The volume of the n-dimensional polytope for arbitrary x := (x 1 ; : : : ; x n ) with x i > 0 for all i de nes a polynomial in variables x i which admits a number of interpretations, in terms of empirical distributions, plane partitions, and parking functions. We interpret the terms of this polynomial as the volumes of chambers in two dierent polytopal subdivisions of n (x). The rst of these subdivisions generalizes to a class of polytopes called sections of order cones. In the second subdivision, the chambers are indexed in a natural way by rooted binary trees with n + 1 vertices, and the con guration of these chambers provides a representation of another polytope with many applications, the associahedron.

Let k be a eld and let G be a nite group. There is a canonical element in the Hochschild cohomology of the Tate cohomology G 2 HH 3; 1 ^ H (G; k) with the following property. Given a graded ^ H (G; k)-module X, the image of G in Ext 3; 1 ^ H (G;k) (X; X) vanishes if and only if X is isomorphic to a direct summand of ^ H (G; M) for some kG-module M . The description of the realizability obstruction works in any triangulated category with direct sums. We show that in the case of the derived category of a dierential graded algebra A, there is also a canonical element of Hochschild cohomology HH 3; 1 H (A) which is a predecessor for these obstructions.

We give a brief introduction to A1-algebras and show three contexts in which they appear in representation theory: the study of Yoneda algebras and Koszulity, the description of categories of ltered modules and the description of triangulated categories. Contents 1. Denitions, the bar construction, the minimality theorem 1 2. Yoneda algebras, Koszulity and ltered modules 5 3. Description of triangulated categories 8 References 10 1. Definitions, the bar construction, the minimality theorem 1.1. A-innity algebras and morphisms. We refer to [11] for a list of references and a topological motivation for the following denition: Let k be a eld. An A1 - algebra over k is a Z-graded vector space A = M p2Z A p endowed with graded maps (=homogeneous k-linear maps) mn : A

Abstract. We present a new and explicit method for lifting a tilting complex to a bimodule complex. The key ingredient of our method is the notion of a strong homotopy action in the sense of Stasheff. 1.

. 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 ...

Greenlees defined an abelian category A whose derived category is equivalent to the rational S 1 -equivariant stable homotopy category whose objects represent rational S 1 - equivariant cohomology theories. We show that in fact the model category of di#erential graded objects in A models the whole rational S 1 -equivariant stable homotopy theory. That is, we show that there is a Quillen equivalence between dgA and the model category of rational S 1 -equivariant spectra, before the quasi-isomorphisms or stable equivalences have been inverted. This implies that all of the higher order structures such as mapping spaces, function spectra and homotopy (co)limits are reflected in the algebraic model. The construction of this equivalence involves calculations with Massey products. In an appendix we show that Toda brackets, and hence also Massey products, are determined by the derived category. 1.

In [3] "mall" 1-homological model H of a commutative differential graded algebra is described. Homological Perturbation Theory (HPT) [7-9] provides an explicit description of an A1-coalgebra structure ( 1 ; 2 ; 3 ; : : :) of H. In this paper, we are mainly interested in the determination of the map 2 : H ! H H as a first step in the study of this structure. Developing the techniques given in [20] (inversion theory), we get an important improvement in the computation of 2 with regard to the first formula given by HPT. In the case of purely quadratic algebras, we sketch a procedure for giving the complete Hopf algebra structure of its 1-homology.