The symplectic representation of mapping classes is not surjective for certain types of mapping classes.
problem The surjectivity of the symplectic representation of mapping classes, particularly pseudo-Anosov ones, is not always preserved.
method Explicit construction of symplectic matrices with a bi-Perron leading eigenvalue that cannot be represented by orientable pseudo-Anosov mapping classes.
result The symplectic representation of orientable pseudo-Anosov mapping classes is not surjective.
The Hilbert map's image is discussed, showing when it's surjective.
problem Understanding when the Hilbert map is surjective.
method Analyzing the Hilbert map's properties to determine surjectivity.
result Necessary and sufficient conditions for the Hilbert map to be surjective.
Simplified proof of K3 surface period map surjectivity.
problem Surjectivity of period map on K3 surfaces.
method Utilizes hyperkähler geometry and collapsing techniques.
result Simple proof of Todorov's result on K3 surfaces.
A map's critical points and surjectivity are linked in this note.
problem Understanding the critical points and surjectivity of smooth maps between manifolds.
method Analyzing the critical points and dimension of their collection for smooth maps between manifolds.
result The dimension of the critical points collection is at least n-1, affecting surjectivity.
Period maps surjective for certain gravitational instantons.
problem Understanding the uniformization of gravitational instantons.
method Uniformization theorems by Chen--Chen, Chen--Viaclovsky, Collins--Jacob--Lin, and Hein--Sun--Viaclovsky--Zhang.
result Period maps surjective for ALH*, ALG, and ALG* gravitational instantons.
Proves surjectivity of certain smooth maps with non-properness sets.
problem Surjectivity of linear operators and global diffeomorphisms of semialgebraic maps.
method Analytic proof involving C∞ semialgebraic local diffeomorphisms and linear partial differential operators. result A new analytic conjecture for polynomial local diffeomorphisms of Rn implies known results. Study surjectivity of Kirwan map for generalized hyperkähler reduction.
problem Establishing surjectivity of Kirwan map for a specific class of Hamiltonian manifolds.
method Defined a close analogue of hyperkähler reduction for manifolds with equivariant functions under semi-linear G-actions. result Surjectivity of Kirwan map proved for the defined class of manifolds.
The paper proves that certain surface mapping class groups do not virtually surject to the integers.
problem Proving that certain mapping class groups do not virtually surject to the integers.
method Analyzing the structure of finite-index subgroups of mapping class groups of surfaces of genus ≥ 3.
result Finite-index subgroups of mapping class groups of surfaces of genus ≥ 3 do not virtually surject to the integers.
Kirwan surjectivity proved for K-contact manifolds with free S^1-action.
problem Proving Kirwan surjectivity for K-contact manifolds.
method Analogue of Kirwan surjectivity in equivariant basic cohomology.
result Symplectic manifold quotient reproduces Kirwan's surjectivity.
Maps between classifying spaces for certain groups are studied, with rational cohomology results.
problem Analyzing maps between classifying spaces for specific groups.
method Study of maps Θ between null-components of homomorphism and based map spaces. result Surjectivity of map Θ in rational cohomology for certain groups, and non-surjectivity for others. Let G be a compact Lie group, and let LG denote the corresponding loop group. Let (X,ω) be a weakly symplectic Banach manifold. Consider a Hamiltonian action of LG on (X,ω), and assume that the moment map $μ: X \to L\fg^*$ is proper. We consider the function ∣μ∣2:X→R, and use a version of Morse theor…
Study differential invariants of objects and their images under surjective maps.
problem Reconstruction of objects from multiple-view images.
method Analysis of differential invariants under surjective maps, considering projectable and non-projectable cases.
result Constructible isomorphism between object and image invariants in the projectable case.
Surjectivity of Cannon-Thurston map proven for metric graph bundles.
problem Proving surjectivity of Cannon-Thurston map in metric graph bundles.
method Generalized Mj-Sardar's result to include more types of fibers.
result Continuous extension map between boundaries is surjective.
Kirwan surjectivity proven for equivariant Dolbeault cohomology.
problem Understanding cohomology of Kähler quotients for Hamiltonian actions.
method Using Cartan-Chern-Weil theory to define and prove a Kirwan map.
result A natural surjective Kirwan map established between equivariant Dolbeault cohomologies.
Introduces conformal Riemannian morphisms generalizing various Riemannian concepts.
problem None explicitly stated; generalizing Riemannian concepts.
method Introduces conformal Riemannian morphisms and proves properties.
result Every injective conformal Riemannian morphism is an injective conformal immersion, and similar results for surjective and bijective cases.
Continuous map from circle to sphere determined by surface bundle triangulation.
problem Understanding the order of filling a sphere by a continuous map from a circle.
method Introduced triangulation of surface bundles by Agol; uses invariant foliations.
result Order of filling sphere is dictated by natural triangulation of surface bundle.
The paper proves neural networks are almost always surjective, impacting model safety.
problem Ensuring neural networks can generate any output, including harmful content.
method Analyzing fundamental neural architectures and generative models.
result Many neural architectures are almost always surjective, allowing for arbitrary outputs.
The paper studies symmetries in quandles and their relative versions.
problem Understanding symmetries in quandle structures and their transformations.
method Introducing relative versions of inner automorphism and transvection groups, and using them to characterize and classify surjective homomorphisms.
result Characterization of connected homomorphisms and classification of quandle structures under certain symmetry assumptions.
This paper proves a map from flow-spines to contact structures is surjective.
problem Mapping flow-spines to contact structures in 3-manifolds.
method Using positive flow-spines and contact structures, proving surjectivity.
result The map from flow-spines to contact structures is surjective.
The period map for 4-manifolds is dense and surjective under certain conditions.
problem Understanding the space of metrics on 4-manifolds and their harmonic forms.
method Constructing families of metrics by stretching along hypersurfaces.
result The period map is dense and surjective for 4-manifolds with b+=1. Compact complex manifolds with trivial canonical bundle and ∂∂ˉ-Lemma have smooth deformations and a surjective Albanese map.
problem Understanding the structure and deformations of compact complex manifolds with specific properties.
method Analyzing the Kuranishi space, showing smooth deformations, and studying the Albanese map.
result Compact complex manifolds with trivial canonical bundle and ∂∂ˉ-Lemma have smooth deformations and a surjective Albanese map. We use Klyachko's methods to prove that the natural map G to G-hat, where G is a torsion-free group and G-hat is obtained by adding a new generator t and a new relator w, is surjective only if w is conjugate to gt or gt^{-1} for some g in G. This solves a special case of the surjectivity problem for group extensions, r…
New rigidity theorem on static manifolds with boundary.
problem Static metrics on manifolds with boundary.
method Obata-type rigidity theorem, sufficient geometric conditions.
result Scalar curvature map can be locally surjective at static metrics on manifolds with boundary.
P. Buser and P. Sarnak showed in 1994 that the maximum, over the moduli space of Riemann surfaces of genus s, of the least conformal length of a nonseparating loop, is logarithmic in s. We present an application of (polynomially) dense Euclidean packings, to estimates for an analogous 2-dimensional conformal systolic i…
Johnson has defined a surjective homomorphism from the Torelli subgroup of the mapping class group of the surface of genus g with one boundary component to ∧3H, the third exterior product of the homology of the surface. Morita then extended Johnson's homomorphism to a homomorphism from the entire mapping cla…
Non-polynomial growth harmonic maps from the complex plane to the hyperbolic space are studied. Some non-surjectivity results are obtained. Moreover, images of such harmonic maps are investigated with reference to their Hopf differentials.
Assembly maps study topological cyclic homology of group algebras at primes.
problem Study topological cyclic homology of group algebras at primes.
method Use assembly maps to analyze TC(A[G];p) for various groups. result Proves various properties of assembly maps for different groups.
This paper uses Morse-theoretic techniques to compute the equivariant Betti numbers of the space of semistable rank two degree zero Higgs bundles over a compact Riemann surface, a method in the spirit of Atiyah and Bott's original approach for semistable holomorphic bundles. This leads to a natural proof that the hyper…
Study of homology commutativity in separable metrizable spaces.
problem Homology commutativity in separable metrizable spaces.
method Computations and analysis of homology approximants and maps.
result Affirmative solution to the surjectivity problem for locally compact spaces in cohomology.
New properties established for SO(3) quantum representations, showing density and surjectivity.
problem Properties of SO(3) quantum representations of mapping class groups.
method Analyzing roots of unity and maximal ideals of Z[ζ_p] to establish properties.
result SO(3) quantum representations have dense image and are surjective modulo unramified maximal ideals.
Study on hyperKähler moment maps and hypertoric manifolds.
problem Classify hyperKähler manifolds with specific properties.
method Tri-Hamiltonian actions of tori, symplectic and hyperKähler geometry.
result HyperKähler moment maps have connected fibers and are surjective.
Let G be a compact connected Lie group, and (M,ω) a Hamiltonian G-space with proper moment map μ. We give a surjectivity result which expresses the K-theory of the symplectic quotient M//G in terms of the equivariant K-theory of the original manifold M, under certain technical conditions on μ. This result is a natural …
The paper shows how surjective homomorphisms between surface braid groups factor and computes their automorphism groups.
problem Characterizing surjective homomorphisms and automorphisms of surface braid groups.
method Analyzing the structure of surface braid groups and their homomorphisms.
result Surjective homomorphisms factor through forgetful maps and automorphisms are geometric.
We use controlled topology applied to the action of the infinite dihedral group on a partially compactified plane and deduce two consequences for algebraic K-theory. The first is that the family in the K-theoretic Farrell-Jones conjecture can be reduced to only those virtually cyclic groups which admit a surjection wit…
Study cohomology of almost complex and symplectic manifolds under proper maps.
problem Understand cohomology relationships in almost-complex and symplectic settings.
method Analyze cohomology groups of de Rham cohomology and pseudo-holomorphic maps.
result Provide new characterizations and relationships in cohomology groups.
Study on ALH manifolds with boundary, showing surjectivity of scalar curvature map and mass rigidity.
problem Characterizing ALH manifolds with boundary and their mass.
method Scalar curvature deformation analysis and mass rigidity study.
result ALH manifolds that minimize mass integrals are characterized.
Research characterizes traces of Sobolev mappings into manifolds, identifying analytical obstructions and proving surjectivity under certain conditions.
problem Characterizing traces of Sobolev mappings into compact manifolds.
method Analytical and topological methods, including homotopy and fundamental groups.
result Identification of an analytical obstruction when p<m is an integer and πp(N) is non-trivial, and proof of surjectivity under specific conditions. Atiyah's formulation of what is nowadays called the convexity theorem of Atiyah-Guillemin-Sternberg has two parts: (a) the image of the moment map arising from a Hamiltonian action of a torus on a symplectic manifold is a convex polytope, and (b) all preimages of the moment map are connected. Part (a) was generalized b…
In this paper, the first of a series of two, we continue the study of higher index theory for expanders. We prove that if a sequence of graphs is an expander and the girth of the graphs tends to infinity, then the coarse Baum-Connes assembly map is injective, but not surjective, for the associated metric space X. Exp…
The paper studies cohomology on incomplete manifolds and stratified spaces.
problem Analyzing cohomology groups on incomplete Riemannian manifolds and stratified spaces.
method Proves injective/surjective maps between Lp and L2 cohomology groups under certain conditions. result Injective/surjective maps between Lp and L2 cohomology groups are established. Defines relative Dolbeault homology and proves its equivalence with Čech-Dolbeault cohomology.
problem Comparing relative Dolbeault cohomology groups of complex manifolds.
method Uses Čech approach to define relative Dolbeault homology and proves equivalence with Čech-Dolbeault cohomology.
result Relative Dolbeault homology and Čech-Dolbeault cohomology are equivalent.
The study characterizes Nash maps between semialgebraic sets and their properties.
problem Existence of surjective Nash maps between semialgebraic sets.
method Characterization of semialgebraic subsets and their images under Nash maps.
result Characterization of semialgebraic sets that are Nash images of the unit ball.
We consider a twisted version of the Hurewicz map on the complement of a hyperplane arrangement. The purpose of this paper is to prove surjectivity of the twisted Hurewicz map under some genericity conditions. As a corollary, we also prove that a generic section of the complement of a hyperplane arrangement has non-tri…
For the oriented 3-dimensional handlebody constructed from a 3-ball by attaching g 1-handles, it is shown that the natural surjection from the group of orientation preserving diffeomorphisms of it to the mapping class group of it has no section when g is at least 6.
The study generalizes cohomology results for hyperbolic groups.
problem Understanding cohomology of hyperbolic groups and their subgroups.
method Generalizing Gersten's theorem on ℓ∞-cohomology and applying it to hyperbolic groups. result Obtained hyperbolicity criteria for specific groups.
Study of liftable mapping class group for superelliptic covers.
problem Understanding mapping class groups of superelliptic covers.
method Computational and algebraic methods to study the liftable mapping class group.
result The liftable mapping class group is independent of the degree of the cover and has finite abelianization.
Finite-gap solutions approximate jets of initial data for certain BKM systems.
problem Approximating jets of initial data for specific PDE systems.
method Using finite-reduction map to finite-gap solutions of Stäckel systems.
result Full jet-surjectivity for KdV and Kaup--Boussinesq, partial for Camassa--Holm.
Modified constraint operator for localized deformation with dominant energy condition.
problem Handling localized deformation with initial data sets under the dominant energy condition.
method Introduced a modified constraint operator to absorb metric changes and established local surjectivity theorem.
result Promoted dominant energy condition to strict inequality through compactly supported variations.