An exact sequence for contact- and symplectic homology

By Frederick J. Bourgeois III, A. Oancea. 2007

115 citations recorded by the source corpus.

A symplectic manifold W with contact type boundary M=∂W induces a linearization of the contact homology of M with corresponding linearized contact homology HC(M). We establish a Gysin-type exact sequence in which the symplectic homology SH(W) of W maps to HC(M), which in turn maps to HC(M), by a map of degree -2, which then maps to SH(W). Furthermore, we give a description of the degree -2 map in terms of rational holomorphic curves with constrained asymptotic markers, in the symplectization of M.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/13db28a6941df3015ac6c4f663bb98a4b5d92b70

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Citations in the corpus

A FLOER HOMOLOGY FOR EXACT CONTACT EMBEDDINGS (2007, background): “And though it is formulated in finite dimensions, it directly carries over to the case of Floer homology; see also [Bourgeois 2003] and [Bourgeois and Oancea 2007] for discussions of Morse–Bott homology in infinite dimensional situations.”
Effect of Legendrian surgery (2009, methodology): “Remark 3.8 F. Bourgeois and A. Oancea proved in [9] that SH+(X) is isomorphic to the reduced symplectic homology of A. Floer and H. Hofer [24], see also [44], and established the exact triangle (3–6).”
Symplectic Cohomology and q-Intersection Numbers (2010, background): “Work in progress by Bourgeois-Oancea, following on from [8], will show that for char(K) = 0, (6.”
Symplectic cohomology and Viterbo's theorem (2013, methodology): “There is a completely different definition of symplectic cohomology using methods from symplectic field theory due to Bourgeois, Ekholm, and Eliashberg [18], generalising the connection between contact homology and equivariant symplectic cohomology due to Bourgeois and Oancea [20].”
Symplectic homology, autonomous Hamiltonians, and Morse-Bott moduli spaces (2007, background): “Moreover, the moduli spaces of Floer trajectories for autonomous Hamiltonians are related to the moduli spaces defining S 1 -equivariant symplectic homology [4, 3 ].”
$S^1$-equivariant symplectic homology and linearized contact homology (2012, methodology): “The goal of the present paper is to explain the relationship between the following symplectic invariants that are associated to (W,ω): symplectic homology SH∗(W ) and its versions SH ± ∗ (W ) [60], linearized contact homology CH lin ∗ (M) [26, 9, 6], S (1)-equivariant symplectic homology SH 1 ∗ (W )”
Symplectic homology and the Eilenberg-Steenrod axioms (2015, methodology): “If W is a Liouville domain we have SH ∗ (W ) = 0, SH ≤0 ∗ (W ) = SH =0 ∗ (W ), SH ≥0 ∗ (W ) = SH∗(W ), and the group SH ∗ (W ) coincides by definition with the group SH + ∗ (W ) of [15].”
Topological quantum field theory structure on symplectic cohomology (2010, unknown): “The untwisted construction is well-known [36, 7].”
Fukaya A∞-structures Associated to Lefschetz Fibrations. II (2014, background): “Most of the technical details are familiar and will be omitted, but our discussion includes two topics which are more commonly encountered in the special version of Floer cohomology known as symplectic cohomology, namely: the contribution of Reeb orbits [53, 10]; and the BV (Batalin-Vilkovisky or lo”
Floer theory for negative line bundles via Gromov–Witten invariants (2011, methodology): “the zero section. There are two ways around this. One can introduce an auxiliary Morse function fon C, and then one deﬁnes CF∗(hk,f) by standard Morse-Bott techniques (see for example Bourgeois-Oancea[4]). The generators will be the critical points of fin C, and the diﬀerential will count rigid traj”
Altering symplectic manifolds by homologous recombination (2010, methodology): “As a representative sample, we consider the definition of reduced symplectic cohomology SH +(M) in an SFT framework, following [6, Sections 2 and 3] (see also [7]).”