Analytic torsion and Reidemeister torsion help show exponential growth of torsion in cohomology.
problem Exponential growth of torsion in the cohomology of arithmetic groups.
method Analytic torsion and Reidemeister torsion, applied to fibered cusp ends of manifolds.
result Exponential growth of torsion in the cohomology of arithmetic groups.
Explicitly expresses torsion functions on lens spaces.
problem Calculating torsion on lens spaces.
method Expressed torsion functions in terms of Hurwitz zeta function and Ray-Singer torsion.
result Found that torsion functions vanish at the origin and determined their values.
The paper compares two torsion invariants in complex vector bundles.
problem Comparing two torsion invariants in complex vector bundles.
method Constructing Bismut-Lott analytic torsion classes and showing they coincide with Igusa-Klein torsions.
result Bismut-Lott analytic torsion classes coincide with Igusa-Klein torsions for trivial flat line bundles.
Analytic torsion matches Ray-Singer for specific nilmanifolds.
problem Matching analytic torsion with Ray-Singer in specific nilmanifolds.
method Examined rank two distributions on 5D nilmanifolds, proving torsion equality.
result Analytic torsion equals Ray-Singer torsion in these specific nilmanifolds.
New spectral torsion defined for rescaled Dirac operators.
problem Defining spectral torsion for rescaled Dirac operators.
method Using three vector fields and noncommutative residue.
result Computed spectral torsion for one form rescaled Dirac operators.
The paper examines torsions in Minkowskian product of Finsler metrics.
problem Investigating Cartan torsion and mean Cartan torsion in Minkowskian product of Finsler metrics.
method Deriving explicit formulas for Cartan torsion and mean Cartan torsion, analyzing their geometric behavior, and providing conditions for bounded norm.
result Both Cartan torsion and mean Cartan torsion decompose additively if and only if the Minkowskian product is Euclidean, and a necessary and sufficient condition for the norm of the mean Cartan torsion to remain bounded in the Euclidean case is provided.
Study on stable torsion length in groups, showing it vanishes in crystallographic groups and providing algorithms for computation.
problem Understanding the stable torsion length in groups, especially in crystallographic and free products of groups.
method Developed linear programming and exact algorithms to compute stable torsion length in free products of groups and finite groups.
result Showed that stable torsion length vanishes in crystallographic groups and provided exact computations for nontrivial examples.
Inspired by the work of Boris Vertman on refined analytic torsion for manifolds with boundary, in this paper we extend the construction of the Cappell-Miller analytic torsion to manifolds with boundary. We also compare it with the refined analytic torsion on manifolds with boundary. As a byproduct of the gluing formula…
Analytic torsion studied for fibred boundary metrics, with applications to conic degeneration.
problem Analytic torsion of fibred boundary metrics and conic degeneration.
method Established invariance and gluing formula for renormalized analytic torsion under deformations of metrics.
result Recovery of a result by Sher and Guillarmou about analytic torsion under conic degeneration.
The refined analytic torsion associated to a flat vector bundle over a closed odd-dimensional manifold canonically defines a quadratic form τ on the determinant line of the cohomology. Both τ and the Burghelea-Haller torsion are refinements of the Ray-Singer torsion. We show that whenever the Burghelea-Haller torsi…
Classifies simply-connected pluriclosed manifolds with parallel Bismut torsion.
problem Classifying specific types of manifolds with parallel Bismut torsion.
method Complete classification through mathematical analysis.
result Established a splitting theorem for certain manifolds.
Hypersymplectic structures with torsion on Lie algebroids are investigated. We show that each hypersymplectic structure with torsion on a Lie algebroid determines three Nijenhuis morphisms. From a contravariant point of view, these structures are twisted Poisson structures. We prove the existence of a one-to-one corres…
Explicit formula for Reidemeister torsion of two-bridge knots.
problem Calculating Reidemeister torsion for two-bridge knots.
method Provided an explicit formula and proved vanishing identities.
result Adjoint Reidemeister torsion satisfies vanishing identities.
In this paper we show that the Ray-Singer complex analytic torsion is trivial for even dimensional Calabi-Yau manifolds. Then we define the quaternionic analytic torsion for quaternionic manifolds and prove that they are metric independent. In dimension four, the quaternionic analytic torsion equals to the self-dual an…
We introduce multi-torsion, a spectral invariant generalizing Ray-Singer analytic torsion. We define multi-torsion for compact manifolds with a certain local geometric product structure that gives a bigrading on differential forms. We prove that multi-torsion is metric-independent in a suitable sense. Our definition of…
In a recent joint work with V. Turaev (cf. math.DG/9810114) we defined a new concept of combinatorial torsion which we called absolute torsion. Compared with the classical Reidemeister torsion it has the advantage of having a well-defined sign. Also, the absolute torsion is defined for arbitrary orientable flat vector …
In the Khovanov homology of links, presence of Z2-torsion is a very common phenomenon. Finite number of examples of knots with Zn-torsion for n>2 were also known, none for n>8. In this paper, we prove that there are infinite families of links whose Khovanov homology contains Zn-t…
Analytic torsion defined for non-compact Lie groups and discrete subgroups.
problem Defining and calculating analytic torsion for non-compact Lie groups and their discrete subgroups.
method Localised analytic torsion and relative analytic torsion defined for Lie groups of type I, using representations and discrete subgroups.
result Relative analytic torsion of (G,Γ) coincides with Lott L2 analytic torsion of a covering space. 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.
A nontrivial element in a group is a generalized torsion element if some nonempty finite product of its conjugates is the identity. We prove that any generalized torsion element in a free product of torsion-free groups is conjugate to a generalized torsion element in some factor group. This implies that the fundamental…
Braverman and Kappeler introduced a refinement of the Ray-Singer analytic torsion associated to a flat vector bundle over a closed odd-dimensional manifold. We study this notion and improve the Braverman-Kappeler theorem comparing the refined analytic torsion with Farber-Turaev refinement of the combinatorial torsion. …
This paper shows hyperbolic knots can have arbitrarily large torsion in knot Floer homology.
problem Understanding the torsion order in knot Floer homology for hyperbolic knots.
method Unified approach using Upsilon torsion function.
result Arbitrarily large torsion orders realized by hyperbolic knots, most of which are twisted torus knots.
Study shows specific states produce Khovanov homology torsion.
problem Understanding torsion in Khovanov homology.
method Proved specific sum of enhanced states produce torsion.
result Proved specific states produce Khovanov homology torsion of order two.
Paper finds generalized torsions in non-bi-orderable 3-manifold groups.
problem Finding non-bi-orderable 3-manifold groups.
method Analyzing fundamental groups of once punctured torus bundles.
result Identifies generalized torsions in non-bi-orderable groups.
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.
Study connects spectral and algebraic torsion in geometric contexts.
problem Relating different torsion concepts in geometric settings.
method Example of product geometry, focusing on spin manifolds and two-point space.
result Established connection between spectral and algebraic torsion.
The abstract discusses transformations on statistical and semi-Weyl manifolds with torsion.
problem Invariance and structure preservation under conformal-projective transformations.
method Proof of invariance and preservation of structures under conformal-projective transformations.
result Semi-Weyl and statistical structures with torsion are invariant under conformal-projective transformations.
Study torsion and curvature in ACYT and AHKT 8-manifolds.
problem Characterize torsion and curvature in ACYT and AHKT 8-manifolds.
method Analyzes properties of Nijenhuis tensors, Ricci tensors, and torsion forms on ACYT and AHKT 8-manifolds.
result Closed torsion condition and Ricci flatness for ACYT 8-manifolds.
Study shows torsion homology growth vanishes for certain free-by-cyclic groups.
problem Understanding torsion homology growth in free-by-cyclic groups.
method Analyzing polynomially growing monodromy and showing vanishing homology torsion.
result Integral torsion equals ℓ2-torsion for these groups, verifying a conjecture. On an odd-dimensional oriented hyperbolic manifold of finite volume with strongly acyclic coefficient systems, we derive a formula relating analytic torsion with the Reidemeister torsion of the Borel-Serre compactification of the manifold. In a companion paper, this formula is used to derive exponential growth of torsi…
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. Ray-Singer torsion is a mathematical concept with applications in physics.
problem No specific problem stated in the abstract.
method No specific method stated in the abstract.
result No specific key result stated in the abstract.
In this text we introduce the torsion of spinor connections. In terms of the torsion we give conditions on a spinor connection to produce Killing vector fields. We relate the Bianchi type identities for the torsion of spinor connections with Jacobi identities for vector fields on supermanifolds. Furthermore, we discuss…
Analytic torsion defined for rank 2 distributions on 5-manifolds.
problem Defining and analyzing analytic torsion for rank 2 distributions.
method Proposed an analytic torsion for Rumin complex associated with rank 2 distributions on 5-manifolds, established anomaly formulas, and showed coincidence with Ray-Singer torsion.
result The proposed torsion coincides with Ray-Singer torsion for certain nilmanifolds.
New hyperbolic manifolds show exponential homology torsion growth.
problem Understanding growth of torsion in homology groups of hyperbolic manifolds.
method Constructed a family of hyperbolic manifolds with specific growth properties.
result Demonstrated that recent bound on homological torsion is asymptotically sharp.
The paper examines L2-torsion in fibration cases relaxing standard conditions.
problem Analyzing L2-torsion in fibrations under relaxed conditions. method Proves sum and product formulas for L2-torsion in an extended abelian category. result Obtains a formula for L2-torsion of a simple fibration with zero Euler characteristic. 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 generalized torsion found in 3-manifolds.
problem Understanding torsion in 3-manifolds.
method Constructing a generalized torsion element.
result Found a new generalized torsion element in a 3-manifold.
New torsion patterns found in Khovanov homology of link diagrams.
problem Identifying and characterizing torsion elements in Khovanov homology.
method Analyzing link diagrams to find new torsion patterns, using Khovanov chain complex submodules.
result Most torsion elements in the same Khovanov module are distinct.
The paper studies curvature identities and solitons on Spin(7)-manifolds.
problem Curvature identities and solitons on Spin(7)-manifolds.
method Analyzes the curvature and torsion of Spin(7)-manifolds, proving identities and conditions.
result Conditions for closed torsion and implications for Ricci flatness and solitons.
Defines a new Upsilon torsion function for knot Floer homology.
problem Obtaining constraints on knot cobordisms.
method Defines a one-parameter family of Heegaard Floer torsion invariants.
result Provides new obstructions related to the Gordian distance between knots.
For a closed manifold equipped with a Riemannian metric, a triangulation, a representation of its fundamental group on an Hilbert module of finite type (over of finite von Neumann algebra), and a Hermitian structure on the flat bundle associated to the representation, one defines a numerical invariant, the relative tor…
We show a relationship between the non-acyclic Reidemeister torsion and a zero of the acyclic Reidemeister torsion for a lambda-regular SU(2) or SL(2, C)-representation of a knot group. Then we give a method to calculate the non-acyclic Reidemeister torsion of a knot exterior. We calculate a new example and investigate…
Paper solves Minkowski problem for k-torsional rigidity.
problem Solving Minkowski problem for k-torsional rigidity.
method Constructing Hadamard variational formula, proposing k-torsional measure, using curvature flow method.
result Existence of smooth non-even solutions to the Minkowski problem.
Paper compares absolute and relative real analytic torsion forms over fibrations.
problem Analytic torsion forms over fibrations with boundaries.
method Gluing formula of analytic torsion forms proved by M. Puchol, Y. Zhang, and the author.
result Establishes a comparison formula of absolute and relative real analytic torsion forms.
New infinite class of hyperbolic knots with high genus and generalized torsion found.
problem Finding knots with high genus and generalized torsion.
method Demonstrated through (2, 2q+1)-torus knots and their twist families.
result Infinitely many hyperbolic knots with arbitrarily high genus and generalized torsion.
Examines a new type of analytic torsion on Riemannian manifolds.
problem Analyzing a new trace formula for Riemannian manifolds.
method Uses residue-trace instead of spectral zeta function quasi-trace.
result Defines and examines the residue analytic torsion.
Paper solves Minkowski problem for q-torsional rigidity using curvature flow.
problem Solving Minkowski problem for q-torsional rigidity.
method Method of curvature flows.
result Existence of smooth non-even solutions.