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21
A combinatorial method for computing Steenrod squares
, 1999
"... We present here a combinatorial method for computing 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 ..."
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We present here a combinatorial method for computing 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. c 1999 Elsevier Science B.V. All rights reserved.
Universal classes for algebraic groups
"... Abstract. We exhibit cocycles representing certain classes in the cohomology of the algebraic group GLn with coefficients in the representation Γ ∗ (gl (1) n). These classes ’ existence was anticipated by van der Kallen, and they intervene in the proof that reductive linear algebraic groups have fin ..."
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Cited by 10 (3 self)
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Abstract. We exhibit cocycles representing certain classes in the cohomology of the algebraic group GLn with coefficients in the representation Γ ∗ (gl (1) n). These classes ’ existence was anticipated by van der Kallen, and they intervene in the proof that reductive linear algebraic groups have finitely generated cohomology algebras [18]. Let�be a field of positive characteristic, let A be a finitely generated �algebra, and let G be a reductive linear algebraic group defined over� and acting rationally on A by algebra automorphisms. Then the rational cohomology H ∗ (G,A) is an algebra, and one can wonder if it is finitely generated. In degree 0, the finite generation of the subalgebra AG = H0 (G,A) is part of Hilbert’s fourteenth problem and was solved positively by the work of Nagata [13] and Haboush [9]. The finite generation of the whole cohomology algebra remained unsolved in general, though much progress had been made in recent years [7, 20].
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 ..."
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Cited by 6 (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.
Chain Homotopies for Object Topological Representations
, 2008
"... This paper presents a set of tools to compute topological information of simplicial complexes, tools that are applicable to extract topological information from digital pictures. A simplicial complex is encoded in a (nonunique) algebraictopological format called AMmodel. An AMmodel for a given o ..."
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This paper presents a set of tools to compute topological information of simplicial complexes, tools that are applicable to extract topological information from digital pictures. A simplicial complex is encoded in a (nonunique) algebraictopological format called AMmodel. An AMmodel for a given object K is determined by a concrete chain homotopy and it provides, in particular, integer (co)homology generators of K and representative (co)cycles of these generators. An algorithm for computing an AMmodel and the cohomological invariant HB1 (derived from the rank of the cohomology ring) with integer coefficients for a finite simplicial complex in any dimension is designed here, extending the work done in [9] in which the ground ring was a field. A concept of generators which are “nicely ” representative is also presented. Moreover, we extend the definition of AMmodels to 3D binary digital images and we design algorithms to update the AMmodel information after voxel set operations (union, intersection, difference and inverse).
Simplicial Perturbation Techniques and Effective Homology
"... In this paper, we deal with the problem of the computation of the homology of a finite simplicial complex after an "elementary simplicial perturbation" process such as the inclusion or elimination of a maximal simplex or an edge contraction. To this aim we compute an algebraic topological ..."
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In this paper, we deal with the problem of the computation of the homology of a finite simplicial complex after an "elementary simplicial perturbation" process such as the inclusion or elimination of a maximal simplex or an edge contraction. To this aim we compute an algebraic topological model that is an special chain homotopy equivalence connecting the simplicial complex with its homology (working with a field as the ground ring).
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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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
A survey of Huebschmann and Stasheff’s paper: Formal solution of the master equation via HPT and deformation theory, math.QA/0704.0432v1
, 2007
"... These notes, based on the paper [8] by Huebschmann and Stasheff, were prepared for a series of talks at Illinois State University with the intention of applying ..."
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These notes, based on the paper [8] by Huebschmann and Stasheff, were prepared for a series of talks at Illinois State University with the intention of applying
"Coalgebra" Structures on 1Homological Models for Commutative Differential Graded Algebras
"... In [3] "mall" 1homological model H of a commutative differential graded algebra is described. Homological Perturbation Theory (HPT) [79] provides an explicit description of an A1coalgebra structure ( 1 ; 2 ; 3 ; : : :) of H. In this paper, we are mainly interested in the determination o ..."
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In [3] "mall" 1homological model H of a commutative differential graded algebra is described. Homological Perturbation Theory (HPT) [79] provides an explicit description of an A1coalgebra 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 1homology.
Computing "Small" 1Homological Models for Commutative Differential Graded Algebras
"... We use homological perturbation machinery specific for the algebra category [13] to give an algorithm for computing the differential structure of a small 1homological model for commutative differential graded algebras (briefly, CDGAs). The complexity of the procedure is studied and a computer packa ..."
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We use homological perturbation machinery specific for the algebra category [13] to give an algorithm for computing the differential structure of a small 1homological model for commutative differential graded algebras (briefly, CDGAs). The complexity of the procedure is studied and a computer package in Mathematica is described for determining such models.