On the classification problem of the quasi-isomorphism classes of free chain algebras

The present paper is devoted to the classification problem of the quasi-isomorphism classes of free differential graded algebras (dgas) over a (P.I.D) R. We introduce the notion of coherent homomorphisms, perfect and quasi-perfect dgas (the Adams-Hilton model of simply connected CW-complex such that H_∗(X,R) is free is a such a dga) and our first main result asserts that two perfect (quasi-perfect) dgas are quasi-isomorphic if and only if their Whitehead exact sequences are coherently isomorphic. Moreover we define the notion of a strong isomorphism between the Whitehead exact sequences and we show that two free R-dgas, of which their Whitehead exact sequences are strongly isomorphic, are quasi-isomorphic.     

The bar complex of an E-infinity algebra

The standard reduced bar complex B(A) of a differential graded algebra A inherits a natural commutative algebra structure if A is a commutative algebra. We address an extension of this construction in the context of E-infinity algebras. We prove that the bar complex of any E-infinity algebra can be equipped with the structure of an E-infinity algebra so that the bar construction defines a functor from E-infinity algebras to E-infinity algebras. We prove the homotopy uniqueness of such natural E-infinity structures on the bar construction.We apply our construction to cochain complexes of topological spaces, which are instances of E-infinity algebras. We prove that the n-th iterated bar complexes of the cochain algebra of a space X is equivalent to the cochain complex of the n-fold iterated loop space of X, under reasonable connectedness, completeness and finiteness assumptions on X.     

同调光滑连通上链DG代数的一个注记

本文给出了有关同调光滑连通上链微分分次(简称DG)代数的两个重要结论.具体地说,当A是同调光滑连通上链DG代数且其同调分次代数H(A)是诺特分次代数时,证明D_(fg)(A)中的任意Koszul DG A-模都是紧致的.另外,当A是Kozul连通上链DG代数且其同调分次代数H(A)是有平衡对偶复形的诺特分次代数时,证明A的同调光滑性质等价于D_(fg)(A)=D~c(A).     

Singular cochains and rational homotopy type

Rational homotopy types of simply connected topological spaces have been classified by weak equivalence classes of commutative cochain algebras (Sullivan) and by isomorphism classes of minimal commutative A _∞-algebras (Kadeishvili). We classify rational homotopy types of the space X by using the (noncommutative) singular cochain complex C*(X, Q), with additional structure given by the homotopies introduced by Baues, {E _1,k } and {F _p,q}. We show that if we modify the resulting B _∞-algebra structure on this algebra by requiring that its bar construction be a Hopf algebra up to a homotopy, then weak equivalence classes of such algebras classify rational homotopy types. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 43, Topology and Its Applications, 2006.     

Homotopy classification of graded Picard categories

Certain low-dimensional symmetric cohomology groups of G-modules, for any given group G, are computed as the cohomology of an explicit cochain complex. This result is used to establish natural one-to-one correspondences between elements of the 3rd symmetric cohomology groups of G-modules, G-equivariant pointed 2-connected homotopy 4-types, and equivalence classes of G-graded Picard categories. The simplicial nerve of a G-graded Picard category is also constructed and studied.     

张量积的代数K－理论及其应用

本文研究了两个代数张量积的Ｇｒｏｔｈｅｎｄｉｅｃｋ群和Ｗｈｉｔｅｈｅａｄ群首先构作三个群同态，并证明：若Ｒ为增广Ａ－代数，则存在的子群Ｃ使得，并存在的子群Ｄ使得．然后给出在群代数和包络代数方面的应用，最后考虑的增广代数的情形．     

Group-like structures and the whitehead product in pro-homotopy and shape

The notion of ‘H-space’ is of considerable importance in the homotopy theory of CW-complexes. This paper studies a similar notion in the framework of pro-homotopy and shape theories. This is achieved by following the general plan set forth by Eckmann and Hilton. Examples of shape H-space are also given; it is observed that every compact connected topological monoid is a shape H-space. The Whitehead product is defined and studied in the pro-homotopy and shape categories; and, it is shown that this Whitehead product vanishes on an H-object in pro-homotopy. These results are the natural extension of some well-known classical results in the homotopy theory of CW-complexes.     

Cochains and homotopy type

Finite type nilpotent spaces are weakly equivalent if and only if their singular cochains are quasi-isomorphic as E_∞ algebras. The cochain functor from the homotopy category of finite type nilpotent spaces to the homotopy category of E_∞ algebras is faithful but not full.     

Vanishing line for the descent spectral sequence

We prove that if $f:X \to Y$ is a surjective, cohomologically (k - 1)-connected and proper map between locally compact spaces, then for the cohomological descent spectral sequence one has $E^{pq}_{2} = 0$ provided $q < pk$.     

Commutative free loop space models at large primes

Let p be a prime number and a natural number. If E is a r-connected finite CW-complex of dimension at most pr, then E is an example of a p -Anick space. For p > 2 we construct a commutative cochain algebra over that is an -model of the free loop space on a p-Anick space, i.e., its cohomology algebra is isomorphic to the mod p cohomology of the free loop space. For p-Anick spaces that are p-formal, such as spheres and projective spaces, we define an even simpler commutative free loop space model that applies for all primes p. We then use the simplified model to compute the cohomology algebras of a number of free loop spaces explicitly. Received: 23 June 1999; in final form: 8 September 2000 // Published online: 7 April 2003     

Frobenius Poisson algebras

This paper is devoted to study Frobenius Poisson algebras. We introduce pseudo-unimodular Poisson algebras by generalizing unimodular Poisson algebras, and investigate Batalin-Vilkovisky structures on their cohomology algebras. For any Frobenius Poisson algebra, all Eatalin-Vilkovisky opera tors on its Poisson cochain complex are described explicitly. It is proved that there exists a Batalin-Vilkovisky operator on its cohomology algebra which is induced from a Batalin-Vilkovisky operator on the Poisson cochain complex, if and only if the Poisson st rue ture is pseudo-unimodular. The relation bet ween modular derivations of polynomial Poisson algebras and those of their truncated Poisson algebras is also described in some cases.     

Quillen-Barr-Beck (co-) homology for restricted Lie algebras

In this paper we use Quillen-Barr-Beck's theory of (co-) homology of algebras in order to define (co-) homology for the category RLie of restricted Lie algebras over a field k of characteristic p≠0. In contrast with the cases of groups, associative algebras and Lie algebras we do not obtain Hochschild (co-) homology shifted by 1.Precisely, we determine for L∈RLie the category of Beck L-modules and the group of Beck derivations of g∈RLie/L to a Beck L-module M. Moreover, we prove a classification theorem which gives a one-to-one correspondence between the one cohomology and the set of equivalent classes of p-extensions. Finally, a universal coefficient theorem is proved, relating the homology to the Hochschild homology via a short exact sequence. This shows that the new homology determines the Hochschild homology.     

Cochain algebra on manifolds and convergence under refinement

In this paper we develop several algebraic structures on the simplicial cochains of a triangulated manifold and prove they converge to their differential-geometric analogues as the triangulation becomes small. The first such result is for a cochain cup product converging to the wedge product on differential forms. Moreover, we show any extension of this product to a C_∞-algebra also converges to the wedge product of forms. For cochains equipped with an inner product, we define a combinatorial star operator and show that for a certain cochain inner product this operator converges to the smooth Hodge star operator.     

A NOTE ON PRODUCTS OF THE WHITEHEAD TYPE

Abstract

A survey is given of various generalizations of the Whitehead product. Carrying the procedure a step further, a generalization is given that specialized in one wry relates to the “Whitehead elements” in the (mod p) kernel of the double suspension (p an odd prime, p > 4). Specialized in another way the product relates to the action of the Q-homomorphism (in the EHQ sequence) on elements that are not double suspensions.     

Properties of comultiplications on a wedge of spheres

In this paper we study the set of comultiplications on a wedge of a finite number of spheres. We are interested in group theoretic properties of these comultiplications such as associativity and commutativity and loop theoretic properties such as inversivity, power-associativity and the Moufang property. Our methods involve Whitehead products in wedges of spheres and the Hopf-Hilton invariants. We obtain extensive results for a restricted class of comultiplications, namely, the one-stage quadratic or cubic comultiplications.     

Minimal varieties of algebras of exponential growth

The exponent of a variety of algebras over a field of characteristic zero has been recently proved to be an integer. Through this scale we can now classify all minimal varieties of given exponent and of finite basic rank. As a consequence, we describe the corresponding T-ideals of the free algebra and we compute the asymptotics of the related codimension sequences, verifying in this setting some known conjectures. We also show that the number of these minimal varieties is finite for any given exponent. We finally point out some relations between the exponent of a variety and the Gelfand-Kirillov dimension of the corresponding relatively free algebras of finite rank.     

Symmetric (co)homologies of Lie algebras

Cohomologies of Lie algebras are usually calculated using the Chevalley-Eilenberg cochain complex of skew-symmetric forms. We consider two cochain complexes consisting of forms with some symmetric properties. First, cochains C^*(L) are symmetric in the last 2 arguments, skew-symmetric in the others and satify moreover some kind of Jacobi condition in the last 3 arguments. In characteristic 0, its cohomologies are isomorphic to the cohomologies of the factor-complex C^*(L,L’)/C^*+1(L,K). Second, a symmetric version C_λ^*(A) is defined for an associative algebra A. It is a subcomplex of the cyclic cochain complex. These symmetric cochain complexes are used for the calculation of 3-cohomologies of Cartan Type Lie algebras with trivial coefficients.     

A one-dimensional embedding complex

We give the first explicit computations of rational homotopy groups of spaces of “long knots” in Euclidean spaces. We define a spectral sequence which converges to these rational homotopy groups whose E¹ term is defined in terms of familiar Lie algebras. For odd k we establish a vanishing line for this spectral sequence, show the Euler characteristic of the rows of this E¹ term is zero, and make calculations of E² in a finite range.     

The plus-construction, Bousfield localization, and derived completion

We define a plus-construction on connective augmented algebras over operads in symmetric spectra using Quillen homology. For associative and commutative algebras, we show that this plus-construction is related to both Bousfield localization and Carlsson’s derived completion.