Unobstructed Immersed Lagrangian Correspondence and Filtered ∞ Functor

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2025
• Main Author: Fukaya, Kenji
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2025
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/213183
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Unobstructed Immersed Lagrangian Correspondence and Filtered ∞ Functor. Kenji Fukaya. SIGMA 21 (2025), 031, 284 pages

In this paper, we 'construct' a 2-functor from the unobstructed immersed Weinstein category to the category of all filtered ∞ categories. We consider arbitrary (compact) symplectic manifolds and their arbitrary (relatively spin) immersed Lagrangian submanifolds. The filtered ∞ category associated with (, ω) is defined by using Lagrangian Floer theory in such generality, see Akaho-Joyce (2010) and Fukaya-Oh-Ohta-Ono (2009). The morphism of an unobstructed immersed Weinstein category (from (₁, ω₁) to (₂, ω₂)) is, by definition, a pair of an immersed Lagrangian submanifold of the direct product and its bounding cochain (in the sense of Akaho-Joyce (2010) and Fukaya-Oh-Ohta-Ono (2009)). Such a morphism transforms an (immersed) Lagrangian submanifold of (₁, ω₁) to one of (₂, ω₂). The key new result proved in this paper shows that this geometric transformation preserves the unobstructedness of the Lagrangian Floer theory. Thus, this paper generalizes earlier results by Wehrheim-Woodward and Mau-Wehrheim-Woodward so that it works in complete generality in the compact case. The main idea of the proofs is based on Lekili-Lipyanskiy's Y diagram and a lemma from homological algebra, together with systematic use of the Yoneda functor. In other words, the proofs are based on a different idea from those that are studied by Bottmann, Mau, Wehrheim, and Woodward, where strip shrinking and figure 8 bubble play the central role.