Abstract. We provide a general definition of higher homotopy operations, encompassing most known cases, including higher Massey and Whitehead products, and long Toda brackets. These operations are defined in terms of the W-construction of Boardman and Vogt, applied to the appropriate diagram category; we also show how some classical families of polyhedra (including simplices, cubes, associahedra, and permutahedra) arise in this way. 1.

Abstract. We give a self-contained introduction to the theory of secondary polytopes and geometric bistellar flips in triangulations of polytopes and point sets, as well as a review of some of the known results and connections to algebraic geometry, topological combinatorics, and other areas. As a new result, we announce the construction of a point set in general position with a disconnected space of triangulations. This shows, for the first time, that the poset of strict polyhedral subdivisions of a point set is not always connected.

Introduction The following text arose from the desire to establish a firm relationship between higher order categories and topological spaces. Our approach combines the algebraic features of Michael Batanin's !-operads [1] with the geometric features of Andr'e Joyal's cellular sets [15] and tries to mimick as far as possible the classical construction of the simplicial nerve of a category. Higher order categories have attracted much attention in the last decade, due to their appearance in several mathematical areas. The ultimate goal is perhaps a faithful algebraic description of homotopy systems [13]. Since a homotopy between homotopies has the shape of a disk, the next higher homotopy the shape of a ball, and so on, we chose "ball compexes", i.e. globular sets [26], as the primitive combinatorial objects. The globular structure is precisely what underlies an !-category [26], once its mu

As yet we are ignorant of an effective method of computing the cohomology of a Postnikov complex from πn and k n+1 [7]. The classical problem of algebraic models for homotopy types is precisely stated, to our knowledge for the first time. Two different natural statements for this problem are produced, the simplest one being entirely solved by the notion of SSEH-structure, due to the authors. Other tentative solutions, Postnikov towers and E∞-chain complexes are considered and compared with the SSEH-structures. In particular, which looks like a severe error about the usual understanding of the k-invariants is explained; which implies we seem far from a solution for the ideal statement of our problem. At the positive side, the problem stated above in the title inscription is solved. 1 Introduction.

Abstract. The paper introduces the notion of a truncating twisting function from a cubical set to a permutahedral set and the corresponding notion of twisted Cartesian product of these sets. The latter becomes a permutocubical set that models in particular the path space fibration on a loop space. The chain complex of this twisted Cartesian product in fact is a comultiplicative twisted tensor product of cubical chains of base and permutahedral chains of fibre. This construction is formalized as a theory of twisted tensor products for Hirsch algebras. 1. introduction The paper continues [12] in which a combinatorial model for a fibration was constructed based on the notion of a truncating twisting function from a simplicial set to a cubical set and on the corresponding notion of twisted Cartesian product of these sets being a cubical set. Applying the cochain functor we obtain a multiplicative twisted tensor product modeling the corresponding fibration. There arises a need to iterate this construction for fibrations over loop or path

. We show that the well known homotopy complementation formula of Bjorner and Walker admits several closely related generalizations on different classes of topological posets (lattices). The utility of this technique is demonstrated on some classes of topological posets including the Grassmannian and configuration posets, e Gn (R) and exp n (X) which were introduced and studied by V. Vassiliev. Among other applications we present a reasonably complete description, in terms of more standard spaces, of homology types of configuration posets exp n (S m ) which leads to a negative answer to a question of Vassilev raised at the workshop "Geometric Combinatorics" (MSRI, February 1997). 1. Introduction One of the objectives of this paper is to initiate the study of topological (continuous) posets and their order complexes from the point of view of Geometric Combinatorics. Recall that finite or more generally locally finite partially ordered sets (posets) already occupy one of privileged ...

. Given an affine projection : P ! Q of a d-polytope P onto a polygon Q, it is proved that the poset of proper polytopal subdivisions of Q which are induced by has the homotopy type of a sphere of dimension d \Gamma 3 if maps all vertices of P into the boundary of Q. This result, originally conjectured by Reiner, is an analogue of a result of Billera, Kapranov and Sturmfels on cellular strings on polytopes and explains the significance of the interior point of Q present in the counterexample to their generalized Baues conjecture, constructed by Rambau and Ziegler. 1. Introduction Motivated by their theory of fiber polytopes [6] [18, Lecture 9], Billera and Sturmfels have associated to any affine projection of convex polytopes : P ! Q the Baues poset !(P ! Q) of proper polytopal subdivisions of Q which are induced by . This poset reduces to the poset of proper cellular strings [7] on P with respect to , if dim(Q) = 1, and can be described in general as the poset of proper ...

Abstract. In the paper the notion of truncating twisting function from a cubical set to a permutahedral set and the corresponding notion of twisted Cartesian product of these sets are introduced. The latter becomes a permutocubical set that models in particular the path fibration on a loop space. The chain complex of this twisted Cartesian product in fact is a comultiplicative twisted tensor product of cubical chains of base and permutahedral chains of fibre. This construction is formalized as a theory of twisted tensor products for Hirsch algebras. 1. introduction The paper continues [13] in which a combinatorial model for a fibration was constructed based on the notion of a truncating twisting function from a simplicial set to a cubical set and on the corresponding notion of twisted Cartesian product of these sets being a cubical set. Applying the cochain functor we obtained a multiplicative twisted tensor product modeling the corresponding fibration. There arises a need to iterate this construction for fibrations over loop or path

Introduction These notes are a detailed account of two lectures I gave during a workshop on operads in Osnabruck (16-19 June 1998). I would like to thank Rainer Vogt for organizing this really stimulating meeting which gave the participants the wonderful chance to exchange their ideas in a very lively atmosphere. The purpose of my lectures is fourfold : 1. to show "on the nose" that the well known configuration space model for \Omega n S n X is homotopy equivalent to Milgram's permutohedral model ; 2. to indicate a "recipe" for constructing cellular decompositions of En - operads ; 3. to give a simplicial splitting of\Omega n S n X using Jeff Smith's filtration of the "symmetric monoidal" operad ; 4. to outline some interaction between En -operads and immersion theory. 1 Configuration spaces and permutohedra. Initial

models for