We derive general Novikov-Morse type inequalities in a Conley type framework for flows carrying cocycles, therefore generalizing our results in [FJ2] derived for integral cocycle. The condition of carrying a cocycle expresses the nontriviality of integrals of that cocycle on flow lines. Gradient-like flows are distingu…
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48 results for “integrable cocycles”
Refines equivariant integral cohomology using Cartan cocycles.
problem Equivariant characteristic forms and cohomology.
method Refinement of equivariant integral cohomology by Cartan cocycles.
result Definition and refinement of equivariant integral cohomology.
Paper calculates indices for group actions using cocycles.
problem Calculating equivariant indices for group actions.
method Construct cyclic cocycles on Harish-Chandra's Schwartz algebra, compute pairings with equivariant indices.
result Index formula completely determines equivariant indices via topological expressions.
In this note we prove that for any finite quandle and any 2-cocycle $φ\in Z^2_{Q\pm}(X; \mathds{Z})$, the cocycle invariant is trivial for all knots .
Defines a new knot integral related to Vassiliev invariants.
problem Understanding knot invariants and their relationships.
method Introduces a 1-cocycle in the space of long knots, generalizing the Kontsevich integral.
result Preserves the relationship between Kontsevich integral and Vassiliev invariants.
The paper studies metrics on Lie groups SO(2) and SO(3) and their integrability.
problem Integrability of gradient systems on Lie groups via Fisher metrics.
method Analysis of Souriau-Fisher metrics and 2-cocycles on Lie groups SO(2) and SO(3).
result Cocycles can locally modify Fisher metrics on Lie group orbits.
From the four normed division algebras--the real numbers, complex numbers, quaternions and octonions, of dimension k=1, 2, 4 and 8, respectively--a systematic procedure gives a 3-cocycle on the Poincare superalgebra in dimensions k+2=3, 4, 6 and 10, and a 4-cocycle on the Poincare superalgebra in dimensions k+3=4, 5, 7…
New Vassiliev invariant of order three derived from Fox-Hatcher cycles.
problem Developing a new Vassiliev invariant of order three.
method Integration of a 1-cocycle over Fox-Hatcher 1-cycles.
result Derives a Vassiliev invariant of order three.
The paper constructs non-trivial cocycles for long embeddings with more than one loop.
problem Constructing non-trivial cocycles for long embeddings with more than one loop.
method Integral over configuration spaces associated with Bott-Cattaneo-Rossi graphs with more than one loop.
result Explicit construction of a non-trivial family of trivial long embeddings for odd dimensions.
Geometrically constructs central extensions of Poisson Lie algebra.
problem Integrability of central extensions of Poisson Lie algebra.
method Prequantization and geometric construction of central S^1-extensions.
result Explicit description of integrable cocycles and nontrivial central extensions.
Associated to a differential character is an integral cohomology class, referred to as the characteristic class, and a closed differential form, referred to as the curvature. The characteristic class and curvature are equal in de Rham cohomology, and this is encoded in a commutative square. In the Hopkins--Singer model…
We present a geometric construction of central extensions of covering groups of the group of volume preserving diffeomorphisms, integrating central extensions of the Lie algebra of divergence free vector fields defined by Lichnerowicz cocycles. Certain covering spaces of non-linear Grassmannians can be realized as preq…
We study the relationship between trivial cocycles on the Torelli group and invariants of oriented integral homology 3-spheres. We give ncecessary and sufficient conditions for a function defined on the union of the Torelli groups to be an invariant of homology spheres. We apply this study to give a new purely algebrai…
Quantum field theory methods yield a physical interpretation of elliptic cohomology.
problem Constructing elliptic cohomology with complex coefficients.
method Using 2D quantum field theory to rigorously construct cocycles.
result Physical interpretation of the elliptic index theorem with complex coefficients.
In this paper we show that via the configuration space integral construction a non-trivalent graph cocycle can also yield a non-zero cohomology class of the space of higher (and even) codimensional long knots. This simultaneously proves that the Browder operation induced by the operad action defined by R. Budney is not…
Constructs integer-valued cohomology classes from graph cocycles.
problem Constructing nontrivial cohomology classes from graph cocycles.
method Gluing compactified configuration spaces to construct integer-valued classes from integer-valued graph cocycles.
result Obtains nontrivial classes from trivalent graph cocycles.
Lie's Third Theorem, asserting that each finite-dimensional Lie algebra is the Lie algebra of a Lie group, fails in infinite dimensions. The modern account on this phenomenon is the integration problem for central extensions of infinite-dimensional Lie algebras, which in turn is phrased in terms of an integration proce…
Given a n-dimensional lamination endowed with a Riemannian metric, we introduce the notion of a multiplicative cocycle of rank d, where n and d are arbitrary positive integers. The holonomy cocycle of a foliation and its exterior powers as well as its tensor powers provide examples of multiplicative cocycles. Next, we …
Singular holomorphic foliations by curves I: Integrability of holonomy cocycle in dimension 2math.DS
We study the holonomy cocycle H of a holomorphic foliation \Fc by Riemann surfaces defined on a compact complex projective surface X satisfying the following two conditions: 1) its singularities E are all hyperbolic; 2) there is no holomorphic non-constant map \C\to X such that out of E the image of \C is locally conta…
A new geometric cocycle measures mass of hyperbolic manifolds.
problem Measuring mass of hyperbolic manifolds.
method Constructing a cocycle mapping ALH metrics to a tractor-valued form field.
result Invariant can be integrated over a sphere boundary.
Recent work applying higher gauge theory to the superstring has indicated the presence of `higher symmetry'. Infinitesimally, this is realized by a `Lie 2-superalgebra' extending the Poincare superalgebra in precisely the dimensions where the classical supersymmetric string makes sense: 3, 4, 6 and 10. In the previous …
New model of differential K-theory using maps to Grassmannians and unitary groups.
problem Constructing a model of differential K-theory.
method Using geometrically defined Chern forms and equivalence classes of maps into Grassmannians and unitary groups.
result Model agrees with spectrum-based models in literature.
Holomorphic functions from knot complements link to quantum modular forms.
problem Analyzing holomorphic functions from knot complements.
method Matrix-valued holomorphic functions, cocycles, and quantum modularity.
result Identifies a matrix-valued holomorphic quantum modular form.
Integral invariants for framed 3-manifolds associated to trivalent graphs possibly with self-loopsmath.GT
New theory defines invariants for framed 3-manifolds using trivalent graphs with self-loops.
problem Defining invariants for framed 3-manifolds with semi-simple Lie groups.
method Constructing graph complexes and cocycles, including self-loops, to define invariants.
result Higher-loop invariants can be defined by graph cocycles with or without self-loops.
We show how the fundamental cocycles on current Lie algebras and the Lie algebra of symmetries for the sigma model are obtained via the current algebra functors. We present current group extensions integrating some of these current Lie algebra extensions.
Local Rigidity for Cocyclesmath.DS
In this paper we study perturbations of constant cocycles for actions of higher rank semi-simple algebraic groups and their lattices. Roughly speaking, for ergodic actions, Zimmer's cocycle superrigidity theorems implies that the perturbed cocycle is measurably conjugate to a constant cocycle modulo a compact valued co…
Extends Borel invariant to measurable cocycles of 3-manifold groups.
problem Defining and analyzing Borel invariant for measurable cocycles.
method Introducing pullback along measurable cocycles and extending Borel invariant.
result Maximal cocycles are trivializable to irreducible representations.
Study projective derivative cocycles for circle diffeomorphisms.
problem Understanding reducibility and almost reducibility in circle diffeomorphisms.
method Computing precise expressions for projective derivative cocycles and extending to 3-torus.
result Generalization of results to diagonal action on 3-torus.
T. Mochizuki determined all 3-cocycles of the third quandle cohomologies of Alexander quandles on finite fields. We show that all the 3-cocycles, except those of 2-cocycle forms, are derived from group 3-cocycles of a meta-abelian group. Further, the quandle cocycle invariant of a link using Mochizuki's 3-cocycle is eq…
We describe natural abelian extensions of the Lie algebra $\aut(P)$ of infinitesimal automorphisms of a principal bundle over a compact manifold and discuss their integrability to corresponding Lie group extensions. Already the case of a trivial bundle is quite interesting. In this case, we show th…
Quandle Cocycle Quiversmath.GT
New graph-based invariants from quandle cocycles.
problem Defining new link invariants from quandle cocycles.
method Integrating quandle cocycle information into quandle coloring quivers to create weighted directed graphs.
result Definition of new link invariants including a 2-variable polynomial.
Virtual index cocycles reformulate virtual link invariants.
problem No specific problem stated; focuses on reformulation.
method Using virtual index cocycles to reformulate invariants.
result Unified reformulation of virtual link invariants.
We give a construction of quandle cocycles from group cocycles, especially, for any integer p \geq 3, quandle cocycles of the dihedral quandle R_p from group cocycles of the cyclic group Z/p. We will show that a group 3-cocycle of Z/p gives rise to a non-trivial quandle 3-cocycle of R_p. When p is an odd prime, since d…
New knot invariants computed without explicit cocycles.
problem Computing knot invariants without explicitly finding cocycles.
method Using generalized Alexander quandles and colorings of 1-tangles.
result The 2-cocycle invariant distinguishes many prime knots.
Study on group cocycles for volume-preserving diffeomorphisms.
problem Understanding group cocycles on volume-preserving diffeomorphisms.
method Constructed two types of group cocycles on the volume-preserving diffeomorphism group.
result One cocycle yields the Euler class of flat sphere bundles for the sphere.
Defines rho numbers for metrics with positive scalar curvature.
problem Computing index classes for manifolds with boundary and cuspidal parabolics.
method Uses orbital integrals, delocalized eta invariants, and index theorems.
result Defines higher rho numbers for metrics with positive scalar curvature.
The paper extends Euler class theory to measurable cocycles.
problem Understanding the structure of measurable cocycles and their cohomology.
method Constructing a parametrized Euler class in bounded cohomology and studying semicohomologous cocycles.
result The parametrized Euler class vanishes if and only if the cocycle can be lifted and admits an equivariant family of points.
On higher Dirac structuresmath.SG
Study of higher-order Dirac structures in field theory.
problem Extending multisymplectic structures to higher-order analogues.
method Define and analyze higher Dirac structures as involutive subbundles of .
result Recover higher Poisson structures as infinitesimal counterparts of multisymplectic groupoids.
New rack and multiple group rack cohomology for surfaces in 3-sphere.
problem Categorizing compact oriented surfaces in 3-sphere based on symmetry.
method Developed cohomology theory for racks and multiple group racks, constructed cocycle invariants.
result Identified new symmetry types of surfaces in 3-sphere.
Researchers compute a residue cocycle for Dirac-type operators using modified Getzler calculus.
problem Computing the residue cocycle for Dirac-type operators.
method Modified Getzler calculus for computation.
result Computed residue cocycle for a class of Dirac-type operators.
We extend the Yang-Baxter cocycle invariants for virtual knots by augmenting Yang-Baxter 2-cocycles with cocycles from a cohomology theory associated to a virtual biquandle structure. These invariants coincide with the classical Yang-Baxter cocycle invariants for classical knots but provide extra information about virt…
New algebraic rules for 5D shapes based on 3D cocycles.
problem Creating rules for 5D shapes.
method Using simplicial 3-cocycles to parameterize heptagon relations.
result Parameterized heptagon relations for 5D shapes.
We consider cocycles of isometries on spaces of nonpositive curvature . We show that the supremum of the drift over all invariant ergodic probability measures equals the infimum of the displacements of continuous sections under the cocycle dynamics. In particular, if a cocycle has uniform sublinear drift, then there…
The theory of quandle (co)homology and cocycle knot invariants is rapidly being developed. We begin with a summary of these recent advances. One such advance is the notion of a dynamical cocycle. We show how dynamical cocycles can be used to color knotted surfaces that are obtained from classical knots by twist-spinnin…
Quandle 2-cocycles yield invariant values for knots under certain algebraic conditions.
problem Defining and understanding invariants of knots using quandle 2-cocycles.
method Analyzing algebraic properties of quandle extensions and their impact on knot invariants.
result The invariant values are constant or follow a restricted form for classical knots under specific conditions.
Quantum cocycle invariants derived from Yang-Baxter cohomology.
problem Constructing stronger quantum knot invariants.
method Developing quantum cocycle invariants using Yang-Baxter cohomology and deformation theory.
result Quantum cocycle invariants yield stronger invariants in certain examples.
New shifting chain map enhances quandle invariants for links.
problem Enhancing quandle invariants for links and surfaces.
method Introducing a shifting chain map σ and its pull-back σ# to transform cocycles.
result Shifting chain map σ# transforms 2-cocycles to 3-cocycles, enhancing invariants.
Develops Patterson-Sullivan theory for coarse cocycles.
problem None explicitly stated in the abstract.
method Theory of Patterson--Sullivan measures for coarse cocycles of convergence groups.
result Existence, uniqueness, and ergodicity results for Patterson-Sullivan measures under geometric assumptions.