Cochain algebra on manifolds and convergence under refinement (Q881459)

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.