Khovanov homology and star-like isotopies

A star-like isotopy for oriented links in 3-space is an isotopy which uses only Reidemeister moves which correspond to the following singularities of planar curves:

, , , .

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Khovanov homology, open books, and tight contact structures

We define the reduced Khovanov homology of an open book (S,?), and identify a distinguished “contact element” in this group which may be used to establish the tightness or non-fillability of contact structures compatible with (S,?). Our construction generalizes the relationship between the reduced Khovanov homology of a link and the Heegaard Floer homology of its branched double cover. As an application, we give combinatorial proofs of tightness for several contact structures which are not Stein-fillable. Lastly, we investigate a comultiplication structure on the reduced Khovanov homology of an open book which parallels the comultiplication on Heegaard Floer homology defined in Baldwin (2008) [4].     

A homology theory for framed links in I-bundles using embedded surfaces

This paper introduces a homology theory for framed links in I-bundles over an orientable surface. The theory is unique in that the elements of the chain groups are embedded surfaces instead of diagrams. It is then shown this theory recaptures the homology theory constructed by Asaeda, Przytycki and Sikora.     

Hochschild and cyclic homology of finite type algebras

We study Hochschild and cyclic homology of finite type algebras using abelian stratifications of their primitive ideal spectrum. Hochschild homology turns out to have a quite complicated behavior, but cyclic homology can be related directly to the singular cohomology of the strata. We also briefly discuss some connections with the representation theory of reductive p-adic groups.     

A Formula for Khovanov Typ e Link Homology of Pretzel Knots

Khovanov type homology is a generalization of Khovanov homology. The main result of this paper is to give a recursive formula for Khovanov type homology of pretzel knots P (?n,?m, m). The computations reveal that the rank of the homology of pretzel knots is an invariant of n. The proof is based on a “shortcut” and two lemmas that recursively reduce the computational complexity of Khovanov type homology.     

Cyclic homology of algebras with one generator

We compute the cyclic homology of A = k[X]/X ⁿ for an arbitrary commutative ring k, and we apply this result to compute the cyclic homology of k[X]/f, when k is a field and f is an arbitrary polynomial.This work was presented at the IX-ELAM, Santigo de Chile, July 1988, and was partially supported by CONICET.The following pepole participated in this research: Jorge A. Guccione, juan José guccione, Maria Julia Redondo, Andrea Solotar, and Orlando E. Villamayor.     

Noncommutative homology of some three-dimensional quantum spaces

We compute the Hochschild and cyclic homology of certain three-dimensional quantum spaces (type A algebras), introduced by Artin and Schelter. We show that the Hochschild homology is determined by the quasi-classical limit.     

Invariance properties of coHochschild homology

The notion of Hochschild homology of a dg algebra admits a natural dualization, the coHochschild homology of a dg coalgebra, introduced in [23] as a tool to study free loop spaces. In this article we prove “agreement” for coHochschild homology, i.e., that the coHochschild homology of a dg coalgebra C is isomorphic to the Hochschild homology of the dg category of appropriately compact C-comodules, from which Morita invariance of coHochschild homology follows. Generalizing the dg case, we define the topological coHochschild homology (coTHH) of coalgebra spectra, of which suspension spectra are the canonical examples, and show that coTHH of the suspension spectrum of a space X is equivalent to the suspension spectrum of the free loop space on X, as long as X is a nice enough space (for example, simply connected.) Based on this result and on a Quillen equivalence established in [24], we prove that “agreement” holds for coTHH as well.     

Leibniz and Hochschild homology

We construct and study the map from Leibniz homology HL_?(𝔥) of an abelian extension 𝔥 of a simple real Lie algebra 𝔤 to the Hochschild homology HH_??1(U(𝔥)) of the universal envelopping algebra U(𝔥). To calculate some homology groups, we use the Hochschild-Serre spectral sequences and Pirashvili spectral sequences. The result shows what part of the non-commutative Leibniz theory is detected by classical Hochschild homology, which is of interest today in string theory.     

Cyclic homology of Dedekind domains

We compute the cyclic homology of the algebraR[t]/(P(t)) relative to an arbitrary integral domainR. As an application, we compute the cyclic homology of number rings.Supported by N.S.F. Grant No. DMS-8807203.     

A relation between Hochschild homology and cohomology for Gorenstein rings

Let ``' stand for Hochschild (co)homology. In this note we show that for many rings there exists such that for an arbitrary -bimodule we have . Such a result may be viewed as an analog of Poincaré duality.

Combining this equality with a computation of Soergel allows one to compute the Hochschild homology of a regular minimal primitive quotient of an enveloping algebra of a semisimple Lie algebra, answering a question of Polo.

     

Cyclic and Hochschild homology of 2-nilpotent algebras

We compute the cyclic homology of a 2-nilpotent algebra, using a reduced way of obtaining it for an algebra containing a separable subalgebra as a direct summand.Supported by the Swiss National Foundation, FNSRS.     

On the colored Jones polynomial, sutured Floer homology, and knot Floer homology

Let K⊂S³, and let denote the preimage of K inside its double branched cover, Σ(S³,K). We prove, for each integer n>1, the existence of a spectral sequence whose E² term is Khovanov's categorification of the reduced n-colored Jones polynomial of (mirror of K) and whose E^∞ term is the knot Floer homology of (when n odd) and of (S³,K#Kr) (when n even). A corollary of our result is that Khovanov's categorification of the reduced n-colored Jones polynomial detects the unknot whenever n>1.     

Comparison theorems for o-minimal singular (co)homology

Here we show the existence of the o-minimal simplicial and singular (co)homology in o-minimal expansions of real closed fields and prove several comparison theorems for o-minimal (co)homology theories.

     

Fibonacci代数的Hochschild同调群

设Λd是Fibonacci代数,基于对Bardzell极小投射双模分解的细致分析,用组合的方法清晰地计算了Fibonacci代数Λd的各阶Hochschild同调群的维数.     

Hochschild (Co)homology of Exterior Algebras

The minimal projective bimodule resolutions of the exterior algebras are explicitly constructed. They are applied to calculate the Hochschild (co)homology of the exterior algebras. Thus the cyclic homology of the exterior algebras can be calculated in case the underlying field is of characteristic zero. Moreover, the Hochschild cohomology rings of the exterior algebras are determined by generators and relations.