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Let C be a differential graded chain coalgebra, U C the reduced cobar construction on C and C-V the dual algebra. We prove that for a large class of coalgebras C there is a natural isomorphism of Gerstenhaber algebras between the Hochschild cohomologies, HH*(C-V, C-V) and HH*(U C; U C). This result yields to a Hodge decomposition of the loop space homology of a closed oriented manifold, when the field of coefficients is of characteristic zero. C 2004 Elsevier B.V. All rights reserved.

In this paper we use May's algebraic approach to Steenrod operation in conjunction with Munkholm's notion of shc-algebra in order to introduce what we call a π-shc-algebra. (π is a cyclic group of order a fixed prime p.) We prove that the homology and the Hochschild homology of a π-shc-algebra have natural Steenrod operations. Our main topological example is the free loop space. We prove that the Jones’ isomorphism preserves Steenrod operations.

Let M be a closed orientable manifold of dimension d and $\mathcal{C}^*(M)$ be the usual cochain algebra on M with coefficients in a field k. The Hochschild cohomology of M, $H\!H^*(\mathcal{C}^*(M);\mathcal{C}^*(M))$ is a graded commutative and associative algebra. The augmentation map $\varepsilon: \mathcal{C}^*(M) \to{\textbf{\textit{k}}}$ induces a morphism of algebras $I : H\!H^*(\mathcal{C}^*(M);\mathcal{C}^*(M)) \to{H\!H^*(\mathcal{C}^*(M);{\textbf{\textit{k}}})}$ . In this paper we produce a chain model for the morphism I. We show that the kernel of I is a nilpotent ideal and that the image of I is contained in the center of $H\!H^*(\mathcal{C}^*(M);{\textbf{\textit{k}}})$ , which is in general quite small. The algebra $H\!H^*(\mathcal{C}^*(M);\mathcal{C}^*(M))$ is expected to be isomorphic to the loop homology constructed by Chas and Sullivan. Thus our results would be translated in terms of string homology.

The purpose of the present article is threefold. First of all, we rebuild the whole theory of cosimplicial models of mapping spaces by using systematically Kan adjunction techniques. Secondly, given two topological spaces X and Y, we construct a cochain algebra which is quasi-isomorphic (as an algebra) to the singular cochain algebra of the corresponding mapping space from X to Y. Here, X has to be homotopy equivalent to the geometric realization a finite simplicial set and of dimension less or equal to the connectivity of Y. At last, we apply these results to the study of finite group actions on mapping spaces that are induced by an action on the source., Comment: 25 pages

Let $X$ be a simply connected space and $\Bbb K$ be any field. The normalized singular cochains $N^*(X; {\Bbb K})$ admit a natural strongly homotopy commutative algebra structure, which induces a natural product on the Hochschild homology $HH_* N^*X$ of the space $X$. We prove that, endowed with this product, $HH_*N^*X$ is isomorphic to the cohomology algebra of the free loop space of $X$ with coefficients in $\Bbb K$. We also show how to construct a simpler Hochschild complex which allows direct computation., Comment: 21 pages, to appear in Topology

International audience; The Hochschild cohomology of a DG algebra A with coefficients in itself is, up to a suspension of degrees, a graded Lie algebra. The purpose of this paper is to prove that a certain DG Lie algebra of derivations appears as a finite codimensional graded sub Lie algebra of this Lie algebra when A is a strongly homotopy commutative algebra whose homology is concentrated in finitely many degrees. This result has interesting implications for the free the loop space homology which we explore here as well.

Let $X$ be a connected CW complex and $[X]$ be its homotopy type. As usual, $\mbox{SNT}(X)$ denotes the pointed set of homotopy types of CW complexes $Y$ such that their $n^{th}$-Postnikov approximations $X^{(n)}$ and $Y^{(n)}$ are homotopy equivalent for all $n$. In this paper we study a particularly interesting subset of \mbox{SNT}$(X)$, denoted SNT$ _{\pi } (X)$, of strong $n$ type; the $n^{th}$-Postnikov approximations $X^{(n)}$ and $Y^{(n)}$ are homotopy equivalent by homotopy equivalences satisfying an extra condition at the level of homotopy groups. First, we construct a CW complex $X$ such that $\mbox{SNT}_\pi(X) \neq \{ [X] \}$ and we establishe a connection between the pointed set $\mbox{SNT}_\pi (X)$ and sub-groups of homotopy classes of self-equivalences via a certain $\displaystyle\lim_{\leftarrow}{}^1 $ set. Secondly, we prove a conjecture of Arkowitz and Maruyama concerning subgroups of the group of self equivalences of a finite CW complex and we use this result to establish a characterization of simply connected CW complexes with finite dimensional rational cohomology such that $\mbox{SNT}_\pi(X) = \{[X]\}$.

In this short paper we try to describe the fundamental contribution of Quillenin the development of abstract homotopy theory and we explain how he uses this theory to lay the foundations of rational homotopy theory.

Nous comparons et calculons trois sous-espaces vectoriels remaquables de la cohomologie d'un CW-complexe.We compare three natural subvectorspaces of the cohomology of a simply connected pointed space. These spaces are related to, respectively, the “last cells” of the space, the “dual Gottlieb groups” and the “evaluation map” defined by the Spivak construction.

A subspace Z of a topological space X is contractible in X if the inclusion i : Z → X is homotopic to a constant map Z → x 0.

In §12 we showed how to model commutative cochain algebras (C,d) by Sullivan algebras, (ΛV, d). Now any cochain algebra comes equipped with the specific morphism k → (C, d), 1→ 1. In this section we introduce relative Sullivan algebras (B ⊗ ΛV,d) and use them to model general morphisms (B.d) → (C,d) of commutative cochain algebras. We shall then prove existence and uniqueness theorems in this setting. This will, in particular, eliminate the hypothesis H 1 = 0 that was frequently supposed in §12.

In applying Sullivan models it is important to be able to compute directly the models of geometric constructions from models for the spaces used in the construction. For example, in §12 we saw that the tensor product of models was a model of the topological product, while in §14 we will see how to model fibrations.

They were introduced, respectively, by Eilenberg-MacLane [48] who showed that \(B(\mathbb{Z}[\Gamma ]) \simeq C_* (K(\Gamma ,{\rm I}))\) and by Adams [1] who showed that \(\Omega C_* (X) \simeq C_* (\Omega X)\).

In this section the ground ring is an arbitrary field k of characteristic zero. Let X be a simply connected topological space with rational homology of finite type. Then the rational homotopy type of X is completely determined by its minimal Sullivan model ( §12, §17) or by any Lie model ( §24). It is thus not unexpected that we should be able to compute cl0 X, cat0 X and e 0X directly from these models. Recall also that a commutative model for X is any commutative cochain algebra connected to Ap L(X) by a chain of quasi-isomorphisms (§10). Thus (A, d) is a commutative model for X if and only if its minimal Sullivan model coincides with that of X

A coalgebra of the form C = will be called one-connected. Such coalgebras are co-augmented by k. The object of this section is to construct Quillen’s functors

Recall from §5 that C * (X; \(\Bbbk \)) denotes the cochain algebra of normalized singular cochains on a topological space X. This algebra is almost never commutative, although it is homotopy commutative.

In this section we see how relative Sullivan algebras model fibrations. In particular, if f : X → Y is a continuous map with homotopy fibre F we construct a Sullivan model for F directly from the morphism A PL ,(f): A PL (Y) → A PL (X), provided Y is simply connected with rational homology of finite type.

Let $\xi=(E,p,B,F)$ be a Hurewicz fibration. In this paper we study the space $\Cal L_G(\xi)$ consisting of fibre homotopy self equivalences of $\xi$ inducing by restriction to the fibre a self homotopy equivalence of $F$ belonging to the group $G$. We give in particular conditions implying that $\pi_1(\Cal L_G(\xi))$ is finitely generated or that $\Cal L_1(\xi)$ has the same rational homotopy type as $\operatorname{aut}_1(F)$.

This chapter discusses the differential graded algebras in topology. The homotopy theory of topological spaces attempts to classify weak homotopy types of spaces and homotopy classes of maps. For a given geometric problem, however, it is often sufficient to replace the homotopy theory of spaces by a weaker “algebraic” homotopy theory as described, for instance, in Baues' book. This chapter considers the specific case when spaces are modeled by the homotopy theory of differential graded algebras.

The purpose of this paper is to study and compare the collapsing of two spectral sequences converging to the cohomology of a configuration space. The noncollapsing of these spectral sequences is related, in some cases, to the existence of Massey products in the cohomology of the manifold M.

In 1999, M.Chas and D.Sullivan have constructed on the homology of free loop space of a manifold a BV-algebra structure. This was the begining of the String topology theory. In this thesis, we show the compatibility of the Serre spectral sequence to Gysin morphism of smooth finite codimensional embeddings between manifolds. Then, we use this technic to compute some examples of structures of string topology. We study essentially the “loop product” and the “loop coproduct” which provide the homology of free loop space of a manifold a stucture of Frobenius algebra without counit., En 1999, M.Chas et D.Sullivan ont mis en évidence sur l'homologie de l'espace des lacets libres d'une variété une structure de BV-algèbre. C'est ce qui fonde la théorie topologique des cordes. Dans cette thèse, nous montrons comment la compatibilité de la suite spectrale de Serre aux morphismes de Gysin de plongements lisses de codimension finie entre vari étés permet d'effectuer des calculs de ces structures de topologie des cordes. Nous étudions essentiellement le ”loop produit” et le ”loop coproduit” qui munissent l'homologie de l'espace des lacets libres d'une vari été d'une structure d'algèbre de Frobenius sans counité.