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.
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.
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.
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.
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.
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.
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+.
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.
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.
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.
Authors prove existence of exotic surfaces and invariants not detecting self-diffeomorphisms.
problem Existence and properties of exotic surfaces and diffeomorphisms.
method Vanishing theorem of family Bauer--Furuta invariant for diffeomorphisms on spin 4-manifolds.
result Family Bauer--Furuta invariants do not detect exotic self-diffeomorphisms on S4 or S2imesS2. The study proves diffeomorphisms can be localized to simpler submanifolds.
problem Localization of exotic diffeomorphisms on compact simply-connected 4-manifolds.
method Localization theorem for diffeomorphisms isotopic to identity after stabilization.
result Diffeomorphisms can be isotoped to simpler submanifolds.
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. 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…
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.
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…
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.
It is known that the long line supports 2ℵ1 many non-diffeomorphic differential structures. We show that the long plane supports a similar number of exotic differential structures, ie structures which are not merely diffeomorphic to the product of two structures on the factor spaces.
New techniques in Khovanov homology help distinguish exotic surfaces in 4-ball.
problem Distinguishing exotic surfaces in the 4-ball that are not diffeomorphic.
method Developed new techniques for distinguishing cobordism maps on Khovanov homology using knot symmetries and braid factorizations.
result Distinguishes smooth surfaces in the 4-ball that are exotically knotted.
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.
Proved boundary Dehn twist is exotic for Milnor fibers with specific conditions.
problem Characterizing exotic diffeomorphisms on Milnor fibers.
method Compared family relative Bauer--Furuta invariants of mapping tori.
result Boundary Dehn twist is an exotic diffeomorphism under given conditions.
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.
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.
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.
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.
The paper studies exotic diffeomorphisms of 4-manifolds with b_+ = 2.
problem Examining exotic diffeomorphisms in 4-manifolds with a specific topological invariant.
method Utilizes Seiberg-Witten invariants for 1-parameter families of 4-manifolds and a gluing formula for connected sums.
result Proves that the mapping class group of certain 4-manifolds is not finitely generated and surjects to Z^∞.
We present a way of constructing and deforming diffeomorphisms of manifolds endowed with a Lie group action. This is applied to the study of exotic diffeomorphisms and involutions of spheres and to the equivariant homotopy of Lie groups.
Detects exotic surfaces without smooth invariants, providing first example of knotted RP².
problem Detecting exotic surfaces without smooth invariants.
method Two versions of families (Seiberg-Witten) generalizations of Donaldson's diagonalization theorem.
result First example of exotically knotted RP² with diffeomorphic complements.
In this article, we construct the first example of a simply connected minimal symplectic 4-manifold homeomorphic but not diffeomorphic to 3CP^2#7CP^2b. We also construct the first exotic symplectic structure on CP^2#5CP^2b.
The paper proves no exotic actions of diffeomorphism groups on 1-manifolds.
problem The study tackles the exotic actions of diffeomorphism groups on 1-dimensional manifolds.
method The approach involves showing any nontrivial homomorphism has a standard form, with countably many embeddings and conjugate actions.
result The groups Diff^r_c(M) have no countable index subgroups, solving a conjecture of Matsumoto.
We provide the first information on diffeotopy groups of exotic smoothings of R^4: For each of uncountably many smoothings, there are uncountably many isotopy classes of self-diffeomorphisms. We realize these by various explicit group actions. There are also actions at infinity by nonfinitely generated groups, for whic…
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…
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 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. 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.
A closed manifold is called a biquotient if it is diffeomorphic to K\G/H for some compact Lie group G with closed subgroups K and H such that K acts freely on G/H. Biquotients are a major source of examples of Riemannian manifolds with nonnegative sectional curvature. We prove several classification results for biquoti…
We construct smooth 4-manifolds homeomorphic but not diffeomorphic to CP^2+6CP^2-bar.
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.
We prove that there exist diffeomorphisms of tori, supported in a disc, which are not isotopic to symplectomorphisms with respect to any symplectic structure. This yields a partial negative answer to a question of Benson and Gordon about the existence of symplectic structures on tori with exotic differential structure.
We construct an infinite family of mutually non-diffeomorphic irreducible smooth structures on the topological 4-manifold S2×S2.
A metric with positive sectional curvature on the Gromoll-Meyer exotic 7-sphere is constructed explicitly. The proof relies on a 2-parameter family of left invariant metrics on Sp(2) and a one-parameter family of conformal deformations via an isoparametric function F on it. One byproduct is a metric with positive secti…
New stable exotic 4-manifolds found with specific topological properties.
problem Identifying conditions for the existence of stable exotic 4-manifolds.
method Investigated fundamental group, Stiefel-Whitney classes, and H_5(π;Z) to determine stable exotic pairs.
result Produced new stable exotica and settings where they do not arise.
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.
Study on four-dimensional Dehn twists and Milnor fibrations, revealing new phenomena.
problem Understanding the monodromy of Milnor fibrations of surface singularities.
method Using Seiberg-Witten invariant and monopole Floer homology.
result Infinite order non-triviality results for boundary Dehn twists.
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 exam…
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…
Study exotic tori and their SL_d(Z) actions, proving many do not admit nontrivial actions.
problem Characterize exotic tori admitting SL_d(Z) actions.
method Compute mapping class groups, analyze homology actions, and prove non-existence of nontrivial actions.
result Many exotic tori do not admit nontrivial SL_d(Z) actions.
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 han…