Study determines Borsuk-Ulam property for maps between torus and Klein bottle.
problem Determining maps with Borsuk-Ulam property between torus and Klein bottle.
method Analyzing homotopy classes of maps and using fundamental groups.
result Identifies specific homotopy classes of maps with Borsuk-Ulam property.
The paper identifies Borsuk-Ulam property for specific map classes between torus and Klein bottle.
problem Determining Borsuk-Ulam property for map classes between torus and Klein bottle.
method Using homotopy classes and free involutions of spaces, the paper analyzes fundamental groups to find the property.
result Homotopy classes of maps from torus to Klein bottle with specific involutions have the Borsuk-Ulam property.
The paper connects sectional category and parametrized Borsuk-Ulam property for fibrations.
problem Parametrized Borsuk-Ulam property for fibrations.
method Investigation of sectional category connections and fibrations.
result The sectional category of q being larger than qp′ implies the parametrized Borsuk-Ulam property for the triple (p,τ;p′). The paper generalizes the Borsuk-Ulam theorem to surfaces and cyclic actions.
problem Generalizing the Borsuk-Ulam theorem to surfaces and cyclic actions.
method Algebraic criterion involving braid groups and homology groups.
result Determines the Borsuk-Ulam property for maps from surfaces to R^2.
We consider spaces with free involutions that satisfy the Borsuk - Ulam theorems (BUT-spaces). There are several equivalent definitions for BUT-spaces that can be considered as their properties. Our main technical tool is Yang's cohomological index.
Let M and N be topological spaces such that M admits a free involution $\τ$. A homotopy class β ∈ [M, N ] is said to have the Borsuk-Ulam property with respect to $\τ$ if for every representative map f : M → N of β, there exists a point x ∈ M such that f ($\τ$ (x)) = f (x). In the case where M i…
Let (X, t, S) be a triple, where S is a compact, connected surface without boundary, and t is a free cellular involution on a CW-complex X. The triple (X, t, S) is said to satisfy the Borsuk-Ulam property if for every continuous map f:X-->S, there exists a point x belonging to X satisfying f(t(x))=f(x). In this paper, …
The Borsuk-Ulam theorem is applied to 3-manifolds with Nil geometry.
problem Determining involutions on 3-manifolds with Nil geometry.
method Applying the Borsuk-Ulam theorem to closed, connected 3-manifolds with Nil geometry.
result All free involutions on the 3-manifolds and their Borsuk-Ulam index are determined.
In this paper we present a Kakutani type theorem that is equivalent to the Borsuk--Ulam theorem for manifolds.
Extends Borsuk-Ulam theorem with applications in sphere coverings and colorings.
problem Complexity bounds and structural insights for triangulated sphere mappings.
method Combinatorial labeling and order type analysis of finite point sets.
result New topological Hall theorem and generalizations of hypergraph Hall theorems.
Let M be a Seifert manifold which belongs to the geometry Flat. In this work we determine all the free involutions τ on M, and the Borsuk-Ulam indice of (M,τ).
Thickenings of a metric space capture local geometric properties of the space. Here we exhibit applications of lower bounding the topology of thickenings of the circle and more generally the sphere. We explain interconnections with the geometry of circle actions on Euclidean space, the structure of zeros of trigonometr…
The paper proves no multiple equichordal points exist in convex bodies.
problem Existence of multiple equichordal points in convex bodies.
method Topological tools like the Borsuk-Ulam theorem and analysis of convex body properties.
result Nonexistence of multiple equichordal points in n-dimensional convex bodies for n≥2. New example solves topological dynamics problem.
problem Embedding compact metric space into cubical shift.
method Borsuk-Ulam theorem, p-adic completions, equivariant Sullivan conjecture.
result Existence of a compact metric space not embeddable into a cubical shift.
New theorem connects distant points and identical points on manifolds.
problem Continuous maps and distant points on manifolds.
method Qualitative extension of Hopf theorem, using topological 'distant' points.
result Existence of connected component containing both distant and identical points.
The paper connects geometric and topological concepts to bound distances between metric spaces.
problem Bounding distances between metric spaces using Gromov-Hausdorff distance.
method Using Borsuk-Ulam theorems and Vietoris-Rips complexes, the paper obstructs the existence of certain continuous maps between complexes to bound discontinuities of functions.
result The paper provides new bounds on Gromov-Hausdorff distances between spheres of different dimensions.
We compute the rings H∗(N;F2) for N a closed Sol3-manifold and then determine the Borsuk-Ulam indices BU(N,φ) with φ=0 in H1(N;F2).
Bounds and constructions for Gromov-Hausdorff distance between spheres.
problem Calculating distances between spheres using Gromov-Hausdorff metric.
method Explicit constructions and topological ideas based on Borsuk-Ulam theorem.
result Lower bounds are tight for specific cases of sphere distances.
We prove several new results around Gromov's waist theorem. We give a simple proof of Vaaler's theorem on sections of the unit cube using the Borsuk--Ulam--Crofton technique. We consider waists of real and complex projective spaces, flat tori, convex bodies in Euclidean space. We establish waist-type results in terms o…
The study connects projective codes to the distribution of zeros of odd maps.
problem Understanding the distribution of zeros of odd maps from spheres to Euclidean space.
method Using the topology of the space of probability measures on the sphere.
result Generalization of the Borsuk-Ulam theorem and its four consequences.
The paper studies which branched covers can be lifted to braided embeddings.
problem Which branched covers lift to braided embeddings?
method Analyzes conditions for liftability using Hansen's criterion and examples in different dimensions.
result Not all branched covers lift to braided embeddings; examples are provided in various dimensions.
Researchers create black holes isometric to extremal Reissner-Nordström.
problem Third law of black hole thermodynamics.
method Novel Ck characteristic gluing procedure and scalar field pulses. result Definitive disproof of the third law of black hole thermodynamics.
Tucker and Ky Fan's lemma are combinatorial analogs of the Borsuk-Ulam theorem (BUT). In 1996, Yu. A. Shashkin proved a version of Fan's lemma, which is a combinatorial analog of the odd mapping theorem (OMT). We consider generalizations of these lemmas for BUT-manifolds, i.e. for manifolds that satisfy BUT. Proofs rel…
We introduce and study a new family of extensions for the Borsuk-Ulam and topological Radon type theorems. The defining idea for this new family is to replace requirements of the form `a subset that is large in some sense goes to a singleton' with requirements of the milder form `a subset that is large in some sense go…
Introduces injective category number for continuous maps, linking classical and contemporary research.
problem Understanding conditions for a continuous map to be injective.
method Defines injective category number and examines its behavior under various operations.
result Provides a cohomological lower bound and expressions for injective category numbers in specific cases.
The paper explores how to reduce classification tasks to optimization problems in Euclidean space.
problem Understanding the minimum dimension needed for reducing classification tasks to optimization problems.
method Developed a generalization of the Borsuk-Ulam Theorem to analyze the expressivity of reductions.
result The minimum Euclidean dimension required can be exponentially larger than the VC dimension, even for slightly non-trivial reductions.
Let V and W be orthogonal representations of G with VG=WG={0}. Let S(V) be the sphere of V and f:S(V)→W be a G-equivariant mapping. We give an estimate for the dimension of the set Zf=f−1{0} in terms of dimV and dimW, if G is the torus Tk, or the p-torus $\ma…
The width w of a curve γ in Euclidean space Rn is the infimum of the distances between all pairs of parallel hyperplanes which bound γ, while its inradius r is the supremum of the radii of all spheres which are contained in the convex hull of γ and are disjoint from γ. We use a mixture of topological and…
It is proved that for a product action of (Zp)k on a product of (mod p) homology spheres Nn1×...×Nnk, where all ni's are assumed to be odd if p is odd, and any continuous map f:Nn1×...×Nnk→Rm the set $A(f)=\{x\in N^{n_1}\times...\times N^{n_k}…
Study of f-neighbors in Riemannian manifolds, proving infinite set of distances.
problem Exploring variations of Hopf theorem in Riemannian manifolds.
method Investigates continuous maps of compact Riemannian manifolds to Rm and introduces f-neighbors. result Set of distances realized as visual f-neighbors is infinite. 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).
We prove recognition theorems for codimension one manifold factors of dimension n≥4. In particular, we formalize topographical methods and introduce three ribbons properties: the crinkled ribbons property, the twisted crinkled ribbons property, and the fuzzy ribbons property. We show that X×R i…
The study shows that several properties are not profinite invariants.
problem Determining which properties are profinite invariants.
method Combining Rips constructions and iterated group-theoretic Dehn filling on hyperbolic virtually special groups.
result Several properties (stable commutator length, quasimorphisms, property NL, property FW∞, property FA, and non-abelian free subgroups) are not profinite invariants. We show that all finite-dimensional resolvable generalized manifolds with the piecewise disjoint arc-disk property are codimension one manifold factors. We then show how the piecewise disjoint arc-disk property and other general position properties that detect codimension one manifold factors are related. We also note …
The paper explores higher property T in lattices and its connections to geometric phenomena.
problem Understanding higher property T in lattices and related geometric phenomena.
method Operator-algebraic characterizations of higher property T and connections to lattice geometry.
result Unified framework for understanding higher property T and related geometric phenomena.
Asymptotic property C was introduced by Dranishnikov to study spaces with infinite asymptotic dimension. We show that asymptotic property C is preserved by infinite products. We also show that countable restricted direct products of countable groups with finite asymptotic dimension have asymptotic property C. Then we i…
Groups of importance in group theory have flexible stability properties.
problem Stability and flexibility of groups in geometric and combinatorial group theory.
method Establishing Kirchberg's Local Lifting Property and Lubotzky--Shalom's Property FD for specific groups.
result Groups like 3-manifold groups, limit groups, and certain one-relator groups are very flexibly stable. This paper generalizes property (QT) to a broader class of groups.
problem Proving property (QT) for a wider range of groups.
method Using projection complex machinery and hierarchical hyperbolic groups.
result Established sufficient conditions for groups to have property (QT).
Proves a vanishing property for symplectic manifold cohomology.
problem Generalizing complex geometry results to symplectic geometry.
method Based on Tseng and Zhou's vanishing property under symplectic flatness.
result Establishes necessity of symplectic flatness for certain results.
Survey on foliations and diffeomorphism groups.
problem Relationship between algebraic and homotopical properties.
method Survey and analysis of existing literature.
result Explains the connection between diffeomorphism groups and foliations.
We prove extension theorems for several geometric properties such as asymptotic property C (APC), finite decomposition complexity (FDC), strict finite decomposition complexity (sFDC) which are weakenings of Gromov's finite asymptotic dimension (FAD). The context of all theorems is a finitely generated group G with a …
Study generalizes property elicitation to imprecise probabilities.
problem Minimizing risk over imprecise probability distributions.
method Maximin risk minimization over a set of imprecise probabilities.
result Conditions for elicitability of IP-properties.
Introduces bounded scale measure and generalizes property A.
problem Defining property A for large scale spaces with bounded geometry.
method Introduces bounded scale measure, shows its coarse invariance, and generalizes property A.
result Definition of property A for large scale spaces with bounded scale measure is a coarse invariant.
Study cohomological and metric properties of non-Kähler complex manifolds.
problem Understanding cohomological invariants and metrics of non-Kähler complex manifolds.
method Partial account of problems through cohomological and metric properties.
result Partial insights into cohomological and metric properties of non-Kähler manifolds.
The paper explores the twisted Rokhlin property in mapping class groups of surfaces.
problem Classifying surfaces whose mapping class groups have the twisted Rokhlin property.
method Generalizing the Rokhlin property to the twisted version, the authors classify surfaces based on their mapping class groups' properties.
result The mapping class groups of connected orientable infinite-type surfaces without boundaries have the twisted Rokhlin property, while those of other surfaces do not.
The balance property is crucial for insurance pricing, ensuring total actuarial price equals loss. Maximum likelihood GLMs fulfill it, but Lindholm-Wüthrich suggests three methods, with constrained GLM being superior.
problem Ensuring the balance property in insurance pricing models
method Using constrained GLM fitting
result Constrained GLM fitting is superior to the two previously discussed balance correction methods
We prove some general results about quasi-actions on trees and define Property (QFA), which is analogous to Serre's Property (FA), but in the coarse setting. This property is shown to hold for a class of groups, including SL(n,Z) for n≥3. We also give a way of thinking about Property (QFA) by breaking it down …
We present examples of metric spaces that are not Riemannian manifolds nor dimensionally homogeneous that satisfy the Tetrahedral Property. In spite of that, Euclidean cones over metric spaces with small diameter do not satisfy this property. We extend Sormani's Tetrahedral Property to a less restrictive property and p…