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

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