Results 1 
8 of
8
Co)homology theories for commutative Salgebras
"... The aim of this paper is to give an overview of some of the existing homology theories for commutative (S)algebras. We do not claim any originality; nor do we pretend to give a complete account. But the results in that field are widely spread in the literature, so for someone who does not actually ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
The aim of this paper is to give an overview of some of the existing homology theories for commutative (S)algebras. We do not claim any originality; nor do we pretend to give a complete account. But the results in that field are widely spread in the literature, so for someone who does not actually work in that subject, it can be difficult to trace all the relationships between the different homology theories. The theories we aim to compare are • topological AndréQuillen homology • Gamma homology • stable homotopy of Γmodules • stable homotopy of algebraic theories • the AndréQuillen cohomology groups which arise as obstruction groups in the GoerssHopkins approach As a comparison between stable homotopy of Γmodules and stable homotopy of algebraic theories is not explicitly given in the literature, we will give a proof of Theorem 2.1 which says that both homotopy theories are isomorphic
Spaces of selfequivalences and free loops spaces
, 2002
"... Let M be a simplyconnected closed oriented Ndimensional manifold. We prove that for any field of coefficients there exists a natural homomorphism of commutative graded algebras Ψ: H∗(Ω aut1M) → H∗+N(M S1) where H∗(M S1) is the loop algebra defined by ChasSullivan, [1]. As usual aut1X (resp. ΩX) ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Let M be a simplyconnected closed oriented Ndimensional manifold. We prove that for any field of coefficients there exists a natural homomorphism of commutative graded algebras Ψ: H∗(Ω aut1M) → H∗+N(M S1) where H∗(M S1) is the loop algebra defined by ChasSullivan, [1]. As usual aut1X (resp. ΩX) denotes the monoid of the selfequivalences homotopic to the identity map (resp. the space of based loops) of the space X. Moreover, if lk is of characteristic zero, Ψ yields isomorphisms πn(Ωaut1M)⊗ lk ∼ = HH n+N (1) where ⊕ ∞ l=1HH n (l) denotes the Hodge decomposition on H ∗ (M S1 AMS Classification: 55P35, 55P62, 55P10 Key words: Free loop space, loop homology, selfhomotopy equivalences, rational homotopy, Hochschild homology, λdecomposition. 1. Introduction. Let X be a path connected space with base point x0. We denote by: XS1 the space of free loops on X, ΩX the space of based loops of X at x0, autX the monoid of self equivalence of X pointed by IdX, aut1X the connected component of IdX
Hochschild homology of structured algebras
, 2011
"... Abstract. We give a general method for constructing explicit and natural operations on the Hochschild complex of algebras over any PROP with A∞–multiplication—we think of such algebras as A∞–algebras “with extra structure”. As applications, we obtain an integral version of the CostelloKontsevichSo ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
Abstract. We give a general method for constructing explicit and natural operations on the Hochschild complex of algebras over any PROP with A∞–multiplication—we think of such algebras as A∞–algebras “with extra structure”. As applications, we obtain an integral version of the CostelloKontsevichSoibelman moduli space action on the Hochschild complex of open TCFTs, the TradlerZeinalian action of Sullivan diagrams on the Hochschild complex of strict Frobenius algebras, and give applications to string topology in characteristic zero. Our main tool is a generalization of the Hochschild complex. The Hochschild complex of an associative algebra A admits a degree 1 selfmap, ConnesRinehart’s boundary operator B. If A is Frobenius, the (proven) cyclic Deligne conjecture says that B is the ∆–operator of a BVstructure on the Hochschild complex of A. In fact B is part of much richer structure, namely an action by the chain complex of Sullivan diagrams on the Hochschild complex [39]. A weaker version of Frobenius algebras, called here A∞–Frobenius algebras, yields instead an action by the chains on the moduli space of Riemann surfaces [8, 21]. In this paper we develop a general method
UNIVERSAL OPERATIONS IN HOCHSCHILD HOMOLOGY
"... Abstract. We provide a general method for finding all natural operations on the Hochschild complex of Ealgebras, where E is any algebraic structure encoded in a PROP with multiplication, as for example the PROP of Frobenius, commutative or A∞algebras. We show that the chain complex of all such nat ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract. We provide a general method for finding all natural operations on the Hochschild complex of Ealgebras, where E is any algebraic structure encoded in a PROP with multiplication, as for example the PROP of Frobenius, commutative or A∞algebras. We show that the chain complex of all such natural operations is approximated by a certain chain complex of formal operations, for which we provide an explicit model that we can calculate in a number of cases. When E encodes the structure of open topological conformal field theories, we identify this last chain complex, up quasiisomorphism, with the moduli space of Riemann surfaces with boundaries, thus establishing that the operations constructed by Costello and KontsevichSoibelman via different methods identify with all formal operations. When E encodes open topological quantum field theories (or symmetric Frobenius algebras) our chain complex identifies with Sullivan diagrams, thus showing that operations constructed by TradlerZeinalian, again by different methods, account for all formal operations. As an illustration of the last result we exhibit two infinite families of nontrivial operations and use these to produce nontrivial higher string topology operations, which had so far been elusive.
Algèbre/Algebra (Topologie/Topology)
, 2002
"... Algebraic braided model of the affine line and difference calculus on a topological space ..."
Abstract
 Add to MetaCart
Algebraic braided model of the affine line and difference calculus on a topological space
A CHEN MODEL FOR MAPPING SPACES AND THE SURFACE PRODUCT
, 905
"... Abstract. We develop a machinery of Chen iterated integrals for higher Hochschild complexes which are complexes whose differentials are modeled by an arbitrary simplicial set much in the same way that the ordinary Hochschild differential is modeled by the circle. We use these to give algebraic model ..."
Abstract
 Add to MetaCart
Abstract. We develop a machinery of Chen iterated integrals for higher Hochschild complexes which are complexes whose differentials are modeled by an arbitrary simplicial set much in the same way that the ordinary Hochschild differential is modeled by the circle. We use these to give algebraic models for general mapping spaces and define and study the surface product operation on the homology of mapping spaces of surfaces of all genera into a manifold, which is an analogue of the loop product in string topology. As an application we show that this product is homotopy invariant. We prove HochschildKostantRosenberg type theorems and use them to give explicit formulae for the surface product of odd spheres and Lie groups. Contents
Covering homology
, 2008
"... We introduce the notion of covering homology of a commutative Salgebra with respect to certain families of coverings of topological spaces. The construction of covering homology is extracted from Bökstedt, Hsiang and Madsen’s topological cyclic homology. In fact covering homology with respect to th ..."
Abstract
 Add to MetaCart
We introduce the notion of covering homology of a commutative Salgebra with respect to certain families of coverings of topological spaces. The construction of covering homology is extracted from Bökstedt, Hsiang and Madsen’s topological cyclic homology. In fact covering homology with respect to the family of orientation preserving isogenies of the circle is equal to topological cyclic homology. Our basic tool for the analysis of covering homology is a cofibration sequence involving homotopy orbits and a restriction map similar to the restriction map used in Bökstedt, Hsiang and Madsen’s construction of topological cyclic homology. Covering homology with respect to families of isogenies of a torus is constructed from iterated topological Hochschild homology. It receives a trace map from iterated algebraic Ktheory and the hope is that the rich structure, and the calculability of covering homology will make covering homology useful in the exploration of J. Rognes ’ “red shift conjecture”. 1