The Gysin exact sequence for $S^1$-equivariant symplectic homology

By Frederick J. Bourgeois III, A. Oancea. 2009

78 citations recorded by the source corpus.

We define $S^1$-equivariant symplectic homology for symplectically aspherical manifolds with contact boundary, using a Floer-type construction first proposed by Viterbo. We show that it is related to the usual symplectic homology by a Gysin exact sequence. As an important ingredient of the proof, we define a parametrized version of symplectic homology, corresponding to families of Hamiltonian functions indexed by a finite dimensional smooth parameter space.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/010c0fe64a0fb4d21dee62180f1f0bba527dd4a3

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In Geometry and Topology

1. A. Oancea
2. P. Seidel
3. Jean Gutt
4. Michael Hutchings
5. Frederick J. Bourgeois III
6. Kai Cieliebak
7. Basak Z. Gurel
8. V. Ginzburg

In Mathematical Physics

1. A. Oancea
2. Jean Gutt
3. P. Seidel
4. Frederick J. Bourgeois III
5. Michael Hutchings
6. Kai Cieliebak
7. V. Ginzburg
8. Basak Z. Gurel

Citations in the corpus

• Symplectic homology and the Eilenberg-Steenrod axioms (2015, background): “Moreover, it follows from the construction that the Gysin and fundamental exact triangles are functorial with respect to the tautological exact triangles and also with respect to the exact triangles of pairs, see also [17, 18] for a basic instance of this phenomenon.”
• Fukaya A∞-structures Associated to Lefschetz Fibrations. II (2014, background): “only 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 loop rotation) operator [42, 47, 11]. (3a) Denition. We will work in the following setting, which benets from strong exactne”
• Lusternik–Schnirelmann theory and closed Reeb orbits (2016, methodology): “eorem 1.2 we give an alternative definition of Dfor which the proof of the Lusternik–Schnirelmann inequality is more natural and show in Section 2.2.4 that this definition is equivalent to the one from [BO13b]. Another consequence of the proofs of Theorems 1.1 and 1.2 is that the operator Dvanishes in”
• Symplectic capacities from positive S1–equivariant symplectic homology (2017, methodology): “ domain, then there is a long exact sequence //SH pX;q /CHpX;q U /CHpX;q SH pX;q / (6.5) Proof. With the above denition of U, this follows as in [BO16, Prop. 2.9]. This was also shown earlier in [BO13] using a slightly dierent denition of positive S1- equivariant symplectic homology. 6.4 Reeb Orbits”
• An exact sequence for contact- and symplectic homology (2007, background): “Remark 20. The similarity between Proposition 11 and Proposition 9 is best explained via the S 1 -equivariant approach in [ 8 ].”
• $S^1$-equivariant symplectic homology and linearized contact homology (2012, methodology): “tisfies the Strong Equivariant Algebraic Weinstein Conjecture (EWC) if, for all k≥0, the element uk ⊗µ2n lies in the kernel of the map HS1 2n+2k(W,∂W) →SH S1 n+2k(W). We proved the following result in [11], and we give here an alternative proof. Proposition 4.2 ([11, Corollary 1.4]). SAWC =⇒EWC. Proo”