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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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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 "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.