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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.
MSC:
58A10Differential forms (global analysis)
49Q15Geometric measure and integration theory, integral and normal currents (optimization)
55N45Products and intersections (algebraic topology)
57R05Triangulating (differential topology)