New complex models differential forms on bigons.
problem Modeling differential forms on bigons.
method Defined a new higher Hochschild Complex with an Iterated Integral map.
result Associated an element in the zigzag Hochschild complex to 2-holonomy of gerbes.
Study Hochschild cohomology of dg manifolds linked to integrable distributions.
problem Understanding Hochschild cohomology of dg manifolds associated with integrable distributions.
method Analyzing the Hochschild cohomology of (F[1],dF) and relating it to the algebra of functions on leaf space. result Established a canonical isomorphism between the Hochschild cohomology of (F[1],dF) and the algebra of functions on leaf space. A formula connects two algebraic structures derived from a category.
problem Connecting two algebraic structures derived from a category.
method Using differential graded modular functors and Calabi-Yau structures.
result The action of a specific mapping class group element transforms one algebraic structure into another.
This article explains A∞-algebras and Hochschild homology.
problem Understanding A∞-algebras and Hochschild homology. method Elementary construction and detailed proofs of results.
result Unified discussion of algebraic results from various sources.
Novel cohomology theories for operadic algebras and spaces.
problem Formulating cohomology theories for operadic algebras.
method Using cotangent complex formalism and spectral Hochschild cohomology.
result Controlled cohomologies of operads and their algebras.
We trade matrix factorizations and Koszul complexes for Hochschild homology of Soergel bimodules to modify the construction of triply-graded link homology and relate it to Kazhdan-Lusztig theory.
Maps geometric deformations to algebraic classes in Lie groupoids and algebroids.
problem Deformation theory of Lie groupoids and algebroids.
method Defining a morphism between deformation complexes and Hochschild complexes, applying to adiabatic groupoids.
result Induced van Est map from geometric to algebraic deformation cohomology.
New mapping class group actions on Hochschild complexes for modular categories.
problem Understanding actions of mapping class groups on Hochschild complexes of modular categories.
method Construction of a symmetric monoidal functor with excision property.
result Homotopy coherent projective action of mapping class groups on Hochschild complexes.
Abstract: Mapping class groups act on cohomology of surfaces via Hochschild cohomology.
problem Understanding the action of mapping class groups on cohomology of surfaces.
method Associate cochain complexes to surfaces, with mapping class groups acting projectively on cohomology.
result Projective action of mapping class groups on Hochschild cohomology of Hopf algebras.
Given a Heegaard splitting of a closed 3-manifold, the skein modules of the two handlebodies are modules over the skein algebra of their common boundary surface. The zeroth Hochschild homology of the skein algebra of a surface with coefficients in the tensor product of the skein modules of two handlebodies is interpret…
Given an n-manifold M and an n-category C, we define a chain complex (the "blob complex") B_*(M;C). The blob complex can be thought of as a derived category analogue of the Hilbert space of a TQFT, and as a generalization of Hochschild homology to n-categories and n-manifolds. It enjoys a number of nice formal properti…
New spectral sequence connects to topological Hochschild homology.
problem Connecting spectral sequences to topological Hochschild homology.
method Developed a spectral sequence and applied Tate diagonal techniques.
result Spectral sequence converges to localized topological Hochschild homology.
Defines Courant pairs and studies their deformations via cohomology.
problem Deforming Courant pairs and understanding their structure.
method Constructs a cohomology bicomplex with coefficients in a module.
result Establishes a connection between Hochschild and Leibniz cohomologies.
We summarize our axioms for higher categories, and describe the blob complex. Fixing an n-category C, the blob complex associates a chain complex B_*(W;C)$ to any n-manifold W. The 0-th homology of this chain complex recovers the usual topological quantum field theory invariants of W. The higher homology groups should …
Global homotopies upgrade classical map in differential geometry.
problem Upgrade classical Hochschild-Kostant-Rosenberg map to a deformation retract.
method Combining symbol calculus and coalgebraic van Est theorem.
result Develop deformation retracts in various settings.
The paper introduces representation homology of topological spaces and its connections to other homology theories.
problem Understanding the algebraic structure of topological spaces through representation homology.
method Developed a geometric relation between representation homology and higher Hochschild homology, constructed maps and spectral sequences, and computed explicit examples.
result Representation homology of the suspension of a space is isomorphic to its higher Hochschild homology.
Researchers compare two homological invariants for mapping classes of surfaces.
problem Comparing two different types of mapping class invariants on surfaces.
method Computed Hochschild homology of an A∞ bimodule and fixed point Floer cohomology, and discussed a potential isomorphism. result Conjectured that the two invariants are isomorphic in the genus two case.
Proves formality conjecture for Hamiltonian actions using equivariant Hochschild-Kostant-Rosenberg quasi-isomorphism.
problem Formality conjecture for Hamiltonian actions.
method Constructs an L∞-quasi-isomorphism using the G-invariant formality. result Equivariant Hochschild-Kostant-Rosenberg quasi-isomorphism extends to an L∞-quasi-isomorphism. The paper compares two notions of invariance in Hochschild cohomology spaces.
problem Comparing two notions of invariance in Hochschild cohomology spaces.
method Analyzing the induced action on Hochschild cohomology of smooth functions and comparing two spaces of invariants.
result For proper group actions, both spaces of invariants are isomorphic.
Researchers compute Hochschild cohomology of Grassmannians.
problem Computing Hochschild cohomology of Grassmannians.
method Explicit description of Gerstenhaber algebra structure, vanishing of higher cohomology.
result Decomposition of Hochschild cohomology concentrated in global sections for certain Grassmannians.
New stable homotopy refinement of quantum annular Khovanov homology.
problem Quantum topological Hochschild homology and annular Khovanov spectra.
method Introducing quantum topological Hochschild homology (qTHH) and constructing a new stable homotopy refinement of quantum annular Khovanov homology.
result The new stable homotopy refinement agrees with qTHH of spectral Chen-Khovanov tangle bimodules and recovers earlier work.
The associator of a non-associative algebra is the curvature of the Hochschild quasi-complex. The relationship ``curvature-associator'' is investigated. Based on this generic example, we extend the geometric language of vector fields to a purely algebraic setting, similar to the context of Gerstenhaber algebras. We int…
The central result here is an explicit computation of the Hochschild and cyclic homologies of a natural smooth subalgebra of stable continuous trace algebras having smooth manifolds X as their spectrum. More precisely, the Hochschild homology is identified with the space of differential forms on X, and the periodic cyc…
We define a "circle Euler characteristic" of a circle action on a compact manifold or finite complex X. It lies in the first Hochschild homology group of ZG where G is the fundamental group of X. It is analogous in many ways to the ordinary Euler characteristic. One application is an intuitively satisfying formula for …
Short note observes quantum Hochschild homology as a composition of known operations.
problem Quantum Hochschild homology as a new invariant of annular links.
method Observes quantum Hochschild homology as a composition of two known operations.
result Quantum Hochschild homology is a valid invariant of annular links.
This paper generalizes Bismut's equivariant Chern character to the setting of abelian gerbes. In particular, associated to an abelian gerbe with connection, an equivariantly closed differential form is constructed on the space of maps of a torus into the manifold. These constructions are made explicit using a new local…
We prove Tsygan's formality conjecture for Hochschild chains of the algebra of functions on an arbitrary smooth manifold M using the Fedosov resolutions proposed in math.QA/0307212 and the formality quasi-isomorphism for Hochschild chains of R[[y_1, ..., y_d]] proposed in paper math.QA/0010321 by Shoikhet. This result …
Categorifies Young symmetrizers and connects to torus link homology.
problem Categorify Young symmetrizers and relate to torus link homology.
method Construct Soergel bimodule complexes and use Hochschild homology.
result Proves conjecture relating Hochschild homology to flag Hilbert scheme.
We show that Khovanov homology and Hochschild homology theories share common structure. In fact they overlap: Khovanov homology of a (2,n)-torus link can be interpreted as a Hochschild homology of the algebra underlining the Khovanov homology. In the classical case of Khovanov homology we prove the concrete connectio…
New proof of Duflo isomorphism for Lie algebras with scalar products.
problem Duflo isomorphism for Lie algebras with scalar products.
method Explicit calculation of first order deformation of differential on Hochschild complex.
result Knot invariant matches Kontsevich integral for unknot.
This paper provides a rational model for fiberwise THH transfer using A-infinity algebras.
problem Rational models for fiberwise THH transfer of fibrations over a base space.
method Explicit description of Hochschild homology transfer in terms of A-infinity algebras.
result Rational models for Becker-Gottlieb transfer and fiberwise THH-simple structures.
The paper simplifies string topology computations using Hochschild chain models.
problem Computing string topology operations efficiently.
method Hochschild chain models to simplify proofs and computations.
result New observations and detection of closed geodesics.
J. Przytycki has established a connection between the Hochschild homology of an algebra A and the chromatic graph homology of a polygon graph with coefficients in A. In general the chromatic graph homology is not defined in the case where the coefficient ring is a non-commutative algebra. In this paper we define a …
In this notes it will be provided a set of techniques which can help one to understand the proof of the Hochschild-Kostant-Rosenberg theorem for differentiable manifolds. Precise definitions of multidiferential operators and polyderivations on an algebra are given, allowing to work on these concepts, when the algebra i…
This is the first of two papers in which we prove that a cell model of the moduli space of curves with marked points and tangent vectors at the marked points acts on the Hochschild co--chains of a Frobenius algebra. We also prove that a there is dg--PROP action of a version of Sullivan Chord diagrams which acts on the …
New homotopy refinements for tangle invariants.
problem Stable homotopy refinements for tangle invariants.
method Refined Khovanov and Chen-Khovanov spectra.
result Induces refinements of platform algebras and invariants.
Open 2D TFTs extend to closed theories with circle value as Hochschild homology.
problem Extending open 2D TFTs to closed theories.
method Using symmetric monoidal ∞-categories and Hochschild homology.
result Open 2D TFTs admit initial open-closed extensions.
Study star products on Poisson manifolds compatible with reduction.
problem Finding star products compatible with coisotropic reduction.
method Compute second constraint Hochschild cohomology of constraint algebra.
result Determine infinitesimal star products on Poisson manifolds.
Proves Serre duality in Khovanov-Rozansky homology.
problem Relating top and bottom Hochschild degrees in Khovanov-Rozansky homology.
method Using type A Soergel bimodules and full twist as a Serre functor.
result Categorifies Kálmán's theorem on Hochschild degrees.
Alternative proof of Khovanov's up-to-sign functoriality for odd Khovanov homology.
problem Functoriality of Khovanov's odd Khovanov homology.
method Extending Hochschild (co)homology to quasi-associative algebras and applying it to Khovanov's homology.
result First proof of functoriality of Naisse and Putyra's tangle theory up to unit.
New spectral sequence connects link homology to Hochschild homology.
problem Computing Hochschild homology of link invariants.
method Uses spectral sequence with E2-page from Khovanov homology of links in S1imesS2. result Spectral sequence converges to Hochschild homology of bordered Floer invariants.
In 2001, Khovanov and Seidel constructed a faithful action of the (m+1)-strand braid group on the derived category of left modules over a quiver algebra, A_m. We interpret the Hochschild homology of the Khovanov-Seidel braid invariant as a direct summand of the sutured Khovanov homology of the annular braid closure.
Study cohomology of Lie algebroids over algebraic spaces using derived functors and Čech cohomology.
problem Cohomology of Lie algebroids over algebraic spaces.
method Express hypercohomology as a derived functor, simplify via Čech cohomology, define Hochschild hypercohomology, present Hochschild-Kostant-Rosenberg theorem.
result Presented a version of Hochschild-Kostant-Rosenberg theorem for locally free Lie algebroids.
Proves Poincaré duality for Hopf algebroids with bijective antipode.
problem Proving Poincaré duality for Hopf algebroids.
method Using twisted Poincaré duality and bijective antipode properties.
result Recovering and extending known Poincaré dualities for Hopf algebroids.
Atiyah classes of DG manifolds of positive amplitude are invariant under weak equivalences.
problem Defining and studying Hochschild cohomology of DG manifolds of positive amplitude.
method Using poly-differential operators and derived intersection, proving invariance under weak equivalences.
result Hochschild cohomology of DG manifolds of positive amplitude is invariant under weak equivalences.
Using a cell model for the little discs operad in terms of spineless cacti we give a minimal common topological operadic formalism for three a priori disparate algebraic structures: (1) a solution to Deligne's conjecture on the Hochschild complex, (2) the Hopf algebra of Connes and Kreimer, and (3) the string topology …
New geometric model for knot homology using monodromic Hecke category.
problem Geometric description of Khovanov-Rozansky knot homology.
method Monodromic model based on Soergel bimodules and Hecke category.
result Geometric description of Khovanov-Rozansky knot homology.
Study on deformation cohomology for braided commutative structures.
problem Classifying and understanding deformations of braided commutative algebras.
method Extending Yang-Baxter Hochschild cohomology to braided commutative deformations.
result Classifies infinitesimal deformations of braided algebras that are braided commutative.