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

TL;DR

This paper generalizes classical algebraic cycle concepts like the Abel-Jacobi map and height pairing to a topological context, connecting these notions via the Poincaré bundle and exploring their relationships.

Contribution

It introduces topological versions of the Abel-Jacobi map and height pairing, extending algebraic geometric ideas to a broader topological framework.

Findings

Topological Abel-Jacobi map constructed for homologically trivial cycles.

Height pairing interpreted as a lift to the Poincaré bundle.

Connections established between topological and algebraic height pairings.

Abstract

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.