The paper explores affine Hirsch foliations on 3-manifolds and classifies their existence.
problem Classifying the existence of affine Hirsch foliations on 3-manifolds.
method Analyzing and constructing 3-manifolds with specific foliations using exchangeable braided links.
result Every closed orientable 3-manifold admits 0, 1, or 2 affine Hirsch foliations, and every case is possible.
Proves Teichmüller space of Hirsch foliation homeomorphic to closed curves.
problem Minimal foliation of a closed 3-manifold by hyperbolic surfaces.
method Proves homeomorphism and uses properties of hyperbolic metrics and homeomorphisms.
result Structure group of bundle is contractible.
The abstract proves a Leray-Hirsch theorem and blow-up formula for Dolbeault cohomology.
problem Cohomology on complex manifolds, especially non-compact ones.
method Proves a Leray-Hirsch theorem and an explicit blow-up formula.
result Explicit blow-up formula for Dolbeault cohomology on complex manifolds.
Abstract: Study of 2-knot groups with restrictions on normal subgroups.
problem Characterizing 2-knot groups based on their normal subgroups.
method Analyzing PD_4-complexes and using properties of π_1(X).
result Characterization of 2-knot groups based on their normal subgroups.
Quantum theory for gerbes connects stack invariants.
problem Quantum theory of gerbes over stacks.
method Proves structure result on Gromov-Witten theory.
result Gromov-Witten invariants of gerbes relate to stack invariants.
In our book on cohomological methods in transformation groups the minimal Hirsch-Brown model was used to good effect. The construction there, however, was rather abstract. Here, for smooth compact connected Lie group actions on smooth closed manifolds, we give a much more explicit construction of the minmal Hirsch-Brow…
Obstruction theory for complex bigraded differential algebras.
problem Understanding extensions and minimal models of bigraded differential algebras with twisted coefficients.
method Development of obstruction theory for Hirsch extensions.
result Proof of uniqueness of relative minimal models and characterization of formality.
New bounds on asymptotic dimension for elementary amenable groups.
problem Bounding the asymptotic dimension of box spaces for elementary amenable groups.
method Using subadditivity of asymptotic dimension in group extensions and properties of box spaces.
result For elementary amenable groups, the asymptotic dimension of box spaces is bounded by their Hirsch length, with equality for a specific subclass.
Study embeddings of 3-manifolds in S4 with nilpotent fundamental groups.
problem Embeddings of 3-manifolds in S4 with specific properties of fundamental groups. method Analyzes embeddings with nilpotent fundamental groups, determines groups with Hirsch length ≤5.
result Identifies all nilpotent groups with Hirsch length ≤5 and torsion-free.
The abstract discusses cohomology of complex manifolds and blow-ups.
problem Calculating cohomologies of complex manifolds and their blow-ups.
method Using sheaf theory and Künneth and Leray-Hirsch theorems.
result Blow-up formulae for complex manifolds are derived.
Rectifies flat singular points for area-minimizing currents.
problem Understanding singularities of area-minimizing currents.
method Analyzes countably (m−2)-rectifiable singular points with flat tangent cones. result The set of singular density-Q points is countably (m−2)-rectifiable and has finite upper Minkowski content. The paper proves a spacetime version of dihedral rigidity for cubes in 3D spacetime.
problem Proving dihedral rigidity for cubic initial data sets in 3D spacetime.
method By studying the level sets of spacetime harmonic functions and extending previous work on dihedral rigidity for prisms in hyperbolic space.
result The paper proves dihedral rigidity for cubes in 3D spacetime, extending previous results.
New insights on nilpotent groups with balanced presentations.
problem Characterizing nilpotent groups with specific properties.
method Analyzing groups with abelian normal subgroups and quotient groups.
result Properties of nilpotent groups with balanced presentations and specific subgroup structures.
Computes monopole Floer homology for three-manifolds.
problem Computing monopole Floer homology for three-manifolds.
method Develops a new framework to study homotopical properties of dga twisted with a specific kind of Maurer-Cartan element, and computes higher operations for the torus.
result Explicit computation of HM∗ for the torus. Paper solves quantum differential equations for projective bundles using Borel multitransforms.
problem Integration of quantum differential equations for P1-bundles. method Introduced Borel (α,β)-multitransforms to reconstruct solutions. result Quantum analog of Leray-Hirsch theorem for quantum cohomology of P1-bundles. We extend Gromov's notion of asymptotic dimension of finitely generated groups to all discrete groups. In particular, we extend the Hurewicz type theorem proven in [B-D2] to general groups. Then we use this extension to prove a formula for the asymptotic dimension of finitely generated solvable groups in terms of their…
In their paper `A new algorithm for recognizing the unknot', in Geometry and Topology', 2 (1998) n. 9, 175-220, the first author and Michael Hirsch presented a then new algorithm for recognizing the unknot. The first part of the algorithm required the systematic enumeration of all discs which support a `braid foliation…
The paper proves a spacetime positive mass theorem for singular initial data sets.
problem Proving the positive mass theorem for initial data sets with corners.
method Extending Hirsch-Kazaras-Khuri's method to singular cases using Hirsch-Miao-Tsang ideas.
result Integral lower bound on spacetime mass and characterisation of zero mass.
The equivariant cohomology ring of a GKM manifold is isomorphic to the cohomology ring of its GKM graph. In this paper we explore the implications of this fact for equivariant fiber bundles for which the total space and the base space are both GKM and derive a graph theoretical version of the Leray-Hirsch theorem. Then…
New metrics with almost nonnegative curvature found on fake RP^6 and RP^14.
problem Finding metrics with specific curvature properties on fake spheres.
method Applying lifting theorem to fake RP^6 and RP^14.
result Metrics of almost nonnegative curvature on fake RP^6 and RP^14.
Let X be a finite CW-complex of dimension q. If its fundamental group π1(X) is polycyclic of Hirsch number h>q we show that at least one of the homotopy groups πi(X) is not finitely generated. If h=q or h=q-1 the same conclusion holds unless X is an Eilenberg-McLane space K(π1(X),1).
In this paper we define two regular homotopy invariants c and i for immersions of oriented 3-manifolds into R^5 in a geometric manner. The pair (c(f),i(f)) completely describes the regular homotopy class of the immersion f. The invariant i corresponds to the 3-dimensional obstruction that arises from Hirsch-Smale theor…
Study on rigidity with non-negative intermediate curvature on low-dimensional manifolds.
problem Extending non-existence theorem of positive scalar curvature to product manifolds.
method Introduced intermediate curvature and studied rigidity conditions.
result Rigidity when intermediate curvature is non-negative in low dimensions.
Holomorphic Koszul-Brylinski homology studied via Dolbeault cohomology.
problem Investigating holomorphic Koszul-Brylinski homology on Poisson manifolds.
method Using Dolbeault cohomology to derive properties of holomorphic Koszul-Brylinski homology.
result Obtained the Leray-Hirsch theorem, Mayer-Vietoris sequence, and Künneth theorem for holomorphic Koszul-Brylinski homology.
Study improves understanding of Ricci curvature in manifolds.
problem Understanding Ricci curvature in manifolds with specific assumptions.
method Exploring m-intermediate Ricci curvature and proving comparison theorems.
result Stable weighted slicing in manifolds with non-negative m-intermediate Ricci curvature has almost non-negative Ricci curvature.
Explains a theorem about curves in the plane.
problem Understanding curves in the plane.
method Simplest case of the Smale-Hirsch theorem.
result Explains the Whitney-Graustein theorem.
Shows flexible sheaves as fibrant objects for Gromov's h-principle.
problem Applying the h-principle to partial differential relations.
method Interprets flexible sheaves as fibrant objects in a model structure.
result Flexible sheaves can be understood as fibrant objects.
We prove that the asymptotic Assouad-Nagata dimension of a connected Lie group G equipped with a left-invariant Riemannian metric coincides with its topological dimension of G/C where C is a maximal compact subgroup. To prove it we will compute the Assouad-Nagata dimension of connected solvable Lie groups and sem…
We show that a flat principal bundle with compact connected structure group and its adjoint bundles of Lie groups have the same cohomology as the trivial bundle, which is done by proving they satisfy the condition for the Leray-Hirsch theorem. This information has been used to construct a cohomology class of the adjoin…
We recognize the Gromoll-Meyer sphere Sigma^7 as the geodesic join of a simple closed geodesic and a minimal subsphere Sigma^5, which can be equivariantly identified with the Brieskorn sphere W^5_3. As applications we in particular determine the full isometry group of Sigma^7, classify all closed subgroups that act fre…
The geometric Hopf invariant of a stable map F is a stable Z_2-equivariant map h(F) such that the stable Z_2-equivariant homotopy class of h(F) is the primary obstruction to F being homotopic to an unstable map. In this paper we express the geometric Hopf invariant of the Umkehr map F of an immersion f:M^m \to N^n in t…
We prove an asymptotic analog of the classical Hurewicz theorem on mappings which lower dimension. This theorem allows us to find sharp upper bound estimates for the asymptotic dimension of groups acting on finite dimensional metric spaces and allows us to prove a useful extension theorem for asymptotic dimension. As a…
The paper studies cohomology of complex manifolds using Morse-Novikov and Dolbeault-Morse-Novikov theories.
problem Analyzing cohomology of complex manifolds using Morse-Novikov theory.
method Establishing invariants, the Leray-Hirsch theorem, and blow-up formula for Dolbeault-Morse-Novikov cohomology.
result Established relations and stabilities of dimensions under complex structure deformations.
Study on minimal foliations in 3D manifolds with specific conditions.
problem Characterizing minimal foliations in 3D manifolds.
method Analyzing Anosov foliations and their intersections.
result Necessary and sufficient conditions for orbit foliation of Anosov flows.
Sharp dimension constraints for positive intermediate curvature metrics are established.
problem Proving sharp dimension constraints for metrics with positive intermediate curvature.
method Constructing counterexamples and extending rigidity results.
result Sharp dimension constraints for positive intermediate curvature metrics are established.
Homogeneous three-spheres have only homogenous foliations.
problem Characterize foliations of homogeneous three-spheres.
method Prove that a three-sphere's metric foliations are homogenous if and only if it is naturally reductive.
result Homogeneous three-spheres have only homogenous foliations.
Integral volume vanishes for manifolds with circle foliations.
problem Integral foliated simplicial volume calculation.
method Regular foliation by circles analysis.
result Integral foliated simplicial volume vanishes.
Proves geometric invariance of signature and cohomology for Riemannian foliations.
problem Defining and proving invariance of geometric invariants for Riemannian foliations.
method Analyzes basic signature and Lichnerowicz cohomology under homotopy equivalence.
result Foliated homotopy invariance of basic signature and cohomology.
Study foliations on Riemannian manifolds with specific vector fields, focusing on geometric properties.
problem Investigate foliations transverse to closed conformal vector fields on Riemannian manifolds.
method Analyze conditions for totally geodesic leaves and geometric constraints on foliations.
result Characterize totally geodesic foliations and classify minimal and constant mean curvature foliations.
The study limits the number of specific foliations with bounded geometry.
problem Bounding the number of isoparametric foliations with bounded geometry.
method Proving finitely many foliations with specific properties and constructing infinite families of non-diffeomorphic foliations.
result There are only finitely many isoparametrically foliated closed connected Riemannian manifolds with bounded geometry, up to foliated diffeomorphism.
Develops deformation theory for symplectic foliations using L∞-algebras.
problem Deformation of symplectic foliations.
method Uses L∞-algebras to control deformation problems. result Establishes a correspondence between small deformations and Maurer-Cartan elements of L∞-algebra. The paper examines a modified Godbillon-Vey class for Reeb foliations and finds it non-trivial for some foliations.
problem Characterizing foliations using the modified Godbillon-Vey class.
method Defined and analyzed the modified Godbillon-Vey class for Reeb foliations.
result The modified Godbillon-Vey class can distinguish non-diffeomorphic foliations and is non-trivial for some foliations.
Complete classification of foliations on spheres from Clifford systems.
problem Classifying foliations on spheres from Clifford systems.
method Classification of homogeneous singular Riemannian foliations of spheres.
result Classification completed for foliations initiated by the second author.
Simple flows on manifold foliations.
problem Transversely oriented foliations on closed manifolds.
method Simple foliated flows on codimension one.
result Existence of simple foliated flows on manifolds.
Study on harmonic maps on weighted Riemannian foliations.
problem Characterize harmonic maps on weighted foliations.
method Analyze transversally f-harmonic and (F,F′)f-harmonic maps. result Equivalence of transversally f-harmonic and (F,F′)f-harmonic maps in minimal foliations. Survey on Killing foliations with technical advantages.
problem Understanding closures of Riemannian foliations.
method Review of Molino's structural theory and transverse isometry theory.
result Closures of Killing foliations described by transverse Killing vector fields.
Proves conjecture about foliations on curved spaces.
problem Completeness of dual foliations on curved spaces.
method Analyzes Riemannian foliations on nonnegatively curved symmetric spaces.
result Foliations split into trivial and single dual leaf foliations.
Simplified proof of foliation closure theorem for linear foliations.
problem Proving the closure of linear foliations on Riemannian manifolds.
method Direct geometric approach, focusing on projectable foliations and compatible connections.
result Smoothness of the closure of linear foliations directly proven.