Liouville hypersurfaces and connect sum cobordisms
In symplectic/contact geometry, the Moser/Gray trick to prove existence of the unique local model (up to symplectomorphism), constructions of isotopies, the flow of a natural vector field provide powerful tools to find nicer local shapes (or the model structure), to decompose a symplectic manifold into building blocks of the model structure, to glue pieces and to assemble topological/homological/categorical data. In [\textit{M. Abouzaid}, Ann. Math. (2) 175, No. 1, 71--185 (2012; Zbl 1244.53089)], the author defined a gluing operation for contact manifolds along nice neighborhoods of Liouville hypersurfaces, called the Liouville connected sum, which generalizes the Weinstein handle attachment for Weinstein (symplectic) manifolds. To do it, he introduced the concept of Liouville hypersurfaces (more generally, Liouville submanifolds) in contact manifolds which generalize ribbons of Legendrian graphs and pages of open book decomposition of contact manifolds. A fundamental construction is that any Liouville hypersurface \(\Sigma\) of a contact manifold can have a standard tubular neighborhood \(\mathcal{N}(\Sigma)\) with smooth, convex boundary, which can be considered as a flattening of \(\Sigma\) by the Reeb vector fields. A local model of Liouville connected sum is convex gluing of unit-cotangent bundles of two smooth compact manifolds with non-empty boundaries with orientation-reversing diffeomorphism on the boundary of the base manifolds (Example 2.14). A Liouville domain is a smooth compact symplectic manifold with boundary with exact symplectic form with Liouville vector field trasversely outward along the boundary. A Liouville hypersurface of a contact manifold \((M^{2n+1}, \xi\subset \text{Ker}(\alpha))\) is the image of an embedding \(i:\Sigma \to M\) of a Liouville domain \((\Sigma^{2n}, d\beta)\) of codimension \(1\) with \(i^{*}\alpha=\beta\). Since a Liouville hypersurface is transverse to the Reeb vector field \(R\) for \(\alpha\) (\(\alpha(R)=1\) and \(d\alpha(R,\cdot)=0\)), a map \(\Phi:[-\epsilon,\epsilon]\times \Sigma\to M\), defined by a time \((z \in [-\epsilon,\epsilon])\) flow of \(R\) starting from \(x\in \Sigma\), gives a small tubular neighborhood \(N(\Sigma)\) of \(\Sigma\) inside \(M\) satisfying \(\alpha|_{N(\Sigma)}=dz+\beta\). After edge-rounding on the neighborhood \(N(\Sigma)\), one get a standard neighborhood \(\mathcal{N}(\Sigma)\) of the Liouville hypersurface (Figure 3, Definition 3.6). The author also proves a neighborhood theorem for Liouville submanifolds of high codimension. The second half of the paper consists of applications of the gluing construction, for example, the construction of a Weinstein handle attachment from a standard neighborhood of a Liouville hypersurface, the symplectic handle attachment to the positive boundary of a weak symplectic cobordism, fillabilities preserved under contact connected sum, the monoid structure on non-vanishing contact homology, extension of contact \((1/k)\)-surgeries to arbitrary dimension, and certain generalized Dehn twists that create exotic structures. The main idea of [\textit{R. Avdek}, J. Symplectic Geom. 19, No. 4, 865--957 (2021; Zbl 07455583)], that was released in the earlier arXiv preprint version, was adapted to glue of Weinstein pairs in [\textit{Y. Eliashberg}, Proc. Symp. Pure Math. 99, 59--82 (2018; Zbl 1448.53083), Section 3.1], where the skeleta of Weinstein pairs are glued along the skeleta of glued hypersurfaces. It would also be interesting to consider the concept of sutured Liouville manifolds and Liouville sectors in [\textit{S. Ganatra} et al., Publ. Math., Inst. Hautes Étud. Sci. 131, 73--200 (2020; Zbl 07209675); ``Sectorial descent for wrapped Fukaya categories'', Preprint, \url{arXiv:1809.03427}; ``Microlocal Morse theory of wrapped Fukaya categories'', Preprint, \url{arXiv:1809.08807}].
scientific article; zbMATH DE number 7455583