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.
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.
In the previous paper the author defined an infinite order plug $(P,φ)$ which gives rise to infinite Fintushel-Stern's knot-surgeries. Here, we give two 4-dimensional infinitely many exotic families $Y_n$, $Z_n$ of exotic enlargements of the plug. The families $Y_n$, $Z_n$ have $b_2=3$, $4$ and the boundaries are 3-manifolds with $b_1=1$, $0$ respectively. We give a plug (or g-cork) twist $(P,φ_{p,q})$ producing the 2-bridge knot or link surgery by combining the plug $(P,φ)$. As a further example, we describe a 4-dimensional twist $(M,μ)$ between knot-surgeries for two mutant knots. The twiste…
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.
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.
Suppose that $X,X'$ are simply-connected closed exotic 4-manifolds. It is well-known that $X'$ is obtained by an order 2 cork twist of $X$. We give an infinite exotic family of 4-manifolds not generated by any infinite order cork. This is the first example admitting such a condition. We prove a necessary condition of 4-dimensional OS-invariants for a family to be generated by an infinite order cork and give non-contractible relatively exotic 4-manifolds that are never induced by any cork. Furthermore, we prove an estimate of the number of OS-invariants of 4-manifolds generated by a cork.
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 "…
Starting with the Dolgachev surface E(1)_{2,3} we construct an infinite family of distinct exotic copies of the rational surface E(1), each of which admits a handlebody decomposition without 1- and 3- handles, and we draw these handlebodies.
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…
We produce infinite families of exotic actions of finite cyclic groups on simply connected smooth 4-manifolds with nontrivial Seiberg-Witten invariants.
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.
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.
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 construct $G$-corks for any extension $G$ of $\mathbb Z^m$ by any finite subgroup of $\mathrm{SO}(4)$ and weakly equivariant $G$-corks for any extension $G$ of $\mathbb Z^m$ by any finite solvable group. In particular, this is the first example of $G$-corks for an infinite nonabelian group $G$ and answers a question by Tange. The construction is a combination of previous results by Auckly-Kim-Melvin-Ruberman, Gompf, and Tange. Using Gompf's results about exotic $\mathbb R^4$'s, we give an application to construct exotic $\mathbb R^4$'s whose diffeotopy group contains all poly-cyclic groups.
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 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.
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…
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.
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.
We construct a new infinite family of models of exotic 7-spheres. These models are direct generalizations of the Gromoll-Meyer sphere. From their symmetries, geodesics and submanifolds half of them are closer to the standard 7-sphere than any other known model for an exotic 7-sphere.
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.
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…
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.
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.
We prove that the Dolgachev surface E(1)_{2,3} (which is an exotic copy of the elliptic surface E(1)=CP^2 # 9(-CP^2)) can be obtained from E(1) by twisting along a simple "plug", in particular it can be obtained from E(1) by twisting along an RP^2.
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 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.
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.
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 $k=6,8,10$.
Florida State University（佛罗里达州立大学）；CNRS, Institut de Mathématiques de Jussieu - Paris Rive Gauche（法国国家科学研究中心，儒勒·让数学家研究所-巴黎左岸）；Facultad de Ciencias, Universidad de la República（乌拉圭共和国大学理学院）
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.
Given a closed simply connected manifold $M$ of dimension $2n\ge6$, we compare the ring of characteristic classes of smooth oriented bundles with fibre $M$ to the analogous ring resulting from replacing $M$ by the connected sum $M\sharpΣ$ with an exotic sphere $Σ$. We show that, after inverting the order of $Σ$ in the group of homotopy spheres, the two rings in question are isomorphic in a range of degrees. Furthermore, we construct infinite families of examples witnessing that inverting the order of $Σ$ is necessary.
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.
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…
It is known that every compact Stein 4-manifolds can be embedded into a simply connected, minimal, closed, symplectic 4-manifold. By using this property, we discuss a new method of constructing corks. This method generates a large class of new corks including all the previously known ones. We prove that every one of these corks can knot infinitely many different ways in a closed smooth manifold, by showing that cork twisting along them gives different exotic smooth structures. We also give an example of infinitely many disjoint embeddings of a fixed cork into a non-compact 4-manifold which pro…
