Constructions of Smooth 4-Manifolds

A note on embeddings of 3-manifolds in symplectic 4-manifolds

We show that any closed oriented 3-manifold can be topologically embedded in some simply-connected closed symplectic 4-manifold, and that it can be made a smooth embedding after one stabilization. As a corollary of the proof we show that the homology cobordism group is generated by Stein fillable 3-manifolds. We also find obstructions on smooth embeddings: there exists 3-manifolds which cannot smoothly embed in a way that appropriately respect orientations in any symplectic manifold with weakly convex boundary. This embedding obstruction can also be used to detect exotic smooth structures on 4…

Exotic Smooth Structures on Small 4-Manifolds with Odd Signatures

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 $k=6,8,10$.

Holomorphic triangles and invariants for smooth four-manifolds

The aim of this article is to introduce invariants of oriented, smooth, closed four-manifolds, built using the Floer homology theories defined in two earlier papers (math.SG/0101206 and math.SG/0105202). This four-dimensional theory also endows the corresponding three-dimensional theories with additional structure: an absolute grading of certain of its Floer homology groups. The cornerstone of these constructions is the study of holomorphic disks in the symmetric products of Riemann surfaces.

在5流形中平滑3流形

Smoothing 3-manifolds in 5-manifolds

该研究证明了在5流形中局部平坦的3流形可通过小同伦变为光滑嵌入，并推导出光滑表面的拓扑同伦与光滑同伦的关系。

Constructing infinitely many smooth structures on small 4-manifolds

The purpose of this article is twofold. First we outline a general construction scheme for producing simply-connected minimal symplectic 4-manifolds with small Euler characteristics. Using this scheme, we illustrate how to obtain irreducible symplectic 4-manifolds homeomorphic but not diffeomorphic to $\CP#(2k+1)\CPb$ for $k = 1,...,4$, or to $3\CP# (2l+3)\CPb$ for $l =1,...,6$. Secondly, for each of these homeomorphism types, we show how to produce an infinite family of pairwise nondiffeomorphic nonsymplectic 4-manifolds belonging to it. In particular, we prove that there are infinitely many …

Normal Smoothings for Smooth Cube Manifolds

We prove that smooth cube manifolds have normal smooth structures.

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.

An invariant of smooth 4-manifolds

We define a diffeomorphism invariant of smooth 4-manifolds which we can estimate for many smoothings of R^4 and other smooth 4-manifolds. Using this invariant we can show that uncountably many smoothings of R^4 support no Stein structure. (Gompf has constructed uncountably many smoothings of R^4 which do support Stein structures.) Other applications of this invariant are given.

光滑化三维流形上的有限群作用

Smoothing finite group actions on three-manifolds

本文证明了有限群在光滑三维流形上的任意连续作用均可由光滑作用一致逼近，解决了三维情形下连续群作用的光滑化问题。

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…

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.

Broken Lefschetz fibrations and smooth structures on 4-manifolds

The broken genera are orientation preserving diffeomorphism invariants of closed oriented 4-manifolds, defined via broken Lefschetz fibrations. We study the properties of the broken genera invariants, and calculate them for various 4-manifolds, while showing that the invariants are sensitive to exotic smooth structures.

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.

Elimination of singularities of smooth mappings of 4-manifolds into 3-manifolds

This paper has been withdrawn. Its new version has been published.

On Smooth Closed 3-manifolds with Finite Fundamental Groups

In this paper, we prove a geometrization conjecture, every orientable smooth closed 3-manifold with finite fundamental group is homeomorphic to $S^3/G$ for some finite cyclic subgroup $G\subset {Isom}^+(S^3)$.

Einstein metrics and the number of smooth structures on a four-manifold

We prove that for every natural number k there are simply connected topological four-manifolds which have at leat k distinct smooth structures supporting Einstein metrics, and also have infinitely many distinct smooth structures not supporting Einstein metrics. Moreover, all these smooth structures become diffeomorphic after connected sum with only one copy of the complex projective plane. We prove that manifolds with these properties cover a large geographical area.

A calculus for shadows of smooth 4-manifolds

The paper has been withdrawn by the author due an error in the proof of Theorem 3.2.

Characterizing algebras of smooth functions on manifolds

Among all $C^\infty$-algebras we characterize those which are algebras of smooth functions on smooth separable Hausdorff manifolds.

Akbulut's corks and h-cobordisms of smooth simply connected 4-manifolds

This is primarily an exposition, combining work of several authors (Curtis, Hsiang, Freedman, Stong, Matveyev, and Bizaca), of the proof that a smooth 5-dimensional h-cobordism between simply connected 4-manifolds is a product off of a contractible piece which itself is diffeomorphic to the 5-ball.

带边界几何对偶的4流形中曲面的光滑拓扑同痕

Smoothing topological isotopy of surfaces in 4-manifolds with a boundary geometric dual

本文证明在带边界几何对偶的光滑可定向4流形中，整洁曲面的拓扑同痕蕴含光滑同痕，从而建立光滑与拓扑同痕类间的双射。

流形上光滑函数的Reeb空间 II

Reeb spaces of smooth functions on manifolds II

本文刻画了闭流形上Reeb空间为有限图的光滑函数，并构造了若干具有特殊性质的显式例子。

定义四流形上的光滑结构：具有无限基本群的四维流形

Smooth structures on definite four-manifolds with infinite fundamental group

本文构造了无限多个非微分同构的光滑结构，其阿贝尔化为Z/2pZ×Z/2Z，针对具有无限基本群的定義四维流形。

Smooth classification of locally standard $T^k$-manifolds

We study locally standard $T^k$-manifolds $M$. In particular, we study the case where there is a continuous section to the orbit map $π: M \rightarrow M/T$. We give a classification of $T^k$-manifolds satisfying these conditions up to equivariant diffeomorphism.

Reeb spaces of smooth functions on manifolds

The Reeb space of a continuous function is the space of connected components of the level sets. In this paper we first prove that the Reeb space of a smooth function on a closed manifold with finitely many critical values has the structure of a finite graph without loops. We also show that an arbitrary finite graph without loops can be realized as the Reeb space of a certain smooth function on a closed manifold with finitely many critical values, where the corresponding level sets can also be preassigned. Finally, we show that a continuous map of a smooth closed connected manifold to a finite …

4-流形中拓扑曲面的灯泡光滑化

Light bulb smoothing for topological surfaces in 4-manifolds

本文提出4-流形中拓扑嵌入曲面的新光滑化技术，实现拓扑到光滑的同痕，并应用于证明维度4中某些圆盘和球面的“拓扑=光滑”结果，同时获得Dax不变量的拓扑版本。

Representing smooth 4-manifolds as loops in the pants complex

We show that every smooth, orientable, closed, connected 4-manifold can be represented by a loop in the pants complex. We use this representation, together with the fact that the pants complex is simply connected, to provide an elementary proof that such 4-manifolds are smoothly cobordant to $\coprod_m \mathbb{C}P^2 \coprod_n \bar{\mathbb{C}P}^2$. We also use this association to give information about the structure of the pants complex. Namely, given a loop in the pants complex, $L$, which bounds a disk, $D$, we show that the signature of the 4-manifold associated to $L$ gives a lower bound on…

Smooth structures on non-orientable four-manifolds and free involutions

In this paper, we investigate existence of inequivalent smooth structures on closed smooth non-orientable 4-manifolds building upon results of Akbulut, Cappell-Shaneson, Fintushel-Stern, Gompf, and Stolz. We add to the number of known constructions and provide new examples of exotic manifolds that are obtained as an application of Gluck twists to the standard smooth structure. Inspection of the smooth structure on the oriented 2-covers yields existence results of orientation-reversing exotic free involutions.

拓扑同痕但光滑不等价的2-球面在单连通4-流形中，其余集具有给定基本群

Topologically isotopic and smoothly inequivalent 2-spheres in simply connected 4-manifolds whose complement has a prescribed fundamental group

本文提出构造单连通4-流形中拓扑同痕但光滑不等价的2-球面无限族的方法，其余集基本群可预设，涵盖有限循环群等，为首批此类纽结现象。\n

关于具有偶数阶基本群的4维流形上的光滑结构

On smooth structures over $4$-manifolds with fundamental group of even order

研究显示，具有特定基本群和二次型条件的4维流形可能拥有零或无限多个光滑结构，并构造了具有零签名的非微分同构光滑结构。

Noncomplex smooth 4-manifolds with genus-2 Lefschetz fibrations

We construct noncomplex smooth 4-manifolds which admit genus-2 Lefschetz fibrations over S^2. The fibrations are necessarily hyperelliptic, and the resulting 4-manifolds are not even homotopy equivalent to complex surfaces. Furthermore, these examples show that fiber sums of holomorphic Lefschetz fibrations do not necessarily admit complex structures.

Group trisections and smooth 4-manifolds

A trisection of a smooth, closed, oriented 4-manifold is a decomposition into three 4-dimensional 1-handlebodies meeting pairwise in 3-dimensional 1-handlebodies, with triple intersection a closed surface. The fundamental groups of the surface, the 3-dimensional handlebodies, the 4-dimensional handlebodies, and the closed 4-manifold, with homomorphisms between them induced by inclusion, form a commutative diagram of epimorphisms, which we call a trisection of the 4-manifold group. A trisected 4-manifold thus gives a trisected group; here we show that every trisected group uniquely determines a…

Smooth Volume Rigidity for Manifolds with Negatively Curved Targets

We establish conditions for a continuous map of nonzero degree between a smooth closed manifold and a negatively curved manifold of dimension greater than four to be homotopic to a smooth cover, and in particular a diffeomorphism when the degree is one. The conditions hold when the volumes or entropy-volumes of the two manifolds differ by less than a uniform constant after an appropriate normalization of the metrics. The results are qualitatively sharp in the sense that the dependencies are necessary. We give a number of corollaries.

The number of smooth 4-manifolds with a fixed complexity

One can define the complexity of a smooth 4-manifold as the minimal sum of the number of disks, strands and crossings in a Kirby diagram. Martelli proved that the number of homeomorphism classes of complexity less than n grows as $n^2$. In this paper we prove that the number of diffeomorphism classes grows at least as fast as $n^{c\sqrt[3]{n}}$. Along the way we construct complete kirby diagrams for a large family of knot surgery manifolds.

Equivariant smoothing of piecewise linear manifolds

We prove that every piecewise linear manifold of dimension up to four on which a finite group acts by piecewise linear homeomorphisms admits a compatible smooth structure with respect to which the group acts smoothly. This solves a challenge posed by Thurston in dimension three and confirms a conjecture by Kwasik and Lee in dimension four in a stronger form.

hyperkähler流形的光滑Birman-Hilden理论

A smooth Birman-Hilden theory for hyperkähler manifolds

本文为容许hyperkähler结构的流形证明了Birman-Hilden定理的光滑类比，借此探究具有非平凡基本群的4维流形的光滑映射类群，并应用于Enriques曲面的Nielsen实现。

非光滑流形分解

Nonsmooth manifold decompositions

该研究探讨光滑函数连续选择诱导的光滑流形结构，分析其通有性下的分层与非退化临界点下的局部拓扑性质，并通过4维流形上三函数最大值的分析为三分法理论提供新视角。

关于光滑4维流形之间h-cobordisms的一些注记

Some remarks on $h$-cobordisms between smooth 4-manifolds

本文探讨了光滑4维流形间h-cobordism定理的实现实质，提出新的条件以解答相关问题，并分析标准方法在特定torsion下的失效条件。

Smooth s-cobordisms of elliptic 3-manifolds

The main result of this paper states that a symplectic s-cobordism of elliptic 3-manifolds is diffeomorphic to a product (assuming a canonical contact structure on the boundary). Based on this theorem, we conjecture that a smooth s-cobordism of elliptic 3-manifolds is smoothly a product if its universal cover is smoothly a product. We explain how the conjecture fits naturally into the program of Taubes of constructing symplectic structures on an oriented smooth 4-manifold from generic self-dual harmonic forms. The paper also contains an auxiliary result of independent interest, which generaliz…

Trisecting Smooth 4-dimensional Cobordisms

We extend the theory of relative trisections of smooth, compact, oriented $4$-manifolds with connected boundary given by Gay and Kirby to include $4$-manifolds with an arbitrary number of boundary components. Additionally, we provide sufficient conditions under which relatively trisected $4$-manifolds can be glued to one another along diffeomorphic boundary components so as to induce a trisected manifold. These two results allow us to define a category $\textrm{Tri}$ whose objects are smooth, closed, oriented $3$-manifolds equipped with open book decompositions, and morphisms are relatively tr…

Minimal handle decomposition of smooth simply connected 5-manifolds

A theorem on the existence of the unique minimal topologic handle decomposition of differentiable simply connected five-dimensional manifolds is proved. For a decomposition of this sort, the number of handles of each index is given.

Niemeier lattices, smooth 4-manifolds and instantons

We show that the set of even positive definite lattices that arise from smooth, simply-connected 4-manifolds bounded by a fixed homology 3-sphere can depend on more than the ranks of the lattices. We provide two homology 3-spheres with distinct sets of such lattices, each containing a distinct nonempty subset of the rank 24 Niemeier lattices.

The Normalized Ricci Flow on Four-Manifolds and Exotic Smooth Structures

In this article, we shall investigate the relationship between the existence or non-existence of non-singular solutions to the normalized Ricci flow and smooth structures on closed 4-manifolds, where non-singular solutions to the normalized Ricci flow are solutions which exist for all time $t \in [0, \infty)$ with uniformly bounded sectional curvature. In dimension 4, there exist many compact topological manifolds admitting distinct smooth structures, i.e., exotic smooth structures. Interestingly, in this article, the difference between existence and non-existence of non-singular solutions to …

Permutations, isotropy and smooth cyclic group actions on definite 4-manifolds

We use the equivariant Yang-Mills moduli space to investigate the relation between the singular set, isotropy representations at fixed points, and permutation modules realized by the induced action on homology for smooth group actions on certain 4-manifolds.

Smooth (non)rigidity of piecewise rank one locally symmetric manifolds

We define \emph{piecewise rank 1} manifolds, which are aspherical manifolds that generally do not admit a nonpositively curved metric but can be decomposed into pieces that are diffeomorphic to finite volume, irreducible, locally symmetric, nonpositively curved manifolds with $π_1$-injective cusps. We prove smooth (self) rigidity for this class of manifolds in the case where the gluing preserves the cusps' homogeneous structure. We compute the group of self homotopy equivalences of such a manifold and show that it can contain a normal free abelian subgroup and thus, can be infinite. Elements o…

Smooth (non)rigidity of cusp-decomposable manifolds

We define cusp-decomposable manifolds and prove smooth rigidity within this class of manifolds. These manifolds generally do not admit a nonpositively curved metric but can be decomposed into pieces that are diffeomorphic to finite volume, locally symmetric, negatively curved manifolds with cusps. We prove that the group of outer automorphisms of the fundamental group of such a manifold is an extension of an abelian group by a finite group. Elements of the abelian group are induced by diffeomorphisms that are analogous to Dehn twists in surface topology. We also prove that the outer automophis…

Amenable groups and smooth topology of 4-manifolds

A smooth five-dimensional s-cobordism becomes a smooth product if stabilized by a finite number n of $S^2xS^2x[0,1]$'s. We show that for amenable fundamental groups, the minimal n is subextensive in covers, i.e., n(cover)/index(cover) has limit 0. We focus on the notion of sweepout width, which is a bridge between 4-dimensional topology and coarse geometry.

关于实力矩-角流形上等变光滑结构的唯一性

On uniqueness of the equivariant smooth structure on a real moment-angle manifold

本文利用微积分论证推导证明：组合等价简单多面体的实力矩-角流形是等变微分同胚的，且多面体作为带角流形也是微分同胚的。

环动作与光滑流形上的悬垂操作

Circle actions and Suspension operations on Smooth manifolds

研究光滑流形上的环动作与悬垂操作，通过环状手术构造并分类具有自由S¹动作的流形。

具有有限循环基本群的四维流形上的光滑结构

Smooth structures on four-manifolds with finite cyclic fundamental groups

本文研究有限循环基本群四维流形的光滑结构分类，证明除七个可能例外外，Z_{4m+2} 且奇相交形式的流形或无光滑结构或有无穷多个，并构造多类非复不可约伪射影平面。

关于界定具有$b_2=1$的光滑4-流形的透镜空间

On lens spaces bounding smooth 4-manifolds with $\boldsymbol{b_2=1}$

该文研究不同拓扑下可界定b₂=1的光滑4-流形的透镜空间，构造出两类无穷族透镜空间并证明其拓扑性质，研究受有理同调射影平面相关问题启发。

3-连通8维流形与球面乘积上的光滑结构

Smooth Structures on the product of 3-connected 8-manifolds with spheres

本文计算了3-连通8维流形与球面乘积的协合惯性群，并分类了同胚于该乘积的光滑流形，特别给出了$H^4(M;\mathbb{Z})=\mathbb{Z}$时与$M\times\mathbb{S}^1$同胚流形的微分同胚分类。

Cartan--Whitney Presentation, Non-smooth Analysis and Smoothability of Manifolds: On a theorem of Kondo--Tanaka

Using tools and results from geometric measure theory, we give a simple new proof of the main result (Theorem 1.3) in K. Kondo and M. Tanaka, Approximation of Lipschitz Maps via Immersions and Differentiable Exotic Sphere Theorems, \textit{Nonlinear Anal.} \textbf{155} (2017), 219--249, as well as the converse statement. It explores the connections between the theory of non-smooth analysis {\it à la} F.~H. Clarke and the existence of special systems of Whitney flat $1$-forms with Sobolev regularity on certain families of homology manifolds.

光滑4维流形的度量拉伸与周期映射

Metric stretching and the period map for smooth 4-manifolds

本文研究光滑闭4维流形的周期映射，利用沿超曲面拉伸构造的度量族，证明其像稠密且在$b^+=1$时满射，并推广至4的倍数维流形。

Semi-free actions with manifold orbit spaces

In this paper, we study smooth, semi-free actions on closed, smooth, simply connected manifolds, such that the orbit space is a smoothable manifold. We show that the only simply connected $5$-manifolds admitting a smooth, semi-free circle action with fixed-point components of codimension $4$ are connected sums of $S^3$-bundles over $S^2$. Furthermore, the Betti numbers of the $5$-manifolds and of the quotient $4$-manifolds are related by a simple formula involving the number of fixed-point components. We also investigate semi-free $S^3$ actions on simply connected $8$-manifolds with quotient a…