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 André-Quillen homology • Gamma homology • stable homotopy of Γ-modules • stable homotopy of algebraic theories • the André-Quillen cohomology groups which arise as obstruction groups in the Goerss-Hopkins approach As a comparison between stable homotopy of Γ-modules 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

Let M be a simply-connected closed oriented N-dimensional 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 Chas-Sullivan, [1]. As usual aut1X (resp. ΩX) denotes the monoid of the self-equivalences 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, self-homotopy equivalences, rational homotopy, Hochschild homology, λ-decomposition. 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

Algebraic braided model of the affine line and difference calculus on a topological space

algebras of mapping spaces and finite group actions.

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 Hochschild-Kostant-Rosenberg type theorems and use them to give explicit formulae for the surface product of odd spheres and Lie groups. Contents

We introduce the notion of covering homology of a commutative S-algebra 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 K-theory 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