Topological versions of Abel-Jacobi, the height pairing, and the Poincar\'e bundle

We extend to the topological setting the classical constructions of the Abel-Jacobi mapping on homologically trivial algebraic cycles and the height pairing between two such cycles. We further interpret the height pairing between homologically trivial topological cycles (with disjoint support) as giving a lifting of their Abel-Jacobi images to the fiber of the Poincar\'e bundle, extending work of R. Hain in the algebraic setting. Part II of the current revision further explores the relationship between the topological height pairing and the classical height pairing in the case of algebraic cycles.