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Cochain algebra on manifolds and convergence under refinement. (English) Zbl 1121.58002
Algebraic structures on the simplicial cochains of a triangulated manifold are considered and it is proved that they converge to the differential-geometric analogous as the triangulation becomes small. Such result is obtained for a cochain cup product converging to the wedge product on differential forms. Also, it is shown that any extension of this product to a 𝒞 -algebra converges to the wedge product of forms. For cochains equipped with an inner product, the notion of combinatorial star operator is defined and it is proved that for a certain cochain inner product this operator converges to the smooth star operator.
58A10Differential forms (global analysis)
49Q15Geometric measure and integration theory, integral and normal currents (optimization)
55N45Products and intersections (algebraic topology)
57R05Triangulating (differential topology)