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Naturality of the contact invariant in monopole Floer homology under strong symplectic cobordisms. (English) Zbl 1445.57011
Let \(Y\) be a closed, oriented, connected \(3\)-manifold. To a contact structure \(\xi\) there is associated a Spin\(^c\)-structure \({\mathfrak s}_\xi\) and a contact invariant \(c(\xi)\in \widehat{HM}^*(Y,{\mathfrak s}_\xi)\cong \widehat{HM}_*(-Y,{\mathfrak s}_\xi)\) in monopole Floer homology. For a symplectic cobordism \(W\) between contact manifolds \((Y,\xi)\) and \((Y^\prime,\xi^\prime)\) one has a homomorphism \(\widehat{HM}_*(W^\dagger,{\mathfrak s}_W)\colon \widehat{HM}_*(-Y,{\mathfrak s}_\xi)\to \widehat{HM}_*(-Y^\prime,{\mathfrak s}_{\xi^\prime})\).
A strong symplectic cobordism is a symplectic cobordism for which the symplectic structure in collar neighborhoods of the convex and concave boundaries is given by symplectizations of the corresponding contact structures. The paper under review proves that under this assumption \(\widehat{HM}_*(W^\dagger,{\mathfrak s}_W)\) sends \(c(\xi)\) to \(c(\xi^\prime)\). This had been conjectured for about ten years.
As corollaries one gets new proofs of \(c(\xi)=0\) for overtwisted contact structures and \(c(\xi)\not=0\) for strongly fillable contact structures. It is also proved that a strong filling of a contact manifold which is an L-space must be negative definite.
57K33 Contact structures in 3 dimensions
57R17 Symplectic and contact topology in high or arbitrary dimension
57R58 Floer homology
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