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31
A cubical model of a fibration
 J. Pure Appl. Algebra
"... Abstract. In the paper the notion of a truncating twisting function from a simplicial set to a cubical set and the corresponding notion of twisted Cartesian product of these sets are introduced. The latter becomes a cubical set. This construction together with the theory of twisted tensor products f ..."
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Cited by 9 (7 self)
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Abstract. In the paper the notion of a truncating twisting function from a simplicial set to a cubical set and the corresponding notion of twisted Cartesian product of these sets are introduced. The latter becomes a cubical set. This construction together with the theory of twisted tensor products for homotopy Galgebras allows to obtain multiplicative models for fibrations. 1.
Higher homotopy operations
"... 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 Wconstruction of Boardman and Vogt, applied to the appropriate diagram categor ..."
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Cited by 6 (3 self)
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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 Wconstruction 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.
Monotone paths on polytopes
 MATH. Z. 235,315–334 (2000)
, 2000
"... We investigate the vertexconnectivity of the graph of fmonotone paths on a dpolytope P with respect to a generic functional f. The third author has conjectured that this graph is always (d − 1)connected. We resolve this conjecture positively for simple polytopes and show that the graph is 2co ..."
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Cited by 6 (4 self)
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We investigate the vertexconnectivity of the graph of fmonotone paths on a dpolytope P with respect to a generic functional f. The third author has conjectured that this graph is always (d − 1)connected. We resolve this conjecture positively for simple polytopes and show that the graph is 2connected for any dpolytope with d ≥ 3. However,we disprove the conjecture in general by exhibiting counterexamples for each d ≥ 4 in which the graph has a vertex of degree two. We also reexamine the Baues problem for cellular strings on polytopes, solved by Billera,Kapranov and Sturmfels. Our analysis shows that their positive result is a direct consequence of shellability of polytopes and is therefore less related to convexity than is at first apparent.
Geometric bistellar flips: the setting, the context and a construction
 In International Congress of Mathematicians. Vol. III
, 2006
"... Abstract. We give a selfcontained 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 n ..."
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Abstract. We give a selfcontained 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.
Algebraic Models for Homotopy Types
 Homology, Homotopy and Applications
"... 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 produce ..."
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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 SSEHstructure, due to the authors. Other tentative solutions, Postnikov towers and E∞chain complexes are considered and compared with the SSEHstructures. In particular, which looks like a severe error about the usual understanding of the kinvariants 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.
The twisted Cartesian model for the double path fibration
, 2004
"... 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 c ..."
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Cited by 2 (0 self)
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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.
The twisted Cartesian model for the double path space fibration
"... 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. T ..."
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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
A Cellular Nerve for Higher Order Categories
, 1999
"... 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 [ ..."
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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
DIAMETER OF REDUCED WORDS
, 2009
"... For finite reflection groups of types A and B, we determine the diameter of the graph whose vertices are reduced words for the longest element and whose edges are braid relations. This is deduced from a more general theorem that applies to supersolvable hyperplane arrangements. ..."
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For finite reflection groups of types A and B, we determine the diameter of the graph whose vertices are reduced words for the longest element and whose edges are braid relations. This is deduced from a more general theorem that applies to supersolvable hyperplane arrangements.
Iterating the bar construction
 Georgian Math. J
, 1998
"... Abstract. For a 1connected space X Adams’s bar construction B(C ∗ (X)) describes H ∗ (ΩX) only as a graded module and gives no information about the multiplicative structure. Thus it is not possible to iterate the bar construction in order to determine the cohomology of iterated loop spaces ΩiX. In ..."
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Abstract. For a 1connected space X Adams’s bar construction B(C ∗ (X)) describes H ∗ (ΩX) only as a graded module and gives no information about the multiplicative structure. Thus it is not possible to iterate the bar construction in order to determine the cohomology of iterated loop spaces ΩiX. In this paper for an nconnected pointed space X a sequence of A(∞)algebra structures {m (k) i}, k = 1, 2,..., n, is constructed, such that for each k ≤ n there exists an isomorphism of graded algebras ∼ = (H(B( · · · (B(B(C ∗ (X); {m (1) i