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 holomorphic and anti-holomorphic
cochains a combinatorial period matrix for surfaces. We show that for a
particularly nice cochain inner product, several of these structures converge
to their continuum analogues as the mesh of a triangulation tends to zero. It
is an open question as to whether or not the combinatorial period matrix
converges to the Riemann period matrix as the mesh of a sequence of
triangulations tends to zero.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.