Small Exotic 4-manifolds with b_2^+=3

Exotic smooth structures on 4-manifolds with zero signature

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 $\CP#\CPb$.

Corks, exotic 4-manifolds and knot concordance

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.

Nuclei and exotic 4-manifolds

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…

软木、奇异4维流形与亏格函数

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。

On certain exotic 4-manifolds of Akhmedov and Park

In an article from 2008, A. Akhmedov and B. D. Park constructed irreducible symplectic 4-manifolds homeomorphic but not diffeomorphic to the manifolds CP^2#3CP^2bar and 3CP^2#5CP^2bar. These manifolds are constructed by using generalized fibre sums. In this note we describe an explicit splitting of the second (co-)homology of these manifolds adapted to their construction as fibre sums. We also calculate the canonical classes of the symplectic structures. This gives a new proof for a formula derived by A. Akhmedov, R. I. Baykur and B. D. Park.

Cork twisting exotic Stein 4-manifolds

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…

具有奇异光滑结构的定号四维流形

Definite four-manifolds with exotic smooth structures

本文研究了具有基本群Z_2和定号相交形式的闭定向4-流形的光滑结构，构造了无限多个第二Betti数为1和2的不可约定号四维流形，并证明了此类正第二Betti数流形若容许光滑结构则必容许无限多个。

带边4-流形的微分同胚与异状嵌入

Diffeomorphisms of 4-manifolds with boundary and exotic embeddings

该研究定义带切触边界4-流形不变量的族版本以检测异状微分同胚，还证明补集微分同胚的光滑闭4-流形中存在首个异状3-球面例子，推进4-流形拓扑研究。

Nonexistence of twists and surgeries generating exotic 4-manifolds

It is well known that for any exotic pair of simply connected closed oriented 4-manifolds, one is obtained from the other by twisting a compact contractible submanifold via an involution on the boundary. By contrast, here we show that for each positive integer $n$, there exists a simply connected closed oriented 4-manifold $X$ such that, for any compact (not necessarily connected) codimension zero submanifold $W$ with $b_1(\partial W)<n$, the set of all smooth structures on $X$ cannot be generated from $X$ by twisting $W$ and varying the gluing map. As a corollary, we show that there exists no…

Exotic Structures on smooth 4-manifolds

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.

On Park's exotic smooth four-manifolds

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.

两个接触3-流形连通和上的奇异 Dehn 扭转

Exotic Dehn twists on sums of two contact 3-manifolds

本文利用推广的 Kronheimer–Mrowka 接触不变量与凸球面 h-原理，证明两个闭接触3-流形连通和上的 Dehn 扭转可形成无限阶奇异接触映射类，并给出 Colin 分解定理的参数化版本及过扭转情形的新现象。

Small Lefschetz fibrations and exotic 4-manifolds

We explicitly construct genus-2 Lefschetz fibrations whose total spaces are minimal symplectic 4-manifolds homeomorphic to complex rational surfaces CP^2 # p (-CP^2) for p=7, 8, 9, and to 3 CP^2 #q (-CP^2) for q =12,...,19. Complementarily, we prove that there are no minimal genus-2 Lefschetz fibrations whose total spaces are homeomorphic to any other simply-connected 4-manifold with b^+ at most 3, with one possible exception when b^+=3. Meanwhile, we produce positive Dehn twist factorizations for several new genus-2 Lefschetz fibrations with small number of critical points, including the smal…

关于4-流形的绝对奇异微分同胚

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。

闭异质4-流形的新构造与不变量

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。

Surgery along star shaped plumbings and exotic smooth structures on 4-manifolds

We define a new 4-dimensional symplectic cut and paste operation which is analogous to Fintushel and Stern's rational blow-down. We use this operation to produce multiple constructions of symplectic smoothly exotic complex projective space blown-up eight times, seven times, and six times. We also show how this operation can be used in conjunction with knot surgery to construct an infinite family of minimal exotic smooth structures on the complex projective space blown-up seven times.

Small Exotic 4-Manifolds

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.

Contact open books with exotic pages

We consider a fixed contact 3-manifold that admits infinitely many compact Stein fillings which are all homeomorphic but pairwise non-diffeomorphic. Each of these fillings gives rise to a closed contact 5-manifold described as a contact open book whose page is the filling at hand and whose monodromy is the identity symplectomorphism. We show that the resulting infinitely many contact 5-manifolds are all diffeomorphic but pairwise non-contactomorphic. Moreover, we explicitly determine these contact 5-manifolds.

Exotic group actions on simply connected smooth 4-manifolds

We produce infinite families of exotic actions of finite cyclic groups on simply connected smooth 4-manifolds with nontrivial Seiberg-Witten invariants.

Absolutely exotic compact 4-manifolds

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…

Exotic open $4$-manifolds which are non-leaves

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 $S^3\times{\mathbb R}$.

Open manifolds, Ozsvath-Szabo invariants and Exotic R^4's

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.

4维流形上的 exotic 结构

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。

An introduction to exotic 4-manifolds

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.

Infinitely many small exotic Stein fillings

We show that there exist infinitely many simply connected compact Stein 4-manifolds with b_2=2 such that they are all homeomorhic but mutually non-diffeomorphic, and they are Stein fillings of the same contact 3-manifold on their boundaries. We also describe their handlebody pictures.

Exotic structures and adjunction inequality

We reprove and strengthen some old difficult theorems of 4-manifolds by the aid of recently discovered modern tools, which involve contact structures on 3-manifolds, compact Stein domains, etc.

A plug with infinite order and some exotic 4-manifolds

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.

Partial twists and exotic Stein fillings

We give an algorithm which produces infinitely many pairwise exotic Stein fillings of the same contact 3-manifolds, applying positive allowable Lefschetz fibrations over the disk. As a corollary, for a large class of Stein fillings, we realize the topological invariants (i.e. fundamental group, homology group, homology group of the boundary, and intersection form) of each filling as those of infinitely many pairwise exotic Stein fillings. Furthermore, applying the algorithm, we produce various contact 3-manifolds of support genus one each of which admits infinitely many pairwise exotic Stein f…

具有小三分亏格的异乎寻常的4-流形

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。

不可约4-流形可容许奇异微分同胚

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。

具有非平凡基本群的奇异定四维流形

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。

签名零的奇异4-流形

Exotic 4-manifolds with signature zero

我们构造了无穷多个签名零的极小辛奇异4-流形，它们同胚于特定的连通和，且是迄今最小的此类流形，方法基于映射类群正分解的特殊 Lefschetz 纤维化。

从小型4-流形中的微分同胚到奇异现象

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。

Corks with large shadow-complexity and exotic 4-manifolds

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 codimension-1 submanifolds in 4-manifolds and stabilizations

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.

具有非循环基本群的奇异定四维流形

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。

Singular maps on exotic 4-manifold pairs

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.

具有自由阿贝尔基本群的小型不可约奇异4-流形的若干例子

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。

Mazur流形的链环与奇异结构

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。

Singularity links with exotic Stein fillings

In a recent paper of Akhmedov, Etnyre, Mark and Smith, it was shown that there exist infinitely many contact Seifert fibered 3-manifolds each of which admits infinitely many exotic (homeomorphic but pairwise non-diffeomorphic) simply-connected Stein fillings. Here we extend this result to a larger set of contact Seifert fibered 3-manifolds with many singular fibers and observe that these 3-manifolds are singularity links. In addition, we prove that the contact structures induced by the Stein fillings are the canonical contact structures on these singularity links. As a consequence, we verify a…

闭的非球面4-流形上的奇异微分同胚

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。

Small exotic 4-manifolds from lines and quadrics in $\mathbb{CP}^{2}$

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.

$b_2^+ = 2$的4-流形上的怪异微分同胚

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 绞合 lasagna 模与 4 维流形中的奇异曲面

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。

The quadratic form E_8 and exotic homology manifolds

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.

Null-homologous exotic surfaces in 4-manifolds

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.

稳定异乎寻常的4维流形

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。

Exotic smooth structures on $S^2\times S^2$

We construct an infinite family of mutually non-diffeomorphic irreducible smooth structures on the topological 4-manifold $S^2 \times S^2$.

4-流形上的奇异Dehn扭

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。

Genus-$3$ Lefschetz Fibrations and Exotic $4$-Manifolds with $b_{2}^{+}=3$

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 …

Exotic Mazur manifolds and knot trace invariants

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…

Deformation of Singular Fibers of Genus $2$ Fibrations and Small Exotic Symplectic $4$-Manifolds

We introduce the $2$-nodal spherical deformation of certain singular fibers of genus $2$ fibrations, and use such deformations to construct various examples of simply connected minimal symplectic $4$-manifolds with small topology. More specifically, we construct new exotic minimal symplectic $4$-manifolds homeomorphic but not diffeomorphic to ${\mathbb{CP}}^{2}\#6({\overline{\mathbb{CP}}^{2}})$, ${\mathbb{CP}}^{2}\#7({\overline{\mathbb{CP}}^{2}})$, and $3{\mathbb{CP}}^{2}\#k({\overline{\mathbb{CP}}^{2}})$ for $k=16, 17, 18, 19$ using combinations of such deformations, symplectic blowups, and (…

Exotic smooth structures on 3{CP}^2#8{-CP}^2

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 同调可以区分 exotic Mazur 流形

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。