| 1. |
- Diogo, Luis, et al.
(author)
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Morse-Bott split symplectic homology
- 2019
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In: Journal of Fixed Point Theory and Applications. - : SPRINGER BASEL AG. - 1661-7738 .- 1661-7746. ; 21:3
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Journal article (peer-reviewed)abstract
- We associate a chain complex to a Liouville domain ((W) over bar ,d lambda) whose boundary Y admits a Boothby-Wang contact form (i.e.is a prequantization space). The differential counts Floer cylinders with cascades in the completion W of (W) over bar, in the spirit of Morse-Bott homology (Bourgeois in A Morse-Bott approach to contact homology, Ph.D. Thesis. ProQuest LLC, Stanford University, Ann Arbor 2002; Frauenfelder in Int Math Res Notices 42:2179-2269, 2004; Bourgeois and Oancea in Duke Math J 146(1), 71-174, 2009). The homology of this complex is the symplectic homology of W (Diogo and Lisi in J Topol 12:966-1029, 2019). Let X be obtained from (W) over bar by collapsing the boundary Y along Reeb orbits, giving a codimension two symplectic submanifold Sigma. Under monotonicity assumptions on X and Sigma, we show that for generic data, the differential in our chain complex counts elements of moduli spaces of cascades that are transverse. Furthermore, by some index estimates, we show that very few combinatorial types of cascades can appear in the differential.
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| 2. |
- Diogo, Luis, et al.
(author)
-
Symplectic homology of complements of smooth divisors
- 2019
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In: Journal of Topology. - : Wiley. - 1753-8416 .- 1753-8424. ; 12:3, s. 967-1030
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Journal article (peer-reviewed)abstract
- If (X,omega) is a closed symplectic manifold, and sigma is a smooth symplectic submanifold Poincare dual to a positive multiple of omega, then X set minus sigma can be completed to a Liouville manifold (W,d lambda). Under monotonicity assumptions on X and on sigma, we construct a chain complex whose homology computes the symplectic homology of W. We show that the differential is given in terms of Morse contributions, Gromov-Witten invariants of X relative to sigma and Gromov-Witten invariants of sigma. We use a Morse-Bott model for symplectic homology. Our proof involves comparing Floer cylinders with punctures to pseudoholomorphic curves in the symplectization of the unit normal bundle to sigma.
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