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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.
On the computability of the plocal homology of twisted cartesian products of EilenbergMac Lane spaces
, 1999
"... Working in the framework of the Simplicial Topology, a method for calculating the plocal homology of a twisted cartesian product X(#, m, #, # # , n) = K(#, m) # K(# # , n) of EilenbergMac Lane spaces is given. The chief technique is the construction of an explicit homotopy equivalence between the ..."
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Cited by 1 (1 self)
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Working in the framework of the Simplicial Topology, a method for calculating the plocal homology of a twisted cartesian product X(#, m, #, # # , n) = K(#, m) # K(# # , n) of EilenbergMac Lane spaces is given. The chief technique is the construction of an explicit homotopy equivalence between the normalized chain complex of X and a free DGAmodule of finite type M , via homological perturbation. If X is a commutative simplicial group (being its inner product the natural one of the cartesian product of K(#, m) and K(# # , n)), then M is a DGAalgebra. Finally, in the special case K(#, 1) ## X K(# # , n), we prove that M can be a small twisted tensor product. 1
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.
unknown title
, 1999
"... Analyzing the transference of the coalgebra structure on the homology of CDGAs ..."
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Analyzing the transference of the coalgebra structure on the homology of CDGAs
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 GonzálezDíaz and Pedro Real
, 2001
"... We present here a combinatorial method for computing cupi products and Steenrod squares of a simplicial set X. This method is essentially based on the determination of explicit formulae for the component morphisms of a higher diagonal approximation (i.e., a family of morphisms measuring the lack of ..."
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We present here a combinatorial method for computing cupi products and Steenrod squares of a simplicial set X. This method is essentially based on the determination of explicit formulae for the component morphisms of a higher diagonal approximation (i.e., a family of morphisms measuring the lack of commutativity of the cup product on the cochain level) in terms of face operators of X. A generalization of this method to Steenrod reduced powers is sketched.