Paper defines conditions for warped product gradient Yamabe solitons.
problem Conditions for warped product gradient Yamabe solitons.
method Investigated necessary and sufficient conditions for warped product M=BimesfF to be a gradient Yamabe soliton. result Obtained solutions in steady case with fiber scalar-flat.
The paper explores properties of Finsler manifolds with specific curvature conditions.
problem Investigating Finsler metrics with various curvature conditions.
method Analyzing non-Riemannian (α,β)-metrics and compact Finsler manifolds with specific curvature properties. result Compact Finsler manifolds with relatively non-negative stretch curvature are Landsberg metrics.
Local holomorphic maps preserving (p,p) forms are shown to be isometries.
problem Preserving (p,p) forms under holomorphic maps between Kähler manifolds.
method Analyzing local holomorphic maps between Kähler manifolds, proving isometries up to scalars.
result Holomorphic maps preserving (p,p) forms are isometries under certain conditions.
Base of fibered correspondence is arbitrary correspondence. Fibered correspondence is interesting when we consider relationship between different bundles. However composition of fibered correspondences may not always be defined. Reduced fibered correspondence is defined only between fibers over the same point of base. …
Extends Frohman and Rannard's result to Seifert fiber spaces with singular surfaces.
problem Characterize essential surfaces in Seifert fiber spaces with singular surfaces.
method Extends Frohman and Rannard's approach to handle surfaces with singular fibers.
result Characterizes essential surfaces in Seifert fiber spaces with singular surfaces.
Investigates conditions for GKM fiber bundles and realizability of fiber bundles of GKM graphs.
problem Conditions for GKM fiber bundles and realizability of fiber bundles of GKM graphs.
method Analysis of GKM graphs and fiber bundles, counterexamples, and classification of twist automorphisms.
result Realizability of fiber bundles of GKM graphs depends on the twist automorphism and can be decided in terms of the classification.
Study shows fibered knots' quandles have consistent structure.
problem Understanding the structure of fibered knots through their quandles.
method Analyzes the fiber bundle structure of fibered knots and applies it to their quandles.
result Finiteness and equivalence of knot quandles of fibered knots are discussed.
Study fiber-preserving, orientation-reversing involutions on Seifert fibered 3-manifolds.
problem Characterize fiber-preserving, orientation-reversing involutions on Seifert fibered 3-manifolds.
method Construct a class of involutions, extend product involutions, and use double covering.
result Any fiber-preserving, orientation-reversing involution factors as a product of an orientation-preserving and a specific class of involutions.
Characterizes decomposable torus fiber bundles in low dimensions.
problem Decomposability of torus fiber bundles over a circle.
method Characterization through properties of matrices.
result Complete characterization of decomposable bundles with fiber dimension less than 4.
We study the relationship between fibered ribbon 1-knots and fibered ribbon 2-knots by studying fibered slice disks with handlebody fibers. We give a characterization of fibered homotopy-ribbon disks and give analogues of the Stallings twist for fibered disks and 2-knots. As an application, we produce infinite families…
Introduces Darboux-Lie derivative for fiber bundles.
problem None explicitly stated; focuses on introducing a new derivative.
method Study of Darboux-Lie derivative for fiber-bundle maps.
result Properties of Darboux-Lie derivative for fiber bundles.
The study classifies vector fields tangent to Seifert fiberings.
problem Classifying vector fields tangent to Seifert fiberings.
method Classification based on Seifert fiberings.
result A classification of vector fields tangent to Seifert fiberings.
The paper proves non-existence of positive scalar curvature on certain fiber bundles.
problem The existence of positive scalar curvature metrics on fiber bundles.
method Analyzing fiber bundles over specific manifolds with incompressible or homotopically nontrivial fibers.
result Non-existence of PSC metrics on certain fiber bundles under specific conditions.
New examples show not all homology fiber bundles are topological.
problem Disprove conjecture about homology fiber bundles.
method Construct flat, projective morphisms that are Z-homology fiber bundles. result Disprove conjecture about homology fiber bundles.
Study shows simplicial volume of certain fiber bundles is zero.
problem Understanding simplicial volume in fiber bundles with nonpositive curvature.
method Proved simplicial volume is zero for specific fiber bundles.
result Simplicial volume of certain fiber bundles is zero.
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.
Study of Seifert fibered spaces using surface complexes.
problem Understanding the topology of Seifert fibered spaces.
method Defined surface complex for 3-manifolds, applied to Seifert fibered spaces.
result Surface complex of Seifert fibered spaces contains a subcomplex isomorphic to the curve complex of the base orbifold.
We construct two infinite families of knots each of which admits a Seifert fibered surgery with none of these surgeries coming from Dean's primitive/Seifert-fibered construction. This disproves a conjecture that all Seifert fibered surgeries arise from Dean's primitive/Seifert-fibered construction. The (-3,3,5)-pretzel…
Study subsurface projections in fibered 3-manifolds and their relation to veering triangulations.
problem Understanding the relationship between subsurface projections and veering triangulations in fibered 3-manifolds.
method Investigate the connections between subsurface projections in curve and arc complexes and Agol's veering triangulation.
result Large-distance subsurfaces in fibers correspond to large simplicial regions in veering triangulations.
We show that a compact complex surface which fibers smoothly over a curve of genus >1 with fibers of genus >1 fibers holomorphically. We deduce an improvement of a result in [D Kotschick, Math. Research Letters, 5 (1998) 227-234], and a characterisation of fibered surfaces with zero signature.
Constructs a family to handle unstable fibers on complex surfaces.
problem Handling unstable fibers on complex surfaces.
method Uses Teichmüller theory to construct a degenerating family over the moduli space.
result Any fibered complex surface with unstable fibers can be pulled back from the constructed family.
The concept of F-algebra and its representation can be extended to an arbitrary bundle. We define operations of fibered F-algebra in fiber. The paper presents the representation theory of of fibered F-algebra as well as a comparison of representation of F-algebra and of representation of fibered F-algebra.
Classifies changes between genus one fibered knots.
problem Understanding changes between genus one fibered knots.
method Analyzed all monodromies and determined manifolds.
result Classified all generalized crossing changes between genus one fibered knots.
Study complex surfaces fibered over Teichmüller curves with Veech fibers.
problem Understanding the geometric and topological properties of complex surfaces.
method Compute invariants via monodromy action on fiber's mod-m homology. result Exact values of invariants for all known algebraically primitive Teichmüller curves.
In this paper, we define the primitive/Seifert-fibered property for a knot in S^3. If satisfied, the property ensures that the knot has a Dehn surgery that yields a small Seifert-fibered space (i.e. base S^2 and three or fewer critical fibers). Next we describe the twisted torus knots, which provide an abundance of exa…
Study surgery obstructions for Seifert fibered homology spheres using knot and Heegaard Floer homology.
problem Determining which knots can yield Seifert fibered homology spheres.
method Utilized Heegaard Floer and Knot Floer homology, along with the mapping cone formula, to find obstructions.
result Showed that genus one knots cannot yield Seifert fibered homology spheres with six or more singular fibers.
A fibered concordance of knots, introduced by Harer, is a concordance between fibered knots that is well-behaved with respect to the fibrations. We consider semi-fibered concordance of two component ordered links L=J⊔K with J fibered. These are concordances that restrict to fibered concordances on the first …
Groups with Property (T) have fiber products with Property (T).
problem When does the fiber product of groups with Property (T) have Property (T)?
method Analyzing fiber products of groups with Property (T).
result Fiber products of groups with Property (T) also have Property (T).
Boundary of fiber convex domains is a cohomological sphere.
problem Understanding the boundary properties of fiber convex domains.
method Analyzing smooth fiber convex domains with smooth boundaries.
result The boundary is a cohomological sphere.
Describes 3-manifolds by families of singularly fibered surfaces.
problem Understanding the structure of 3-manifolds through fibered surfaces.
method Explicitly describes a fibration of the complement of a homogeneous braid.
result Each fiber intersects every cross-section of S3. Stability of volume forms on noncompact fiber bundles.
problem Volume form stability on noncompact fiber bundles.
method Generalization of Moser's stability result to noncompact fibers.
result Stability of volume forms proven for noncompact fiber bundles.
Study Veech groups in fibered 3-manifolds, proving no parabolics for fibers.
problem Characterizing Veech groups in fibered 3-manifolds.
method Analyzing pseudo-Anosov monodromies and foliations, proving properties of Veech groups.
result Veech groups in fibers typically contain no parabolic elements.
Paper connects homologically fibered links to solutions of equations.
problem Existence of homologically fibered links in 3-manifolds.
method Interprets existence in terms of first homology group and torsion linking form, or surgery diagrams.
result Computes hc for specific 3-manifolds.
Defines products for fibered corners manifolds, generalizing resolutions.
problem Resolving fibered corners manifolds.
method Introduces a category of fibered corners manifolds with products and transverse fiber products, defining the 'ordered product' for wedge metrics.
result The 'ordered product' is a natural product for wedge metrics.
Study simplicial volume in fiber bundles with connected groups.
problem Understanding simplicial volume in fiber bundles.
method Investigated fiber bundles with connected structure groups.
result Simplicial volume of total space matches trivial bundle under certain conditions.
New research finds infinitely many hyperbolic knots not almost-fibered.
problem Understanding the properties of knots and their complements.
method Examined hyperbolic knots and their Seifert surfaces.
result Existence of infinitely many hyperbolic genus one knots that are not almost-fibered.
The paper constructs trisections for fiber bundles over a circle.
problem Building trisections for fiber bundles over the circle.
method Using sutured Heegaard splittings and diagrams, the paper provides an algorithm to construct relative trisection diagrams.
result Explicit construction of relative trisection diagrams for fiber bundles over the circle.
Simpler proof for Kielak's virtual fibering criterion.
problem Virtual fibering criterion for RFRS groups
method Simpler proof
result Simplified proof of Kielak's criterion
Study non-fibered links' relation to tight contact structures.
problem Understanding non-fibered links and their tight contact structures.
method Analyze non-fibered links with induced partial open books and contact structures.
result Strongly quasipositive non-fibered links induce tight contact structures, but the converse is not always true.
Around 1960, R. Palais and J. Cerf proved a fundamental result relating spaces of diffeomorphisms and imbeddings of manifolds: If V is a submanifold of M, then the map from Diff(M) to Imb(V,M) that takes f to its restriction to V is locally trivial. We extend this and related results into the context of fibered manifol…
We characterize cutting arcs on fiber surfaces that produce new fiber surfaces, and the changes in monodromy resulting from such cuts. As a corollary, we characterize band surgeries between fibered links and introduce an operation called Generalized Hopf banding. We further characterize generalized crossing changes bet…
Knots can only be fibered if each component is fibered and their genus is at least the connected sum.
problem Understanding the fibered nature of band-connected sums of knots.
method Analyzing the genus of fibered knots and their connected sums.
result The genus of a fibered knot is greater than or equal to the genus of its connected sum components.
This paper computes the number of fiberings for non-product manifolds.
problem Computing the number of fiberings for non-product manifolds.
method Analyzes specific manifolds and uses bundle properties.
result Fib(M)=2 for the Atiyah-Kodaira manifold and its finite covers, and Fib(M)=4 for an example. New bases found for Kauffman bracket skein module of fibered torus.
problem Computing Kauffman bracket skein module of fibered 3-manifolds.
method Constructed bases for Kauffman bracket skein module of product annulus and circle, then applied to (β,2)-fibered torus. result Found a new basis for KBSM of (β,2)-fibered torus. We construct an explicit topological model (similar to the topological Springer fibers appearing in work of Khovanov and Russell) for every two-row Springer fiber associated with the even orthogonal group and prove that the respective topological model is homeomorphic to its corresponding Springer fiber. This confirms …
Extends fibering theorems for 3-manifolds.
problem Fibering compact 3-manifolds over the circle.
method Generalizes Moon and Griffiths' theorems.
result New fibering results for 3-manifolds.
Classifies fibering of state surfaces for various knot families.
problem Determining which state surfaces are fibered.
method Algebraic characterization of fibers from state graphs, decomposing graphs into planar components.
result Characterizes fibering for many families of state surfaces.
Study ribbon concordance and minimal compressions, proving new results about fibered knots.
problem Understanding ribbon concordance and minimal compressions of surface homeomorphisms.
method Proving monotonicity of simplicial volume and dilatation under ribbon concordance, algorithmic enumeration of minimal compressions.
result Every fibered knot has only finitely many predecessors in the ribbon-concordance partial order.