Unobstructed Immersed Lagrangian Correspondence and Filtered ∞ Functor
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 ass...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2025 |
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| Sprache: | Englisch |
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Інститут математики НАН України
2025
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/213183 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Unobstructed Immersed Lagrangian Correspondence and Filtered ∞ Functor. Kenji Fukaya. SIGMA 21 (2025), 031, 284 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862592155315339264 |
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| author | Fukaya, Kenji |
| author_facet | Fukaya, Kenji |
| citation_txt | Unobstructed Immersed Lagrangian Correspondence and Filtered ∞ Functor. Kenji Fukaya. SIGMA 21 (2025), 031, 284 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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.
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| first_indexed | 2026-03-21T12:00:00Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-213183 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T12:00:00Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
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| spelling | Fukaya, Kenji 2026-02-16T16:34:51Z 2025 Unobstructed Immersed Lagrangian Correspondence and Filtered ∞ Functor. Kenji Fukaya. SIGMA 21 (2025), 031, 284 pages 1815-0659 2020 Mathematics Subject Classification: 53D35; 53D40; 57R56; 53D12; 53D37; 57R17 arXiv:1706.02131 https://nasplib.isofts.kiev.ua/handle/123456789/213183 https://doi.org/10.3842/SIGMA.2025.031 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. The author would like to thank the Simons Center for Geometry and Physics, where most of the research written in this paper was performed. The author would like to thank J. Evans, Y. Lekili, and Y.-G. Oh, H. Ohta, K. Ono, for helpful discussions while he was working on the contents of this paper. He also would like to thank anonymous referees for careful and serious reading, which is a heavy and painstaking work, and for a huge number of important comments, which improve this paper significantly compared to its earlier version. Special thanks are due to K. Ono, who agreed to write an article [68] on sign and orientation, which we need in this paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Unobstructed Immersed Lagrangian Correspondence and Filtered ∞ Functor Article published earlier |
| spellingShingle | Unobstructed Immersed Lagrangian Correspondence and Filtered ∞ Functor Fukaya, Kenji |
| title | Unobstructed Immersed Lagrangian Correspondence and Filtered ∞ Functor |
| title_full | Unobstructed Immersed Lagrangian Correspondence and Filtered ∞ Functor |
| title_fullStr | Unobstructed Immersed Lagrangian Correspondence and Filtered ∞ Functor |
| title_full_unstemmed | Unobstructed Immersed Lagrangian Correspondence and Filtered ∞ Functor |
| title_short | Unobstructed Immersed Lagrangian Correspondence and Filtered ∞ Functor |
| title_sort | unobstructed immersed lagrangian correspondence and filtered ∞ functor |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/213183 |
| work_keys_str_mv | AT fukayakenji unobstructedimmersedlagrangiancorrespondenceandfilteredfunctor |