Morse-Bott Split Symplectic Homology

Symplectic homology of complements of smooth divisors

“…This enables us to obtain a bijection between moduli spaces of Floer trajectories and of pseudoholomorphic curves (rather than just a continuation map relating symplectic homology and a non-equivariant version of contact homology), which is then used to compute the symplectic homology differential explicitly. As studied in our paper [DL18], this also allows us to achieve transversality by geometric arguments involving monotonicity and automatic transversality. For more on the relation between this work and [BO09a], see Remark 9.8.…”

“…Thanks to the monotonicity assumptions we impose on X and on Σ, the cascades that have the correct Fredholm index to appear in the differential turn out to be relatively simple. This is analyzed in detail in [DL18], see in particular [DL18, Section 6.1]. We now provide a brief summary of the cascades that contribute to the differential in split symplectic homology.…”

“…We say that a Floer cylinder is simple if ṽ : R ˆS1 zΓ Ñ R ˆY projects to a somewhere injective curve in Σ, or if it projects to a constant. Using the notation from [DL18], we write M H,l,RˆY ;k´,k`p A; J Y q for the space of simple Floer cylinders ṽ : R ˆS1 zΓ Ñ R ˆY , where Γ is a set of l augmentation punctures, lim sÑ˘8 ṽps, .q are Hamiltonian orbits of multiplicity k ȃnd π Σ ˝ṽ represents the homology class A P H 2 pΣ; Zq. The spaces of Floer cylinders with cascades in Case 1 are unions of fibre products (4.5)…”

“…This is also related to Remark 9.8. This paper is complemented by [DL18] and is organized as follows. Part 1 provides more details about our geometric setup and the almost complex structures that we use.…”

“…We use a Morse-Bott approach, so the differentials count Floer cylinders with cascades. The full discussion of the stretched case (which we refer to as split) is given in [DL18], including the relevant transversality results and an explanation of how our monotonicity assumptions imply that the Floer cylinders with cascades that can contribute to the split differential can only be of four simple types. Part 3 is the most technical of this paper, where we explain why counting punctured Floer cylinders in R ˆY is equivalent to counting punctured pseudohomolorphic cylinders in R ˆY , and how the latter can be expressed in terms of counts of pseudoholomorphic spheres in Σ (which are in turn related to Gromov-Witten invariants).…”