On the algebra and geometry of a manifold's chains and cochains

This dissertation consists of two parts, each of which describes new algebraic and geometric structures defined on chain complexes associated to a manifold.; In the first part we define, on the simplicial cochains of a triangulated manifold, analogues of certain objects in differential geometry. In particular, we define a cochain product and prove several results on its convergence to the wedge product of 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 a triangulated Riemann surface. There are several convergence theorems here as well; for a particularly nice cochain inner product, both of these combinatorial structures converge to their continuum analogues as the mesh of the triangulation tends to zero.; In the second part, we describe an algebraic structure on the chains of a manifold, induced by the transversal intersection of chains. We prove that, up to quasi-isomorphism, the chains form an Einfinity algebra (a generalization of a commutative algebra). This chain algebra induces the usual intersection product on homology. This result follows from a general theorem that we prove, cast in the language of operads, on partially defined algebraic structures. We also describe an application of this theorem to string topology.