Anar Akhmedov
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.
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We produce infinite families of exotic actions of finite cyclic groups on simply connected smooth 4-manifolds with nontrivial Seiberg-Witten invariants.
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.
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.
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…
We study the relationship between exotic R^4's and Stein surfaces as it applies to smoothing theory on more general open 4-manifolds. In particular, we construct the first known examples of large exotic R^4's that embed in Stein surfaces. This relies on an extension of Casson's Embedding Theorem for locating Casson handles in closed 4-manifolds. Under sufficiently nice conditions, we show that using these R^4's as end-summands produces uncountably many diffeomorphism types while maintaining independent control over the genus-rank function and the Taylor invariant.
Classical work of Lee, Schultz, and Stolz relates the smooth transformation groups of exotic spheres to the stable homotopy groups of spheres. In this note, we apply recent progress on the latter to produce smooth circle and cyclic group actions on certain exotic spheres.
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 construct an infinite family of mutually non-diffeomorphic irreducible smooth structures on the topological 4-manifold $S^2 \times S^2$.
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。
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.
We construct smooth 4-manifolds homeomorphic but not diffeomorphic to CP^2+6CP^2-bar.
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 $S^3\times{\mathbb R}$.
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。
Motivated by a construction of Fintushel and Stern, we show that the topological 4--manifold $CP^2#5{\bar CP^2}$ supports infinitely many distinct smooth structures.
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 construct series of examples of exotic smooth structures on compact locally symmetric spaces of noncompact type. In particular, we obtain higher rank examples, which do not support Riemannian metric of nonpositive curvature. The examples are obtained by taking the connected sum with an exotic sphere. To detect the change of the smooth structure we use a tangential map from the locally symmetric space its dual compact type twin.
Exotic Dehn twists and homotopy coherent group actions
本文研究将光滑4-流形边界上的同伦相干有限循环群作用延拓至内部的问题，并证明沿任意Seifert同调球面（3-球面除外）在单连通正定填充上的Dehn扭转具有无限阶奇异性。
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.
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 $\CP#\CPb$.
We study finite group actions on smooth manifolds of the form $M\#Σ$, where $Σ$ is an exotic $n$-sphere and $M$ is a closed aspherical space form. We give a classification result for free actions of finite groups on $M\#Σ$ when $M$ is 7-dimensional. We show that if $\mathbb Z/p\mathbb Z$ acts freely on $T^n\#Σ$, then $Σ$ is divisible by $p$ in the group of homotopy spheres. When $M$ is hyperbolic, we give examples $M\#Σ$ that admit no nontrivial smooth action of a finite group, even though Isom($M$) is arbitrarily large. Our proofs combine geometric and topological rigidity results with smooth…
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 $k=6,8,10$.
Since the first work on exotic smoothness in physics, it was folklore to assume a direct influence of exotic smoothness to quantum gravity. In the second paper, we calculate the "smoothness structure" part of the path integral in quantum gravity for the exotic R^4 as non-compact manifold. We discuss the influence of the "sum over geometries" to the "sum over smoothness structure". There are two types of exotic R^4: large (no smooth embedded 3-sphere) and small (smooth embedded 3-sphere). A large exotic R^4 can be produced by using topologically slice but smoothly non-slice knots whereas a smal…
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.
As an application of `reverse engineering' technique introduced by R. Fintushel, D. Park and R. Stern \cite{FPS}, we construct an infinite family of fake (2n+2l-1)CP^2#(2n+4l-1)(-CP^2)'s for all n \ge 0, l \ge 1.
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。
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.
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。
Smooth and symplectic symmetries of an infinite family of distinct exotic $K3$ surfaces are studied, and comparison with the corresponding symmetries of the standard $K3$ is made. The action on the $K3$ lattice induced by a smooth finite group action is shown to be strongly restricted, and as a result, nonsmoothability of actions induced by a holomorphic automorphism of a prime order $\geq 7$ is proved and nonexistence of smooth actions by several $K3$ groups is established (included among which is the binary tetrahedral group $T_{24}$ which has the smallest order). Concerning symplectic symme…
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。
We study the set $\widehat{\mathcal S}_M$ of framed smoothly slice links which lie on the boundary of the complement of a 1-handlebody in a closed, simply connected, smooth 4-manifold $M$. We show that $\widehat{\mathcal S}_M$ is well-defined and describe how it relates to exotic phenomena in dimension four. In particular, in the case when $X$ is smooth, with a handle decompositions with no 1-handles and homeomorphic to but not smoothly embeddable in $D^4$, our results tell us that $X$ is exotic if and only if there is a link $L\hookrightarrow S^3$ which is smoothly slice in $X$, but not in $D…
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 …
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.
Since the first work on exotic smoothness in physics, it was folklore to assume a direct influence of exotic smoothness to quantum gravity. Thus, the negative result of Duston (arXiv:0911.4068) was a surprise. A closer look into the semi-classical approach uncovered the implicit assumption of a close connection between geometry and smoothness structure. But both structures, geometry and smoothness, are independent of each other. In this paper we calculate the "smoothness structure" part of the path integral in quantum gravity assuming that the "sum over geometries" is already given. For that p…
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.
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…
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.
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。
We show that any simply connected topological closed $4$-manifold punctured along any compact, totally disconnected tame subset $Λ$ admits a continuum of smoothings which are not diffeomorphic to any leaf of a $C^{1,0}$ codimension one foliation on a compact manifold. This includes the remarkable case of $S^4$ punctured along a tame Cantor set. This is the lowest reasonable regularity for this realization problem. These results come from a new criterion for nonleaves in $C^{1,0}$ regularity. We also include a new criterion for nonleaves in the $C^2$-category. Some of our smooth nonleaves are "…
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 present an infinite sequence of smooth embeddings of a connected sum of 6 projective planes in the 4-sphere, which are all ambient homeomorphic, but pairwise ambient non-diffeomorphic. The double covers of the 4-sphere ramified along these surfaces form a family of the exotic $\Bbb CP^2#5\bar{\Bbb CP^2}$ constructed recently by Park, Stipsicz and Szabó.
We show that the minimal volume entropy of closed manifolds remains unaffected when nonessential manifolds are added in a connected sum. We combine this result with the stable cohomotopy invariant of Bauer-Furuta in order to present an infinite family of four-manifolds with the following properties: 1) They have positive minimal volume entropy. 2) They satisfy a strict version of the Gromov-Hitchin-Thorpe inequality, with a minimal volume entropy term. 3) They nevertheless admit infinitely many distinct smooth structures for which no compatible Einstein metric exists.
Exotic non-orientable four-manifolds with prescribed fundamental group
本文证明具有指数2子群的有限表示群可作为闭光滑不可定向四维流形的基本群，该流形通过Gluck扭结获得奇异光滑结构，其定向2-覆盖微分同胚，且两种结构在添加不同流形后表现出不等价与稳定化特性。
Distinguishing exotic $\mathbb{R}^4$'s with Heegaard Floer homology
通过Heegaard Floer同调区分exotic $\mathbb{R}^4$'s，证明不同切结绳生成的exotic $\mathbb{R}^4$'s不相抵，并重新证明Y×$\mathbb{R}$具有无限多个光滑结构。
By using the relation between foliations and exotic R^4, orbifold $K$-theory deformed by a gerbe can be interpreted as coming from the change in the smoothness of R^4. We give various interpretations of integral 3-rd cohomology classes on S^3 and discuss the difference between large and small exotic R^4. Then we show that $K$-theories deformed by gerbes of the Leray orbifold of S^3 are in one-to-one correspondence with some exotic smooth R^4's. The equivalence can be understood in the sense that stable isomorphisms classes of bundle gerbes on S^{3} whose codimension-1 foliations generates the …
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 …
This is the next step of uncovering the relation between string theory and exotic smooth R^4. Exotic smoothness of R^4 is correlated with D6 brane charges in IIA string theory. We construct wild embeddings of spheres and relate them to a class of topological quantum Dp-branes as well to KK theory. These branes emerge when there are non-trivial NS-NS H-fluxes where the topological classes are determined by wild embeddings S^2 -> S^3. Then wild embeddings of higher dimensional $p$-complexes into S^n correspond to Dp-branes. These wild embeddings as constructed by using gropes are basic objects t…
In this article we prove that any unitary, axiomatic topological quantum field theory in four-dimensions can not detect changes in the smooth structure of M, a simply connected, closed (compact without boundary), oriented smooth manifold. However, as Donaldson-Witten theory (a topological quantum field theory but not an axiomatic one) is able to detect changes in the smooth structure of such an M, this seemingly leads to a contradiction. This seeming contradiction is resolved by introducing a new set of axioms for a "differential quantum field theory", which in truth only slightly modify the n…
We show that the cobordism maps on Khovanov homology can distinguish smooth surfaces in the 4-ball that are exotically knotted (i.e., isotopic through ambient homeomorphisms but not ambient diffeomorphisms). We develop new techniques for distinguishing cobordism maps on Khovanov homology, drawing on knot symmetries and braid factorizations. We also show that Plamenevskaya's transverse invariant in Khovanov homology is preserved by maps induced by positive ascending cobordisms.
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…
In this paper, we present the idea that the formalism of string theory is connected with the dimension 4 in a new way, not covered by phenomenological or model-building approaches. The main connection is given by structures induced by small exotic smooth R^4's having intrinsic meaning for physics in dimension 4. We extend the notion of stable quantum D-branes in a separable noncommutative C* algebras over convolution algebras corresponding to the codimension-1 foliations of S^3 which are mainly connected to small exotic R^4. The tools of topological K-homology and K-theory as well KK-theory de…
Over the last two decades, many unexpected relations between exotic smoothness, e.g. exotic $\mathbb{R}^{4}$, and quantum field theory were found. Some of these relations are rooted in a relation to superstring theory and quantum gravity. Therefore one would expect that exotic smoothness is directly related to the quantization of general relativity. In this article we will support this conjecture and develop a new approach to quantum gravity called \emph{smooth quantum gravity} by using smooth 4-manifolds with an exotic smoothness structure. In particular we discuss the appearance of a wildly …
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.
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…
The paper shows deep connections between exotic smoothings of small R^4, noncommutative algebras of foliations and quantization. At first, based on the close relation of foliations and noncommutative C*-algebras we show that cyclic cohomology invariants characterize some small exotic R^4. Certain exotic smooth R^4's define a generalized embedding into a space which is K-theoretic equivalent to a noncommutative Banach algebra. Furthermore, we show that a factor III von Neumann algebra is naturally related with nonstandard smoothing of a small R^4 and conjecture that this factor is the unique hy…
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 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 show that the smooth $4$-manifold $M$ obtained by attaching a $2$-handle to $B^4$ along a certain knot $K\subset \partial B^4$ admits infinitely many absolutely exotic copies $M_n$, $n=0,1,2..$, such that each copy $M_n$ is obtained by attaching $2$-handle to a fixed compact smooth contractible manifold $W$ along the iterates $f^{n}(c)$ of a knot $c\subset \partial W$ by a diffeomorphism $f:\partial W \to \partial W$. This generalizes the example in author's 1991 paper, which corresponds to $n=1$ case.
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。
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.
Positive scalar curvature and exotic structures on simply connected four manifolds
本文针对单连通光滑四流形，证明了Gromov带宽不等式与Rosenberg的$S^1$-稳定性猜想在同胚意义下成立，并推广至非单连通情形。
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