  In this paper we develop several algebraic structures on the simplicial
cochains of a triangulated manifold that are analogues of objects in
differential geometry. We study a cochain product and prove several statements
about its convergence to the wedge product on differential forms. Also, for
cochains with an inner product, we define a combinatorial Hodge star operator,
and describe some applications, including a combinatorial period matrix for
surfaces. We show that for a particularly nice cochain inner product, these
combinatorial structures converge to their continuum analogues as the mesh of a
triangulation tends to zero.
