Dedicated to the memory of V.K.A.M. Gugenheim Abstract. Our main objective is to demonstrate how homological perturbation theory (HPT) results over the last 40 years immediately or with little extra work give some of the Koszul duality results that have appeared in the last decade. Higher homotopies typically arise when a huge object, e. g. a chain complex defining various invariants of a certain geometric situation, is cut to a small model, and the higher homotopies can then be dealt with concisely in the language of sh-structures (strong homotopy structures). This amounts to precise ways of handling the requisite additional structure encapsulating the various coherence conditions. Given e. g. two augmented differential graded algebras A1 and A2, an sh-map from A1 to A2 is a twisting cochain from the reduced bar construction BA1 of A1 to A2 and, in this manner, the class of morphisms of augmented differential graded algebras is extended to that of sh-morphisms. In the present paper, we explore small models for equivariant (co)homology via differential homological algebra techniques including homological perturbation theory which, in turn, is a standard tool to handle

In this paper, working over Z (p) and using algebra perturbation results from [18], p-minimal homological models of twisted tensor products (TTPs) of Cartan 's elementary complexes are obtained. Moreover, making use of the notion of indecomposability of a TTP, we deduce that a homological model of a indecomposable p-minimal TTP of length # (# 2) of exterior and divided power algebras is a tensor product of k-indecomposable (k #) p-minimal TTPs of exterior and divided power algebras.

Higher homotopy generalizations of Lie-Rinehart algebras, Gerstenhaber, and Batalin-Vilkovisky algebras are explored. These are defined in terms of various antisymmetric bilinear operations satisfying weakened versions of the Jacobi identity, as well as in terms of operations involving more than two variables of the Lie triple systems kind. A basic tool is the Maurer-Cartan algebra—the algebra of alternating forms on a vector space so that Lie brackets correspond to square zero derivations of this algebra—and multialgebra generalizations thereof. The higher homotopies are phrased in terms of these multialgebras. Applications to foliations are discussed: objects which serve as replacements for the Lie algebra of vector fields on the “space of leaves ” and for the algebra of multivector fields are developed, and the spectral sequence of a foliation is shown to arise as a special case of a more general spectral sequence including as well the Hodge-de Rham spectral sequence.

Higher homotopies are nowadays playing a prominent role in mathematics as well as in certain branches of theoretical physics. The purpose of the talk is to recall some of the connections between the past and the present developments. Higher homotopies were isolated within algebraic topology at least as far back as the 1940’s. Prompted by the failure of the Alexander-Whitney multiplication of cocycles to be commutative, Steenrod developed certain operations which measure this failure in a coherent manner. Dold and Lashof extended Milnor’s classifying space construction to associative H-spaces, and a careful examination of this extension led Stasheff to the discovery of An-spaces and A∞-spaces as notions which control the failure of associativity in a coherent way so that the classifying space construction can still be pushed through. Algebraic versions of higher homotopies have, as we all know, led Kontsevich eventually to the proof of the formality conjecture. Homological perturbation theory (HPT), in a simple form first isolated by Eilenberg and Mac Lane in the early 1950’s, has nowadays become a standard tool to handle algebraic incarnations of higher homotopies. A basic

To the memory of G. Chogoshvili Abstract. Let R be a local ring and A a connected differential graded algebra over R which is free as a graded R-module. Using homological perturbation theory techniques, we construct a minimal free multi model for A having properties similar to that of an ordinary minimal model over a field; in particular the model is unique up to isomorphism of multialgebras. The attribute ‘multi ’ refers to the category of multicomplexes. 2000 Mathematics Subject Classification. 18G10, 18G35, 18G55, 55P35, 55P62, 55U15, 57T30. Key words and phrases. Models for differential graded algebras, minimal models for differential graded algebras over local rings, multicomplex, multialgebra, homological perturbations. 2 JOHANNES HUEBSCHMANN

. We establish an algorithm computing the homology of commutative dierential graded algebras (briey, CDGAs). The main tool in this approach is given by the Homological Perturbation Theory particularized for the algebra category (see [21]). Taking into account these results, we develop and rene some methods already known about the computation of the Hochschild and cyclic homologies of CDGAs. In the last section of the paper, we analyze the p-local homology of the iterated bar construction of a CDGA (p prime). 1. Introduction. The description of eÆcient algorithms of homological computation might be considered as a very important question in Homological Algebra, in order to use those processes mainly in the resolution of problems on algebraic topology; but this subject also inuence directly on the development of non so closedareas as Cohomological Physics (in this sense, we nd useful references in [12], [24], [25]) and Secondary Calculus ([14], [27], [28]). Working in the context ...

Abstract. We propose a new point of view of the Spencer cohomology appearing in the formal theory of differential equations based on a dual approach via comodules. It allows us to relate the Spencer cohomology with standard constructions in homological algebra and, in particular, to express it as a Cotor. We discuss concrete methods for its construction based on homological perturbation theory. Appears in Georgian Math. J., vol. 9, No. 4, 2002, 723-772. 1.

Abstract. We determine the mod 2 cohomology algebra of the holomorph of any finite cyclic group whose order is a power of 2. 2000 Mathematics Subject Classification. 20J05 20J06. Key words and phrases. Holomorph of a group, mod 2 cohomology of the holomorph of a cyclic group, homological perturbations and group cohomology. 2 JOHANNES HUEBSCHMANN 1. Outline The holomorph of a group is the semi-direct product of the group with its automorphism group, with respect to the obvious action. The automorphism group of a non-trivial finite cyclic group of order r is well known to be cyclic if and only if the number r is of the kind r = 4, r = pρ, r = 2pρ where p is an odd prime; in these cases, the holomorph is thus a split metacyclic group. The mod p cohomology algebra of an arbitrary metacyclic group has been determined in [2]. In this note we will determine the mod 2 cohomology of the holomorph of a cyclic group whose order is a power of 2. Since for odd p the p-primary part of the automorphism group of any cyclic group is cyclic, we thus get a complete

Abstract. We explain the essence of perturbation problems. The key to understanding is the structure of chain homotopy equivalence – the standard one must be replaced by a finer notion which we call a strong chain homotopy equivalence. We formulate an Ideal Perturbation Lemma and show how both new and classical (including the Basic Perturbation Lemma) results follow from this ideal statement. 1. Introduction and

Abstract. We extend homological perturbation theory to encompass algebraic structures governed by operads and cooperads. Specifically, for an operad O, we define the notion of an ‘O-algebra contraction ’ and we prove that the formulas of the Basic Perturbation Lemma preserve O-algebra contractions. Over a ground ring containing the rational numbers, we give explicit formulas for constructing an O-algebra contraction from any given contraction, generalizing the so called ‘Tensor Trick’. As an illustration of our results we use them to give short proofs of the transfer and minimality theorems for O∞-algebras, where O is any Koszul operad. This subsumes, but is not restricted to, the cases of A∞, C ∞ and L∞-algebras.