An exotic smooth structure on CP^2+6CP^2-bar

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.

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。

Symmetries and exotic smooth structures on a $K3$ surface

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 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.

Knot concordances in $S^1\times S^2$ and exotic smooth $4$-manifolds

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…

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.

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.

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.

Some smooth circle and cyclic group actions on exotic spheres

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.

Construction of Exotic Smooth Structures

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.

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…

Exotic Smoothness and Quantum Gravity II: exotic R^4, singularities and cosmology

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…

Exotic smoothings via large R^4's in Stein surfaces

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.

Exotic smooth structures on nonpositively curved symmetric spaces

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.

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…

从零手术到奇异定四维流形的候选者

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。

Exotic Smoothness and Quantum Gravity

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…

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 …

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.

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

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。

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。

奇异 Dehn 扭转与同伦相干群作用

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。

Gerbes on orbifolds and exotic smooth R^4

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 …

A note on exotic families of 4-manifolds

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.

Symmetries of exotic aspherical space forms

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…

Topological quantum D-branes and wild embeddings from exotic smooth R^4

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…

An essential relation between Einstein metrics, volume entropy, and exotic smooth structures

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.

具有指定基本群的奇异不可定向四维流形

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。

Exotic smooth R^4, noncommutative algebras and quantization

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…

Quantum D-branes and exotic smooth R^4

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…

Smooth quantum gravity: Exotic smoothness and Quantum gravity

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 …

Axiomatic TQFT, Axiomatic DQFT, and Exotic 4-Manifolds

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…

怪异非球面4-流形

Exotic aspherical 4-manifolds

本研究构造了同胚但不微分同胚的闭非球面光滑4-流形，为四维Borel猜想的光滑类比提供反例，核心方法是将反射群技巧应用于带边怪异4-流形对。

Exotic smooth R^4 and certain configurations of NS and D branes in string theory

In this paper we show that in some important cases 4-dimensional data can be extracted from superstring theory such that a) the data are 4 Euclidean geometries embedded in standard $\mathbb{R}^{4}$, b) these data depend on NS and D brane charges of some string backgrounds, c) it is of potential relevance to 4-dimensional physics, d) the compactification and stabilization techniques are not in use, but rather are replaced. We analyze certain configurations of NS and D-branes in the context of $SU(2)$ WZW model and find the correlations with different exotic smoothings of $\mathbb{R}^{4}$. First…