Finite order corksmath.GT
New method creates finite order Stein corks.
problem Creating Stein corks of finite order.
method Cyclic branched covers of slice knots.
result Examples of finite order corks produced.
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48 results for “Stein corks”
New corks found with Stein structures and hyperbolic boundaries.
problem Existence of boundary-sum irreducible finite order corks with Stein structures.
method Proved for any positive integer n, constructed corks with Stein structures and verified hyperbolic boundaries.
result Found boundary-sum irreducible -corks with Stein structures and verified hyperbolic boundaries.
Planar Lefschetz fibrations and Stein structures with distinct Ozsvath-Szabo invariants on corksmath.GT
New examples show Stein structures with distinct Ozsvath-Szabo invariants can't always imply distinct contact structures.
problem Understanding when Stein structures induce distinct contact structures.
method Using Mazur type corks to construct examples.
result Converse of the Lisca-Matic-Plamenevskaya theorem does not hold in general.
Study of 3-manifolds with non-trivial boundary groups, showing they cannot be corks.
problem Investigating non-trivial boundary groups of 3-manifolds and their implications.
method Combination of 3 and 4-manifold techniques.
result Every diffeomorphism of the boundary extends to the full manifold, proving they cannot be corks.
Stein 4-manifolds and corksmath.GT
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 th…
We first construct a genus zero positive allowable Lefschetz fibration over the disk (a genus zero PALF for short) on the Akbulut cork and describe the monodromy as a positive factorization in the mapping class group of a surface of genus zero with five boundary components. We then construct genus zero PALFs on infinit…
Study of contractible manifolds and their twists to determine if they are .
problem Determine if a specific manifold is using twists and handlebody descriptions.
method Analyze two contractible manifolds, one Stein and one non-Stein, with non-trivial boundary mapping classes. Use a specific homotopy sphere to test if it is .
result Show that a specific manifold is indeed .
An infinite order corkmath.GT
We construct an infinite order cork (W,f), which means that W is a smooth compact contractible 4-manifold with Stein structure, and f is a self diffeomorphism of the boundary of W, such that the n-fold composition maps f^{n}=f o f o... o f give rise to smoothly distinct corks (W, f^{n}) for sufficiently large values of…
Small exotic Stein manifoldsmath.GT
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 ; we do this by enlarging corks and plugs.
Knotting corksmath.GT
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 t…
New method finds exotic 4-manifolds from knots, disproving conjecture.
problem Finding exotic 4-manifolds from knots.
method Introduce new cork twists and satellite maps.
result Disprove Akbulut-Kirby conjecture about knots and concordance.
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-manifo…
A simple characterization is given of open subsets of a complex surface that smoothly perturb to Stein open subsets. As applications, complex 2-space C^2 contains domains of holomorphy (Stein open subsets) that are exotic R^4's, and others homotopy equivalent to the 2-sphere but cut out by smooth, compact 3-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 …
Strong corks derived from previous work are proven.
problem Understanding and constructing strong corks.
method Instanton-theoretic invariant and linear combinations of previous corks.
result Proves nontrivial linear combinations of previous corks are strong corks.
Corksmath.HO
Abstract discusses different corks.
problem None specified in abstract.
method None specified in abstract.
result None specified in abstract.
On infinite order corksmath.GT
Infinite order loose corks created.
problem Creating loose corks of infinite order.
method Constructing an infinite order loose cork.
result Demonstrated construction of an infinite order loose cork.
The Akbulut cork cannot transform all exotic 4-manifolds.
problem The Akbulut cork's universality in transforming exotic 4-manifolds.
method Exhibited infinitely many exotic pairs of 4-manifolds not related by any cork.
result The Akbulut cork is not universal.
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.…
Concrete description of infinite order cork automorphism.
problem Describing the infinite order loose-cork automorphism.
method Concatenating the defining ribbon disk by an infinite order isotopy.
result Concrete description of the infinite order cork automorphism.
New corks found that are not strong and exotic.
problem Finding non-strong corks and exotic examples.
method Constructing new corks and analyzing their properties.
result First non-strong corks and new exotic examples.
Surface corks modify 4-manifold structures without changing their homeomorphism type.
problem Understanding how to change 4-manifold structures using surface corks.
method Introducing surface corks as compact, contractible submanifolds intersecting smoothly embedded surfaces in 4-manifolds.
result Explicit construction of a transverse surface cork that is diffeomorphic to a 4-ball.
New infinite order corks found that cannot twist exotic 4-manifolds.
problem Finding new infinite order corks to twist exotic 4-manifolds.
method Proved necessary conditions for generating 4-manifolds by infinite order corks and found non-contractible examples.
result First example of infinite order corks not generating exotic 4-manifolds.
Infinite order corks are described with detailed handle decompositions.
problem Understanding the structure of infinite order corks in 4-manifolds.
method Detailed handle decompositions and embeddings of corks in .
result The twisted doubles of infinite order corks are Gluck twists and log transforms of .
The paper proves trisection genus of Akbulut cork and constructs many corks with trisection genus 3.
problem Determining the trisection genus of contractible 4-manifolds and exotic pairs.
method Proved trisection genus of Akbulut cork and constructed corks with trisection genus 3.
result First examples of contractible 4-manifolds with determined trisection genus except for 4-ball.
Infinite nonabelian corksmath.GT
New corks constructed for infinite nonabelian groups, answering a question.
problem Constructing corks for infinite nonabelian groups.
method Combination of previous results by Auckly-Kim-Melvin-Ruberman, Gompf, and Tange.
result First example of -corks for an infinite nonabelian group .
New method produces corks using Heegaard Floer homology.
problem Creating corks for certain knots.
method Floer-theoretic condition and formalism.
result Many new examples of corks produced.
Found the smallest 4-manifold with a specific Betti number.
problem Finding a 4-manifold with a specific Betti number.
method Provided an explicit example of a cork for a 4-manifold.
result First explicit example of a cork with second Betti number 9.
Corks transform complex curves without changing topology.
problem Transforming complex curves without changing their topological properties.
method Using branched covers of holomorphic disks in the 4-ball and exotic factorizations of quasipositive braids.
result Properly embedded, smooth complex curves that are isotopic through homeomorphisms but not diffeomorphisms.
The paper constructs corks and exotic 4-manifolds with controlled shadow-complexity.
problem Constructing exotic 4-manifolds with controlled shadow-complexity.
method Using corks of Mazur type, the authors construct exotic pairs of 4-manifolds with controlled shadow-complexity.
result An infinite family of exotic pairs of 4-manifolds with controlled shadow-complexity.
New method uses Chern-Simons filtration to study corks and bounding.
problem Studying strong corks and equivariant bounding invariants.
method Utilizes Chern-Simons filtration in instanton Floer homology.
result Examples of corks with specific properties.
Corks and exotic ribbons in math.GT
New ribbon disks in 4D space, non-isotopic to each other.
problem Non-isotopic ribbon disks in 4D.
method Using corks to construct diffeomorphic ribbon disks.
result Non-isotopic ribbon disks constructed in 4D.
Higher order corksmath.GT
New method to separate and create different 4-manifolds.
problem Creating and separating different 4-manifolds.
method Using a single cork and boundary diffeomorphisms.
result Finite families of corks can be separated.
New handle diagrams show infinite order corks in 4-manifolds.
problem Existence of infinite order corks in 4-manifolds.
method Translated earlier proof into handle diagrams and conditions on torus twists.
result First handle diagrams of infinite order corks.
New insights into exotic smoothings of R^4 via group actions and corks.
problem Understanding diffeotopy groups and exotic smoothings of R^4.
method Explicit group actions and cork theory.
result Uncountably many isotopy classes of self-diffeomorphisms for exotic smoothings of R^4.
The paper finds infinitely many Mazur type manifolds and corks with shadow complexity one.
problem Finding manifolds and corks with specific shadow complexity.
method Using Turaev's shadow and contractible special polyhedra with one true vertex.
result The discovery of infinitely many Mazur type manifolds and corks with shadow complexity one.
Homotopy 4-spheres created by iterated Gluck twists of S^4.
problem Understanding the creation and properties of homotopy 4-spheres.
method Using iterated Gluck twists and handle cancellations to show homotopy equivalence to S^4.
result Homotopy spheres Σ_n formed by doubling an infinite order loose-cork are homotopy equivalent to S^4.
On a homotopy -spheremath.GT
Standard proved to be diffeomorphic to a curious homotopy sphere.
problem Determining the diffeomorphism of a curious homotopy sphere to the standard .
method Proof based on properties of homotopy spheres and loose corks.
result The curious homotopy sphere is diffeomorphic to the standard .
We consider a family of corks, denoted , constructed by Akbulut and Yasui. Each cork gives rise to an exotic structure on a smooth 4-manifold via a twist on its boundary . We compute the instanton Floer homology of and show that the map induced on the instanton Floer homology by $τ: Σ…
New invariants detect corks obstructing homology ball extensions.
problem Detecting corks obstructing homology ball extensions.
method Adapting Heegaard Floer homology formalism for local equivalence, using monotonicity and surgery.
result Established new families of corks and proved various examples are strongly non-extendable.
Disproves the Smale Conjecture for S^4 by showing Diff(S^4) is not SO(5).
problem Disproving the Smale Conjecture for S^4.
method Directly showing π₀Diff(S^4) ≠ 0 by proving a loose-cork cannot be a loose-cork.
result Diff(S^4) ≠ SO(5).
On Shake Slice Knotsmath.GT
Study on shake slice knots and proves 0-shake slice knots are slice.
problem Understanding and characterizing shake slice knots.
method Exploring the relation to corks and proving slice properties.
result Proves 0-shake slice knots are slice.
Study relates trisected 4-manifolds to cork twists via γ-curves.
problem Understanding relationships between trisected 4-manifolds and cork twists.
method Analyzes γ-curves derived from topology of relative trisected 4-manifolds and their connections to Torelli and Johnson kernels.
result Establishes a connection between γ-curves and cork twists in relative trisected 4-manifolds.
We utilize the Ozsvath-Szabo contact invariant to detect the action of involutions on certain homology spheres that are surgeries on symmetric links, generalizing a previous result of Akbulut and Durusoy. Potentially this may be useful to detect different smooth structures on 4-manifolds by cork twisting operation.
Protocorks link exotic 4-manifolds, with applications to Floer homology.
problem Understanding exotic 4-manifolds and their transformations.
method Introducing protocorks and proving theorems on their properties and Floer homology.
result Protocorks link exotic 4-manifolds and contribute to understanding Seiberg-Witten invariants.
The Dolgachev Surfacemath.GT
We prove that the Dolgachev surface E(1)_{2,3} admits a handlebody decomposition without 1- and 3- handles, and we draw the explicit picture of this handlebody. We also locate a "cork" inside of E(1)_{2,3}, so that E(1)_{2,3} is obtained from E(1) by twisting along this cork.
Gluck twisting certain knots results in standard 4-spheres.
problem Understanding when Gluck twists yield standard 4-spheres.
method Analyzing smooth homotopy 4-spheres and diffeomorphisms.
result Infinite collection of twisted doubles of corks are standard.
4-manifolds can be exotic after connected sum with S^2 x S^2.
problem Existence of exotic contractible 4-manifolds.
method Constructed a cork that remains exotic after connected sum with S^2 x S^2.
result Existence of exotic pair of contractible 4-manifolds.