Paper shows unique decomposition of 3-manifolds and multiplicative property of Reidemeister torsion.
problem Unique decomposition of compact 3-manifolds.
method Study of adjoint Reidemeister torsion on disk sum decompositions.
result Adjoint Reidemeister torsion has a multiplicative property on unique decompositions.
Study 2-complexes' homology properties and torsion growth.
problem Quantitative connections between 1-cycle filling inequalities and homology complexities.
method Geometric lower bounds on first homology of finite covers.
result Geometric lower bound on first homology size of finite covers.
The study explores properties of metric connections with skew torsion and their curvature identities.
problem Investigating curvature properties of metric connections with skew torsion.
method Analyzing the curvature and torsion properties of metric connections with skew torsion.
result Necessary and sufficient conditions for a metric connection with skew torsion to satisfy the Riemannian first and second Bianchi identities are presented.
New connections found with specific torsion properties.
problem Understanding metric connections with specific torsion properties.
method Described Lorentzian manifolds with metric connections having parallel, skew-symmetric torsion.
result Found new Lorentzian manifolds with metric connections having parallel, skew-symmetric torsion.
Recently, Cappell and Miller extended the classical construction of the analytic torsion for de Rham complexes to coupling with an arbitrary flat bundle and the holomorphic torsion for ∂ˉ-complexes to coupling with an arbitrary holomorphic bundle with compatible connection of type (1,1). Cappell and Mil…
A natural connection with totally skew-symmetric torsion on almost contact manifolds with B-metric is constructed. The class of these manifolds, where the considered connection exists, is determined. Some curvature properties for this connection, when the corresponding curvature tensor has the properties of the curvatu…
Study shows conjugacy of torsion in genus 2 surfaces.
problem Torsion elements in mapping class groups of surfaces.
method Proved congruence subgroup property for centralizers of finite subgroups.
result Torsion elements in surfaces of genus ≤ 2 are conjugacy distinguished.
Classifies Riemannian manifolds with specific torsion properties.
problem Classifying Riemannian manifolds with parallel, non-twistorial torsion.
method Classifies complete simply connected Riemannian manifolds with a metric connection having parallel torsion, non-zero vectorial component, and zero twistorial component.
result Classifies complete simply connected Riemannian manifolds with the specified torsion properties.
Bi-orderable groups from left-orderable ones, showing non-profinite properties.
problem Non-profinite properties in bi-orderability and generalized torsion elements.
method Using fiber products to construct bi-orderable groups, showing non-profinite properties.
result Bi-orderability and generalized torsion elements are not profinite properties.
Study of twisted L2-torsion on 3-manifold character varieties.
problem Analyzing L2-torsion on character varieties of 3-manifolds. method Definition and study of twisted L2-torsion on character varieties, proving analyticity in hyperbolic cases. result Twisted L2-torsion is a real analytic function in a neighborhood of holonomy representations of hyperbolic 3-manifolds. Knots without 2-torsion have minimal Khovanov homology rank.
problem Proving the minimality of Khovanov homology rank for knots without 2-torsion.
method Analyzing the effect of rational tangle replacements on Khovanov homology rank.
result Knots without 2-torsion have minimal Khovanov homology rank.
We prove for the Reidemeister-Turaev torsion of closed oriented three-manifolds some finiteness properties in the sense of Goussarov and Habiro, that is, with respect to some cut-and-paste operations which preserve the homology type of the manifolds. In general, those properties require the manifolds to come equipped w…
Study computability of real numbers from group properties.
problem Computability of real numbers from group properties.
method Analyzing L2-Betti numbers and L2-torsion of groups. result Real numbers as L2-Betti numbers or L2-torsion are computable. We construct a higher Whitehead torsion map, using algebraic K-theory of spaces, and show that it satisfies the usual properties of the classical Whitehead torsion. This is used to describe a "geometric assembly map" defined on stabilized structure spaces in purely homotopy theoretic terms.
A geometric interpretation of curvature and torsion of linear transports along paths is presented. A number of (Bianchi type) identities satisfied by these quantities are derived. The obtained results contain as special cases the corresponding classical ones concerning curvature and torsion of linear connections.
The paper explores algebraic properties of Alexander polynomials and Reidemeister torsions for torus knots.
problem Investigating algebraic properties of Alexander polynomials and Reidemeister torsions for torus knots.
method Analyzing twisted Alexander polynomials and Reidemeister torsions of torus knots associated with irreducible SLn(C)-representations. result Proves that coefficients of twisted Alexander polynomials are locally constant functions on the SLn(C)-character variety. Groups with specific properties have vanishing ℓ2-Betti numbers.
problem Understanding ℓ2-Betti numbers for certain groups. method Introduced cheap 1-rebuilding property and used structure theorem of Tucker-Drob.
result First ℓ2-Betti numbers vanish for specified groups. Study on instantons in G2 and Spin(7) manifolds.
problem Investigate instanton properties in G2 and Spin(7) manifolds. method Analyze the characteristic and torsion connections on G2 and Spin(7) manifolds, focusing on parallel torsion and Lee forms. result Conditions for the curvature of the characteristic connection to be a G2 instanton and for the torsion connection to be a Spin(7) instanton are established. Researchers identify knot groups with generalized torsion of order two.
problem Understanding knot groups with specific algebraic properties.
method Analyzing knot groups through generalized torsion, unique root property, and Baumslag-Solitar relations.
result Knot groups with generalized torsion of order two are R-groups and $ar{R}$-groups. We propose a definition for analytic torsion of the contact complex on contact manifolds. We show it coincides with Ray-Singer torsion on any 3-dimensional CR Seifert manifold equipped with a unitary representation. In this particular case we compute it and relate it to dynamical properties of the Reeb flow. In fact th…
Characterizes Hermitian manifolds with Bismut parallel torsion.
problem Understanding curvature properties of Hermitian manifolds with specific torsion.
method Analyzes Bismut curvature tensor and parallel torsion conditions.
result Characterizes Bismut torsion parallel manifolds using Bismut curvature alone.
Abstract: Review and properties of Gieseker stability for Higgs sheaves.
problem Defining and studying Gieseker stability for Higgs sheaves.
method Review and prove properties similar to classical Gieseker stability for coherent sheaves.
result Properties of Gieseker stability for Higgs sheaves extend to Higgs case, including direct sums and morphisms.
New physics-based method calculates Reidemeister torsions for hyperbolic 3-manifolds.
problem Calculating Reidemeister torsions for cusped hyperbolic 3-manifolds.
method Physics of wrapped M5-branes on the manifold.
result Rigorous proof and numerical verification for figure-eight knot complement and several knots.
The curvature properties of a specific type of 6-manifold are explored.
problem Curvature identities on almost Calabi-Yau 6-manifolds with torsion.
method Analysis of the Nijenhuis tensor, torsion connection, and Ricci solitons.
result The curvature of the torsion connection is symmetric and vanishes if the norm of the torsion or scalar curvature is constant.
Curvature and torsion of linear transports along paths in, respectively, vector bundles and the tangent bundle to a differentiable manifold are defined and certain their properties are derived.
Introduces formal frames for manifolds and their properties.
problem Understanding and generalizing frames and connections on manifolds.
method Introduces formal frames, canonical forms, and torsions.
result Equivalence of vanishing torsions to realizability of formal frames as ordinary frames.
The paper studies relationships between twisted torsion and connected sums of 3-manifolds.
problem Understanding the topology and geometry of 3-manifolds through twisted torsion.
method Investigates the multiplicative property of twisted Reidemeister torsion and its connection to the connected sum operation.
result Establishes a relationship between the multiplicative property of twisted torsion and connected sums of 3-manifolds.
We define analytic torsion of Z_2-graded elliptic complexes as an element in the graded determinant line of the cohomology of the complex, generalizing most of the variants of Ray-Singer analytic torsion in the literature. It applies to a myriad of new examples, including flat superconnection complexes, twisted analyti…
We review the Reidemeister torsion, Ray-Singer's analytic torsion and the Cheeger-M"uller theorem. We describe the analytic torsion of the de Rham complex twisted by a flux form introduced by the current authors and recall its properties. We define a new twisted analytic torsion for the complex of invariant differentia…
Paper investigates double coverings and torsions in homology groups.
problem Relationship between cohomology ranks and twisted cohomology groups.
method Analyzes double coverings and torsions in homology groups of Milnor fiber.
result Strict version of Papadima-Suciu's inequality and non-trivial 2-torsion in homology are equivalent.
In this paper, we introduce the notion of Reidemeister torsion for quasi-isomorphisms of based chain complexes over a field. We call a chain map a quasi-isomorphism if its induced homomorphism between homology is an isomorphism. Our notion of torsion generalizes the torsion of acyclic based chain complexes, and is a ch…
In a recent paper, Dimca and Nemethi pose the problem of finding a homogeneous polynomial f such that the homology of the complement of the hypersurface defined by f is torsion-free, but the homology of the Milnor fiber of f has torsion. We prove that this is indeed possible, and show by construction that, for each pri…
We study Dirac-harmonic maps from surfaces to manifolds with torsion, which is motivated from the superstring action considered in theoretical physics. We discuss analytic and geometric properties of such maps and outline an existence result for uncoupled solutions.
Study torsion's impact on 2D affine Killing vectors on homogeneous surfaces.
problem Effects of torsion on affine Killing vectors on homogeneous surfaces.
method Complete description of Lie algebras of affine Killing vector fields on homogeneous surfaces.
result Complete description of Lie algebras of affine Killing vector fields on homogeneous surfaces.
Study curvature properties of G2 connections with skew-symmetric torsion.
problem Investigate curvature identities and solitons on G2 manifolds. method Analyzes curvature identities and properties of G2 connections with skew-symmetric torsion. result Characterizes conditions for curvature to be symmetric and Ricci flat.
New dynamical torsion for contact Anosov flows connects to Reidemeister torsion.
problem Understanding contact Anosov flows and their properties.
method Introducing dynamical torsion and showing its properties.
result Locally constant ratio between dynamical and Turaev torsion.
Reidemeister torsion is algebraic for most 3-manifolds.
problem Characterizing algebraic properties of Reidemeister torsion.
method Defined algebraic numbers and proved them to be algebraic integers for specific 3-manifolds.
result Reidemeister torsions are algebraic integers for most Seifert fibered spaces and infinitely many hyperbolic 3-manifolds.
New group Q has unusual properties like no uniform non-amenability.
problem Properties of acylindrically hyperbolic groups.
method Proving a common quotient and applying to a specific group.
result Found a group Q with strong fixed point properties and unusual characteristics. Paper introduces a new multilinear functional for spectral triples and computes its properties.
problem Computing properties of spectral triples and their associated Hodge operators.
method Introduces a new multilinear functional for spectral triples and computes its properties using noncommutative residue and perturbed de-Rham Hodge operators.
result Recover two forms, torsion of the linear connection, and four forms by the noncommutative residue and perturbed de-Rham Hodge Dirac triple.
Paper explores topological invariance in torsion-sensitive intersection homology.
problem Torsion-sensitive intersection homology's topological invariance.
method Analogous to classical intersection homology, investigates invariance of sheaf complexes.
result Provides torsion-sensitive versions of existing invariance theorems and new results.
Classifies 7D manifolds with specific geometric properties.
problem Classifying 7D manifolds with parallel skew-symmetric torsion and G2 holonomy. method Extending Friedrich's work, using classification techniques for naturally reductive spaces and nearly parallel G2-structures. result Complete classification of 7D manifolds with the specified properties.
Quaternionic curves with specific torsion properties don't exist.
problem Existence of quaternionic Bertrand curves with non-zero torsion and bitorsion.
method Definition of quaternionic (1,3)-Bertrand curves using Type 2-Quaternionic Frame and Matsuda-Yorozu method.
result No quaternionic Bertrand curves with non-zero torsion and bitorsion exist.
Study moduli spaces of 3D homogeneous manifolds with specific properties.
problem Characterize moduli spaces of oriented Type A manifolds in 3D.
method Analyze manifolds with symmetric Ricci tensor, torsion, and varying dimensions.
result Determine all symmetry groups in 3D torsion-free manifolds.
Defines universal L2-torsion for 3-manifolds, linking to polytopes.
problem Calculating L2-torsion for 3-manifolds. method Defining universal L2-torsion in terms of universal covering chain complex, studying its properties. result Identifies universal L2-torsion with polytopes and various invariants. Constructs equivariant analytic torsion for proper actions on manifolds.
problem Analytic torsion for proper actions on Riemannian manifolds.
method Equivariant construction of Ray-Singer analytic torsion for proper, isometric actions.
result Generalizes earlier results to noncompact conjugacy classes under suitable conditions.
The purpose of this article is to give an explicit formula for all curves of constant torsion τ in the unit two-sphere S2(1). These curves and their basic properties have been known since the 1890's, and some of these properties are discussed in the Appendix. Some example curves, computed with a standard ODE packa…
We study the irreducible decomposition under Sp(2n, R) of the space of torsion tensors of almost symplectic connections. Then a description of all symplectic quadratic invariants of torsion-like tensors is given. When applied to a manifold M with an almost symplectic structure, these instruments give preliminary insigh…
The equations for a G_2-structure with torsion on a product M^7 = N^6 x S^1 are studied in relation to the induced SU(3)-structure on N^6. All solutions are found in the case when the Lee-form of the G_2-structure is non-zero and N^6 is a six-dimensional nilmanifold with half-integrable SU(3)-structure. Special propert…