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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 $\mathcal C _{\infty}$-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:
 58A10 Differential forms (global analysis) 49Q15 Geometric measure and integration theory, integral and normal currents (optimization) 55N45 Products and intersections (algebraic topology) 57R05 Triangulating (differential topology)
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