A Künneth Theorem for Lagrangian Quantum Homology

Abstract

In 1996, Yong-Geun Oh introduced the pearl complex as a means of computing the Lagrangian intersection Floer homology of a monotone Lagrangian with itself. This complex and its homology, the Lagrangian quantum homology, were later studied in detail by Paul Biran and Octav Cornea. We explain the construction of the pearl complex and, under a genericity assumption, prove a Künneth theorem for its homology. The proof consists of two parts: An algebraic part, which is a Künneth theorem for differential graded modules, and a geometric part, whose proof closely resembles the proof of the Künneth theorem for Morse homology. We present the algebraic part at the outset and the geometric part at the end after establishing the necessary prerequisites from local and global symplectic geometry.