The paper explores slice disks and their properties using satellite operations and knot Floer homology.
problem Tackles the conjecture about slice disks and their positive Whitehead doubles.
method Uses techniques from knot Floer homology, Seiberg-Witten theory, and Khovanov homology.
result Provides evidence for the conjecture and constructs exotic disks.
New exotic 4D spaces found using knot slicing techniques.
problem Finding non-diffeomorphic exotic 4D spaces.
method Using RBG links to create slice knots with non-diffeomorphic complements.
result Distinguished new exotic 4D spaces using end Floer homology.
New method uses Floer homology to create exotic disk pairs.
problem Creating exotic disk pairs in 4-manifolds.
method Singular instanton Floer homology with Chern--Simons filtration.
result Constructs exotic pairs of slice disks and knots.
New invariants prove exotic slice disks for knots.
problem Proving exotic slice disks for knots.
method Involutive Khovanov homology and derived invariants.
result Reprove exotic slice disks for knots Jn. New surgery operation preserves monotonicity of Lagrangians.
problem Preserving monotonicity of Lagrangians in surgery operations.
method BSP surgery, wall-crossing formula for disk-potentials.
result BSP surgery can preserve monotonicity of Lagrangians.
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.
We give an algorithm which produces infinitely many pairwise exotic Stein fillings of the same contact 3-manifolds, applying positive allowable Lefschetz fibrations over the disk. As a corollary, for a large class of Stein fillings, we realize the topological invariants (i.e. fundamental group, homology group, homology…
The paper constructs exotic knotted surfaces and curves in 4-manifolds.
problem Understanding exotic knotted surfaces and curves in 4-manifolds.
method Local knotting and construction of surfaces and curves.
result First examples of exotically knotted complex curves and symplectic 2-spheres.
The study distinguishes exotic R^4's using knot Floer homology and Casson handles.
problem Distinguishing exotic R^4's with different knot Floer homologies.
method Using Gadgil's end Floer homology and Casson handles.
result A countably infinite family of pairwise nondiffeomorphic chiral exotic R^4's.
Single stabilization is not always enough to make exotic surfaces isotopic.
problem Determining if a single stabilization is sufficient to make exotic surfaces isotopic.
method Study of stabilization distance with satellite operations using Floer theoretic techniques.
result Found examples of exotic disks in the four-ball with arbitrarily large stabilization distance.
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.
Study framed links in simply connected 4-manifolds, linking to exotic phenomena.
problem Understanding framed links in simply connected 4-manifolds and their relation to exotic phenomena.
method Analyzing the set of framed smoothly slice links on the boundary of a 1-handlebody in a closed, simply connected, smooth 4-manifold.
result Exotic 4-manifolds are distinguished by links that are slice in the exotic manifold but not in the standard D4. This paper studies the rational homotopy groups of the group Diff(S4) of self-diffeomorphisms of S4 with the C∞-topology. We present a method to prove that there are many `exotic' non-trivial elements in π∗Diff(S4)⊗Q parametrized by trivalent graphs. As a corollary of…
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…
The paper bounds the genus of surfaces in four-manifolds with indefinite forms.
problem Bounding the genus of surfaces in four-manifolds with indefinite intersection forms.
method Using adjunction inequalities and the 10/8 + 4 theorem, the paper provides methods to bound the genus of surfaces.
result The set of H-slice knots can detect exotic smooth structures on closed 4-manifolds.
The linear slice of quasi-Fuchsian once-punctured torus groups is defined by fixing the complex length of some simple closed curve to be a fixed positive real number. It is known that the linear slice is a union of disks, and it always has one standard component containing Fuchsian groups. Komori and Yamashita proved t…
New 4-manifolds with exotic diffeomorphisms found.
problem Existence of exotic diffeomorphisms in 4-manifolds.
method Proves existence of infinitely many contractible 4-manifolds with exotic diffeomorphisms.
result Infinitely many contractible 4-manifolds with absolutely exotic diffeomorphisms.
Exotic submanifolds in 4-manifolds remain exotic after stabilizations.
problem Constructing exotic codimension-1 submanifolds in 4-manifolds.
method Constructing pairs of exotic codimension-1 submanifolds with diffeomorphic complements and showing they remain exotic after stabilizations.
result Exotic submanifolds remain exotic after any number of stabilizations.
Study knot Floer homology to create concordance invariants and slice genus bounds.
problem Developing concordance invariants using knot Floer homology.
method Using knot Floer homology, define and analyze equivariant concordance invariants.
result Showed a family of strongly invertible slice knots with arbitrarily large equivariant slice genus.
Study exotic Dehn twists in 4-manifolds, producing first known exotic diffeomorphisms.
problem Detecting exotic diffeomorphisms in 4-manifolds.
method 2-parameter families Seiberg-Witten theory over RP2. result Constructed the smallest closed 4-manifold with exotic diffeomorphisms.
New methods distinguish exotic 4-manifolds using Heegaard Floer homology.
problem Distinguishing exotic 4-manifolds from standard ones.
method Using Heegaard Floer homology to define new invariants.
result New invariants remain distinct in covers and can distinguish exotic manifolds.
Constructs exotic proper 2-knots from open 2-handles.
problem Proving topological equivalence of exotic proper knotted surfaces.
method Flexible construction of exotic open 2-handles, distinguishing through genus functions and end Floer homology.
result Constructs infinite families of irreducible exotic proper knotted surfaces.
The paper presents a new method to create exotic 4-manifolds and surfaces that remain exotic after stabilization.
problem Stabilization of exotic 4-dimensional phenomena and knotted surfaces.
method Elementary approach to constructing exotic 4-manifolds and surfaces, including examples in closed, simply connected 4-manifolds.
result The construction yields exotic surfaces in the 4-ball that remain exotic after stabilization, detected by Khovanov homology.
New actions found on exotic spheres using group theory.
problem Understanding smooth transformations on exotic spheres.
method Recent progress in stable homotopy groups of spheres and group theory.
result Smooth circle and cyclic group actions on exotic spheres produced.
Paper finds exotic diffeomorphisms on specific 4-manifolds.
problem Understanding exotic diffeomorphisms on 4-manifolds with b2+=2. method Comparing winding numbers of parameter families.
result 2\mathbb{C}\mathbb{P}^2 \# 10 (-\mathbb{C}\mathbb{P}^2) admits exotic diffeomorphisms.
Detects exotic embeddings in 4-manifolds with boundary.
problem Detecting exotic diffeomorphisms in 4-manifolds with boundary.
method Defined family versions of Kronheimer-Mrowka invariants and used them to find exotic embeddings.
result First example of exotic 3-spheres in a smooth closed 4-manifold with diffeomorphic complements.
Study exotic phenomena in small 4-manifolds using diffeomorphisms.
problem Understanding exotic smooth structures and embeddings in 4-manifolds.
method Using constraints from Seiberg-Witten theory on diffeomorphisms.
result New methods to detect and bound exotic phenomena.
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 t…
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 th…
New exotic structures found on 4-manifolds with specific groups.
problem Finding exotic smooth structures on 4-manifolds with certain fundamental groups.
method Constructing new smooth structures using definite intersection forms and fundamental groups.
result Infinitely many exotic smooth structures on 4-manifolds with fundamental group Z/2Z.
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.
New exotic 4-manifolds found with small trisection genus.
problem Finding exotic 4-manifolds with small trisection genus.
method Constructing genus-3 relative trisections for contractible 4-manifolds.
result Exotic 4-manifolds with boundary have trisection genus of 4.
This paper constructs exotic diffeomorphisms on reducible 4-manifolds with odd b+.
problem Detecting exotic diffeomorphisms on reducible 4-manifolds with odd b+.
method Using a families Bauer-Furuta invariant and pin-cobordism.
result Constructs exotic diffeomorphisms on reducible 4-manifolds with odd b+.
In this paper we introduce exotic twisted T-equivariant K-theory of loop space LZ depending on the (typically non-flat) holonomy line bundle LB on LZ induced from a gerbe with connection B on Z. We also define exotic twisted T-equivariant Chern character that maps the exotic tw…
Classifies geodesic planes outside convex core of geometrically finite 3-manifolds.
problem Classifying geodesic planes in geometrically finite 3-manifolds.
method Constructive proof involving exotic rays and roofs.
result Existence of exotic roofs depends on the existence of exotic rays and bending lamination properties.
Exotic diffeomorphisms found on complex surfaces and 4-manifolds.
problem Finding exotic diffeomorphisms on 4-manifolds.
method Minimal complex surfaces and spin 4-manifolds with S3 boundary. result First known instances of exotic diffeomorphisms of irreducible 4-manifolds.
Study of torus surgeries on knot traces, finding exotic surfaces and traces.
problem Understanding exotic surfaces and traces through torus surgeries.
method Realizing annulus twisting as torus surgery, using key technical insight.
result Exotic elliptic surfaces and traces discovered, improving known geography.
Paper defines end Khovanov homology to detect exotic planes.
problem Detecting exotic Lagrangian and symplectic planes in 4-space.
method Define and apply end Khovanov homology to surfaces in 4-space.
result First known examples of exotic Lagrangian and symplectic planes found.
Paper explores using 0-surgery to find exotic 4-manifolds.
problem Distinguishing smooth structures on 4-manifolds.
method Uses 0-surgery homeomorphisms to generate exotic pairs.
result Found 5 topologically slice knots potentially leading to exotic spheres.
New results on localization of exotic diffeomorphisms in 4-manifolds.
problem Understanding when groups of exotic diffeomorphisms can be localized to smaller embedded submanifolds.
method Analyzing families Seiberg-Witten theory, Dehn twists, and properties of Seifert fibered homology spheres.
result Exotic diffeomorphisms cannot be localized to topologically embedded rational homology balls or homology spheres.
New exotic 4D spaces with nontrivial mappings.
problem Creating exotic copies of R4 with nontrivial mapping class groups. method Modification of diffeomorphism corks to make exteriors simply-connected.
result Construct exotic copies of R4 with mapping class groups of arbitrarily large rank. Construct non-split 2-component links in S4 to produce exotic pairs of 4-manifolds
problem Construct non-split 2-component links in S4 method Construct non-split 2-component links in S4 result Obtain exotic pairs of simply connected, definite 4-manifolds with boundary
New method uses Khovanov homology to distinguish exotic 4-manifolds.
problem Distinguishing exotic compact, orientable 4-manifolds.
method Introduces a new, simple way of using Khovanov homology.
result First analysis-free proof of existence of exotic Mazur manifolds.
The paper creates exotic 4-manifold structures with a specific group.
problem Producing exotic structures on 4-manifolds with infinite dihedral fundamental group.
method Using specific conditions on b2+ and b2−, the paper constructs these structures. result The existence of infinite exotic structures on 4-manifolds with infinite dihedral fundamental group.
We prove that every smoothly embedded surface in a 4--manifold can be isotoped to be in bridge position with respect to a given trisection of the ambient 4--manifold; that is, after isotopy, the surface meets components of the trisection in trivial disks or arcs. Such a decomposition, which we call a \emph{generalized …
Exotic diffeomorphism survives stabilizations on a contractible 4-manifold.
problem Finding exotic diffeomorphisms on contractible 4-manifolds.
method Developed a Pin(2) × Z_2-equivariant refinement for computing Seiberg-Witten Floer homotopy types.
result Constructed an exotic diffeomorphism that survives two stabilizations.
Exotic hypercomplex structures on a torus are proven to not exist.
problem Existence of exotic hypercomplex structures on a torus.
method Classification of complete flat affine structures on real tori using the Obata connection.
result Exotic hypercomplex structures on a torus do not exist.
An explicit example of an exotic symplectic R6 is given. Together with an earlier known example on R4, this yields an explicit exotic symplectic form on R2n for all n≥2.