Results 1  10
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11
Homological Perturbation Theory and Associativity
, 2000
"... In this paper, we prove various results concerning DGAalgebras in the context of the Homological Perturbation Theory. We distinguish two class of contractions for algebras: full algebra contractions and semifull algebra contractions. A full algebra contraction is, in particular, a semifull algebr ..."
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Cited by 16 (10 self)
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In this paper, we prove various results concerning DGAalgebras in the context of the Homological Perturbation Theory. We distinguish two class of contractions for algebras: full algebra contractions and semifull algebra contractions. A full algebra contraction is, in particular, a semifull algebra contraction. Taking a full algebra contraction and an "algebra perturbation" as data of the Basic Perturbation Lemma, the Algebra Perturbation Lemma (or simply, FAPL) of [20] and [29] appears in a natural way. We establish here a perturbation machinery, the SemiFull Algebra Perturbation Lemma (or, simply, SFAPL) that is a generalization of the previous one in the sense that the application range of SFAPL is wider than that of FAPL. We show four important applications in which this result is essential for the construction of algebra or coalgebra structures in various chain complexes.
Computation of Cohomology Operations on Finite Simplicial Complexes
 Homology, Homotopy and Applications
, 2003
"... We propose a method for calculating cohomology operations for finite simplicial complexes. ..."
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Cited by 12 (9 self)
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We propose a method for calculating cohomology operations for finite simplicial complexes.
Geometric Objects and Cohomology Operations
 Proc. of the 5th Workshop on Computer Algebra in Scientific Computing
, 2002
"... Cohomology operations (including the cohomology ring) of a geometric object are finer algebraic invariants than the homology of it. In the literature, there exist various algorithms for computing the homology groups of simplicial complexes ([Mun84], [DE95,ELZ00], [DG98]), but concerning the algorith ..."
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Cited by 4 (4 self)
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Cohomology operations (including the cohomology ring) of a geometric object are finer algebraic invariants than the homology of it. In the literature, there exist various algorithms for computing the homology groups of simplicial complexes ([Mun84], [DE95,ELZ00], [DG98]), but concerning the algorithmic treatment of cohomology operations, very little is known. In this paper, we establish a version of the incremental algorithm for computing homology given in [ELZ00], which saves algebraic information, allowing us the computation of the cup product and the effective evaluation of the primary and secondary cohomology operations on the cohomology of a finite simplicial complex. The efficient combinatorial descriptions at cochain level of cohomology operations developed in [GR99,GR99a] are essential ingredients in our method. We study the computational complexity of these processes and a program in Mathematica for cohomology computations is presented.
Computing Cocycles on Simplicial Complexes
"... In this note, working in the context of simplicial sets [17], we give a detailed study of the complexity for computing chain level Steenrod squares [20,21], in terms of the number of face operators required. This analysis is based on the combinatorial formulation given in [5]. As an application, we ..."
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Cited by 3 (1 self)
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In this note, working in the context of simplicial sets [17], we give a detailed study of the complexity for computing chain level Steenrod squares [20,21], in terms of the number of face operators required. This analysis is based on the combinatorial formulation given in [5]. As an application, we give here an algorithm for computing cupi products over integers on a simplicial complex at chain level. 1
Simplification Techniques for Maps in Simplicial Topology
 ARTICLE SUBMITTED TO JOURNAL OF SYMBOLIC COMPUTATION
"... This paper offers an algorithmic solution to the problem of obtaining "economical" formulae for some maps in Simplicial Topology, having, in principle, a high computational cost in their evaluation. In particular, maps of this kind are used for defining cohomology operations at the cochain level. As ..."
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Cited by 3 (2 self)
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This paper offers an algorithmic solution to the problem of obtaining "economical" formulae for some maps in Simplicial Topology, having, in principle, a high computational cost in their evaluation. In particular, maps of this kind are used for defining cohomology operations at the cochain level. As an example, we obtain an explicit combinatorial description of all Steenrod kth powers exclusively in terms of face operators.
HPT and Cocyclic Operations
 HOMOLOGY, HOMOTOPY AND APPLICATIONS
, 2004
"... We reinterpret the classical theory of cocyclic operations in terms of permutations and homotopy equivalences of explicit chains. The essential tools we use are Homological Perturbation Theory and EilenbergZilber Theorem. The main objective of this technique is the final identification of cohomolog ..."
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Cited by 1 (1 self)
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We reinterpret the classical theory of cocyclic operations in terms of permutations and homotopy equivalences of explicit chains. The essential tools we use are Homological Perturbation Theory and EilenbergZilber Theorem. The main objective of this technique is the final identification of cohomology operations at cochain level.
Article Submitted to Journal of Symbolic Computation Simplification Techniques for Maps in Simplicial Topology
"... This paper o#ers an algorithmic solution to the problem of obtaining "economical " formulae for some maps in Simplicial Topology, having, in principle, a high computational cost in their evaluation. In particular, maps of this kind are used for defining cohomology operations at the cochain level. As ..."
Abstract
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This paper o#ers an algorithmic solution to the problem of obtaining "economical " formulae for some maps in Simplicial Topology, having, in principle, a high computational cost in their evaluation. In particular, maps of this kind are used for defining cohomology operations at the cochain level. As an example, we obtain an explicit combinatorial description of all Steenrod kth powers exclusively in terms of face operators. 1.
Homology, Homotopy and Applications, vol.5(2), 2003, pp.8393
 Homology, Homotopy and Applications
, 2003
"... We propose a method for calculating cohomology operations on finite simplicial complexes. ..."
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We propose a method for calculating cohomology operations on finite simplicial complexes.
Computing Adem Cohomology Operations
, 2006
"... We deal with the problem of obtaining explicit simplicial formulae defining the classical Adem cohomology operations at the cochain level. Having these formulae at hand, we design an algorithm for computing these operations for any finite simplicial set. ..."
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We deal with the problem of obtaining explicit simplicial formulae defining the classical Adem cohomology operations at the cochain level. Having these formulae at hand, we design an algorithm for computing these operations for any finite simplicial set.
Rocío González–Díaz, Pedro Real Universidad de Sevilla, Depto. de Matemática Aplicada I,
, 2001
"... In this note, working in the context of simplicial sets [17], we give a detailed study of the complexity for computing chain level Steenrod squares [20, 21], in terms of the number of face operators required. This analysis is based on the combinatorial formulation given in [5]. As an application, we ..."
Abstract
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In this note, working in the context of simplicial sets [17], we give a detailed study of the complexity for computing chain level Steenrod squares [20, 21], in terms of the number of face operators required. This analysis is based on the combinatorial formulation given in [5]. As an application, we give here an algorithm for computing cup–i products over integers on a simplicial complex at chain level. 1