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.
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…
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…
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.
Study exotic smoothness structures on R^4 using set-theoretic forcing.
problem Characterize small exotic smooth R^4 structures.
method Assign set-theoretic forcing to Casson handles and use models in Grothendieck toposes.
result Obtain deformation of usual complex functions to noncommutative operators.
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.
New 4D shapes found that defy smoothness rules.
problem Smoothness rules in 4D shapes don't always hold.
method Applied reflection group trick to exotic 4-manifolds.
result Found counterexamples to smooth Borel conjecture.
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 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.
Quantum gravity linked to exotic smoothness structures.
problem Quantum gravity and exotic smoothness structures.
method Developed smooth quantum gravity using exotic smoothness structures.
result Identified a wildly embedded 3-manifold as a quantum state.
We construct Anosov diffeomorphisms on manifolds that are homeomorphic to infranilmanifolds yet have exotic smooth structures. These manifolds are obtained from standard infranilmanifolds by connected summing with certain exotic spheres. Our construction produces Anosov diffeomorphisms of high codimension on infranilma…
We produce infinite families of exotic actions of finite cyclic groups on simply connected smooth 4-manifolds with nontrivial Seiberg-Witten invariants.
The paper finds infinite families of exotic spheres with free actions.
problem Detecting smooth free S1 and S3 actions on exotic spheres. method Topological modular forms.
result Infinite families of very exotic spheres with free actions.
Study on exotic smooth embeddings of surfaces in 4-manifolds, revealing different properties and complexities.
problem Understanding exotic smooth embeddings of surfaces in 4-manifolds.
method Analyzing smooth, proper embeddings of noncompact surfaces in 4-manifolds, focusing on exotic planes and annuli.
result Exotic planes and annuli exhibit radically different properties, with one class being simple enough to draw explicit level diagrams.
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 c…
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 trans…
Study shows exotic Dehn twists on certain 3-sphere fillings.
problem Extending group actions from boundaries to interiors of 4-manifolds.
method Analyzing Dehn twists on Seifert homology spheres and their fillings.
result Dehn twists on certain fillings are infinite order exotic.
Study on 4-manifolds with exotic smooth structures and Z_2 fundamental group.
problem Exploring smooth structures on 4-manifolds with specific fundamental groups.
method Construction of irreducible, smooth, oriented, closed, definite 4-manifolds with Z_2 fundamental group and specific Betti numbers.
result Proves existence of infinitely many smooth structures on definite 4-manifolds with positive second Betti number and Z_2 fundamental group.
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 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.
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 ext…
Study shows how any group can be fundamental of a non-orientable 4-manifold with exotic structure.
problem Realizing any finitely presented group as the fundamental group of a non-orientable 4-manifold with exotic structure.
method Performing a Gluck twist to obtain an exotic smooth structure, and using 2-covers to stabilize the structure.
result Any finitely presented group can be realized as the fundamental group of a non-orientable 4-manifold with exotic structure.
Introduces slice knots and concordance, linking to exotic smooth structures.
problem Understanding slice knots and their equivalence through concordance.
method Explains connections and introduces filtrations, satellite operations.
result Connections between slice knots and exotic smooth structures on \(\mathbb{R}^4\).
Constructs Riemannian foliations with exotic tori as leaves.
problem Creating Riemannian foliations with exotic tori.
method Smooth fiber bundles with exotic tori fibers and finite abelian fundamental group total space.
result Examples of Riemannian foliations with exotic tori leaves and finite abelian fundamental group total space.
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 de…
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…
The paper finds small exotic 4-manifolds with free abelian groups.
problem Finding small exotic 4-manifolds with specific fundamental groups.
method Pairwise non-diffeomorphic closed irreducible 4-manifolds with free abelian fundamental group of rank less than three.
result Explicit mechanism to compute equivariant intersection form and stabilization result for exotic structures.
Software simplifies triangulations of 4-manifolds, revealing exotic structures.
problem Understanding smooth 4-manifolds from discrete and algorithmic perspectives.
method New software tools, including an algorithm for triangulations from Kirby diagrams and a heuristic for simplification.
result Presented new triangulations of exotic pairs, corks, and plugs, including the smallest known K3 surface.
New tools prove smooth actions on exotic spheres.
problem Existence of smooth actions on exotic spheres.
method Homotopy-theoretic tools, complex and quaternionic Mahowald invariants.
result Existence of smooth U(1)- and Sp(1)-actions on exotic spheres. The study identifies criteria for 3-manifolds to be boundaries of exotic 4-manifolds.
problem Determining which 3-manifolds can be boundaries of exotic 4-manifolds.
method Provided criteria and examples of 3-manifolds that can be boundaries of 4-manifolds with infinitely many distinct smooth structures.
result Identified specific types of 3-manifolds (weakly fillable contact, non-vanishing Heegaard Floer invariant) that are boundaries of 4-manifolds with infinitely many distinct smooth structures.
Study finite group actions on exotic aspherical space forms.
problem Classify finite group actions on M#Σ where M is a closed aspherical space form and Σ is an exotic n-sphere. method Combines geometric and topological rigidity results with smoothing theory and spectral sequence computations.
result Classification of free actions of finite groups on M#Σ when M is 7-dimensional. 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.
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.
New definite 4-manifolds found with non-cyclic groups.
problem Finding exotic smooth structures on 4-manifolds with specific fundamental groups.
method Constructing infinitely many non-diffeomorphic structures.
result Infinitely many pairwise non-diffeomorphic definite 4-manifolds with Z/2imesZ/2 fundamental group. 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.
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.
New exotic 4-manifolds found with fiber bundles.
problem Finding non-diffeomorphic exotic 4-manifolds.
method Smooth fiber bundles over the circle.
result Exotic 4-manifolds with non-simply-connected fiber.
Paper constructs exotic spacetimes with same physical properties.
problem Whether two topologically identical manifolds can have different geometries.
method Computational approach to produce physical models on exotic spheres.
result Lorentzian metrics on homeomorphic but not diffeomorphic manifolds with same physical properties.
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. The paper investigates exotic smooth structures on manifolds with group actions.
problem Existence of homeomorphic but not diffeomorphic smooth manifolds with shared basic spectra.
method Investigates Riemannian Laplacian eigenvalues and eigenfunctions on manifolds with compact Lie group actions.
result Establishes the existence of homeomorphic yet not diffeomorphic manifolds with shared basic spectra.
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 operatio…
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 ap…
New manifold copies found through knot iterations.
problem Finding infinitely many exotic smooth structures on a 4-manifold.
method Using a specific knot and its iterates to attach 2-handles to a fixed compact manifold.
result Infinitely many absolutely exotic copies of a 4-manifold are constructed.
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 w…
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.
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.…
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.
The study proves exotic smooth structures and equivalent genus functions in 4-manifolds.
problem Understanding exotic smooth structures and genus functions in 4-manifolds.
method Proving the existence of infinitely many exotic structures and equivalence of genus functions.
result Genus functions of exotic 4-manifolds are pairwise equivalent and stable under certain operations.