$
针对$b_2^+=2$的4-流形怪异微分同胚研究因参数化规范理论不变量定义不完善而受限的问题，本文通过比较参数族环绕数的方法，证明$2\bC\bP^2\n#10(-\bC\bP^2)$存在怪异微分同胚，是目前已知最小的相关闭4-流形例子。
Bar-Natan skein lasagna modules and exotic surfaces in 4-manifolds
本文利用 Bar-Natan Khovanov 同调构造绞合 lasagna 模，并通过连通和粘合映射在 4 维流形中生成带边界的奇异曲面，证明一次内部稳定化通常不足以消除其奇异性。
In this paper we exhibit infinite families of embedded tori in 4-manifolds that are topologically isotopic but smoothly distinct. The interesting thing about these tori is that they are topologically trivial in the sense that each bounds a topologically embedded solid handlebody. This implies that there are stably ribbon surfaces in 4-manifolds that are not ribbon.
Stably exotic 4-manifolds
该研究针对4维流形的稳定异乎寻常性问题，探究基本群与Stiefel-Whitney类的取值对其存在性的影响，在H₅(π;ℤ)=0条件下给出完整刻画，构造了新例子也明确了不存在的新情形。
We explicitly construct a genus-$3$ Lefschetz fibration over $\mathbb{S}^{2}$ whose total space is $\mathbb{T}^{2}\times \mathbb{S}^{2}\# 6\overline{\mathbb{C} P^{2}}$ using the monodromy of Matsumoto's genus-$2$ Lefschetz fibration. We then construct more genus-$3$ Lefschetz fibrations whose total spaces are exotic minimal symplectic $4$-manifolds $3 \mathbb{C} P^{2} \# q\overline{\mathbb{C} P^{2}}$ for $q=13,\ldots,19$. We also generalize our construction to get genus-$3k$ Lefschetz fibration structure on the $4$-manifold $Σ_{k}\times \mathbb{S}^{2}\# 6\overline{\mathbb{C} P^{2}}$ using the …
From a handlebody-theoretic perspective, the simplest compact, contractible 4-manifolds, other than the 4-ball, are Mazur manifolds. We produce the first pairs of Mazur manifolds that are homeomorphic but not diffeomorphic. Our diffeomorphism obstruction comes from our proof that the knot Floer homology concordance invariant $ν$ is an invariant of the smooth 4-manifold associated to a knot in the 3-sphere by attaching an n-framed 2-handle to the 4-ball along the knot. In contrast, we also show (modulo forthcoming work of Ozsváth and Szabó) that the concordance invariants $τ$ and $ε$ are not in…
Motivated by Stipsicz and Szabó's exotic 4-manifolds with b_2^+=3 and b_2^-=8, we construct a family of simply connected smooth 4-manifolds with b_2^+=3 and b_2^-=8. As a corollary, we conclude that the topological 4-manifold 3{CP}^2#8{-CP}^2 admits infinitely many distinct smooth structures.
Khovanov homology can distinguish exotic Mazur manifolds
本文提出利用 Khovanov 同调区分 exotic Mazur 流形的新方法，无需依赖 skein lasagna 模块，并首次给出了 exotic Mazur 流形存在的分析自由证明。
We present a pair of smooth fiber bundles over the circle with a common $4$-dimensional fiber with the following properties: (1) their total spaces are diffeomorphic to each other; (2) they are isomorphic to each other as topological fiber bundles; (3) they are not isomorphic to each other as smooth fiber bundles. In particular, we exhibit an example with non-simply-connected fiber.
Stably exotic fillings of 3-manifolds
该研究探讨三维流形界定四维流形的稳定奇异填充问题，证明闭定向三维流形及部分非定向三维流形存在此类填充，而含双侧$\text{RP}^2$的三维流形不存在。
From zero surgeries to candidates for exotic definite four-manifolds
本文研究通过0-手术同胚构造具有相同零手术的结对，以系统生成奇异四维流形候选者，并通过计算机实验找到5个拓扑切片结，若其中任一切片则可得到奇异四维球面。
New Exotic four-manifolds with $\mathbb{Z}/2\mathbb{Z}$ fundamental group
本文扩展了Stipsicz-Szabó的构造，利用双节点手术和有理吹落技术，为具有偶数$b_2^+$和基本群$\mathbb{Z}/2\mathbb{Z}$的闭四维流形构造了无穷多不可约异构光滑结构，并通过泛覆盖的Seiberg-Witten不变量进行区分。
In this article, we construct infinitley many simply connected, nonsymplectic and pairwise nondiffeomorphic 4-manifolds starting from E(n) and applying the sequence of knot surgery, ordinary blowups and rational blowdown. We also compute the Seiberg-Witten invariants of these manifolds.
We explore transformation groups of manifolds of the form $M\times S^n$, where $M$ is an asymmetric manifold, i.e. a manifold which does not admit any non-trivial action of a finite group. In particular, we prove that for $n=2$ there exists an infinite family of distinct non-diagonal effective circle actions on such products. A similar result holds for actions of cyclic groups of prime order. We also discuss free circle actions on $M \times S^1$, where $M$ belongs to the class of "almost asymmetric" manifolds considered previously by V. Puppe and M. Kreck.
Heegaard Floer knot trace invariants, exotic 4-manifolds, and symplectic obstructions
本研究证明纽结Floer同调的数值不变量$ν$和$|ε|$是任意整数$n$的纽结$n$-迹不变量，拓展后可构造新怪异对族，还通过新不变量给出4-流形作为边界切触结构强辛填充的拓扑阻碍。
We construct an infinite family of simply connected, pairwise nondiffeomorphic 4-manifolds, all homeomorphic to 3CP^2 blown up at 9 points.
Let $N$ be a smooth manifold that is homeomorphic but not diffeomorphic to a closed hyperbolic manifold $M$. In this paper, we study the extent to which $N$ admits as much symmetry as $M$. Our main results are examples of $N$ that exhibit two extremes of behavior. On the one hand, we find $N$ with maximal symmetry, i.e. Isom($M$) acts on $N$ by isometries with respect to some negatively curved metric on $N$. For these examples, Isom($M$) can be made arbitrarily large. On the other hand, we find $N$ with little symmetry, i.e. no subgroup of Isom($M$) of "small" index acts by diffeomorphisms of …
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.
The Rasmussen s-invariant and exotic 4-manifolds
本文通过改进的 Rasmussen s-invariant 方法，无需 skein lasagna 模块，证明了 exotic 4-维流形的存在性。
Let M be either CP^2#3CP^2bar or 3CP^2#5CP^2bar. We construct the first example of a simply-connected symplectic 4-manifold that is homeomorphic but not diffeomorphic to M.
We give several criteria on a closed, oriented 3-manifold that will imply that it is the boundary of a (simply connected) 4-manifold that admits infinitely many distinct smooth structures. We also show that any weakly fillable contact 3-manifold, or contact 3-manifolds with non-vanishing Heegaard Floer invariant, is the boundary of a simply connected 4-manifolds that admits infinitely many distinct smooth structures each of which supports a symplectic structure with concave boundary, that is there are infinitely many exotic caps for any such contact manifold.
The main theorem of the paper shows that a smooth manifold which is homeomorphic to S^2xS^2 and has nonvanishing Ozsvath-Szabo invariant does not admit a perfect Morse function. I am withdrawing the paper because it is unclear to me if such a manifold exists.
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.
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。
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.
Internal and external stabilization of exotic surfaces in 4-manifolds
本文研究光滑4-流形中奇异曲面的稳定化问题，证明多种显式构造的奇异曲面对在一次外部稳定化后变得光滑同位，并建立内部与外部稳定化之间的联系，同时证明特定奇异族中的2-links具有brunnian性质。
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。
Exotic Dehn twists on 4-manifolds
该研究率先探索4-流形内沿非$S^3$ 3-流形的奇异Dehn扭，借助$\text{RP}^2$上的2参数族Seiberg-Witten理论，构造出多类新的奇异微分同胚例子，还证明了相关微分同胚群基本群同态的非满射性。
Let $M$ be $\CP#2\CPb$, $3\CP#4\CPb$ or $(2n-1)\CP#2n\CPb$ for any integer $n\geq 3$. We construct an irreducible symplectic 4-manifold homeomorphic to $M$ and also an infinite family of pairwise non-diffeomorphic irreducible non-symplectic 4-manifolds homeomorphic to $M$. We also construct such exotic smooth structures when $M$ is $\CP#4\CPb$ or $3\CP# k \CPb$ for $k=6,8,10$.
It is known that there is a unique concordance class in the free homotopy class of $S^1\times pt \subset S^1 \times S^2$. The constructive proof of this fact is given by the second author. It turns out that all the concordances in this construction are invertible. The knots $K\subset S^{1}\times S^{2}$ with hyperbolic complements and trivial symmetry group are special interest here, because they can be used to generate absolutely exotic compact 4-manifolds by the recipe given by Akbulut and Ruberman. Here we built absolutely exotic manifold pairs by this construction, and show that this constr…
Wall-type stabilization problems investigate the collapse of exotic 4-dimensional phenomena under stabilization operations (e.g., taking connected sums with $S^2 \times S^2$). We propose an elementary approach to these problems, providing a construction of exotic 4-manifolds and knotted surfaces that are candidates to remain exotic after stabilization -- including examples in the setting of closed, simply connected 4-manifolds. As a proof of concept, we show this construction yields exotic surfaces in the 4-ball that remain exotic after (internal) stabilization, detected by the cobordism maps …
We construct a contact 5-manifold supported by infinitely many distinct open books with the identity monodromy and pairwise exotic Stein pages (i.e. pages are pairwise homeomorphic but non-diffeomorphic Stein fillings of a fixed contact 3-manifold), moreover we describe a process of generating infinitely many such examples. In contrast to this result, on each of $\#_{n} S^{2}\times S^{3}$ and $\#_{n} S^{2}\tilde{\times} S^{3}$ $(n\geq 2)$ we construct infinitely many open books with pairwise exotic Stein pages (and identity monodromy) supporting mutually distinct contact structures.
For every integer $k\geq 2$, we construct infinite families of mutually nondiffeomorphic irreducible smooth structures on the topological $4$-manifolds $(2k-1)(S^2\times S^2)$ and $(2k-1)(\CP#\CPb)$, the connected sums of $2k-1$ copies of $S^2\times S^2$ and $\CP#\CPb$.
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。
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.
Exotic diffeomorphisms of reducible $4$-manifolds with odd $b_+$
利用族Bauer-Furuta不变量，在具有奇数$b_+$的可约$4$流形上构造奇异微分同胚，并发现映射类群非有限生成的单连通$4$流形新例子。
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…
Harer, Kas and Kirby have conjectured that every handle decomposition of the elliptic surface $E(1)_{2,3}$ requires both 1- and 3-handles. In this article, we construct a smooth 4-manifold which has the same Seiberg-Witten invariant as $E(1)_{2,3}$ and admits neither 1- nor 3-handles, by using rational blow-downs and Kirby calculus. Our manifold gives the first example of either a counterexample to the Harer-Kas-Kirby conjecture or a homeomorphic but non-diffeomorphic pair of simply connected closed smooth 4-manifolds with the same non-vanishing Seiberg-Witten invariants.
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