A Weinstein \(2n\)-manifold \(X\) can be built by attaching Weinstein \(k\)-handles with \(0\leq k\leq n\). The wrapped Fukaya category of \(X\) is known to be generated by the cocore disks of the critical Weinstein handles (i.e., the \(n\)-handles) [B. Chantraine et al., “Geometric generation of the wrapped Fukaya category of Weinstein manifolds and sectors”, Preprint, arXiv:1712.09126]. The endomorphism algebra of the wrapped Fukaya category is \(A_{\infty}\)-quasi-isomorphic to the Chekanov-Eliashberg dg-algebra \(CE^*\) of the attaching (Legendrian) spheres. For \(\Lambda\) an attaching Legendrian sphere in the contact boundary of a Weinstein domain, \(CE^*(\Lambda)\) is generated by the Reeb chords of \(\Lambda\), supplied with the Conley-Zehnder indices and the differential counts the appropriate holomorphic disks in the symplectisation of the boundary of the domain.
A cover of a Weinstein manifold by Weinstein sections is said to be good if the neighbourhood of every intersection of \(m+1\) Weinstein sections is a smaller Weinstein manifold of codimension \(2m\) times the cotangent bundle of \(\mathbb{R}^m\). The main result of the present paper is to use these good sectorial covers to “decompose the Chekanov-Eliashberg dga into smaller pieces to obtain a Seifert-van Kampen-type result”. See [S. Ganatra et al., “Sectorial descent for wrapped Fukaya categories”, Preprint, arXiv:1809.03427] for a similar result in wrapped Fukaya categories.
To achieve this, the author constructs, starting from any good sectorial cover of \(X\) – which always exists – a simplicial decomposition of \(X\) so that the Weinstein manifolds and Weinstein hypersurfaces in question are in bijection with the faces and face inclusions of a fixed simplicial complex \(C\). “Roughly speaking, using each Weinstein manifold” he constructs “a simplicial handle, such that when glued together according to \(C\) gives \(X\) up to Weinstein isomorphism”. Then the main theorem in the article assures an isomorphism of dg-algebras \[ CE^* (\Sigma; X_0 ) \cong \mathrm{colim} A_{\sigma_k},\] where \(\Sigma\) is the union of the Legendrian attaching spheres of \(X\) in its subcritical Weinstein manifold \(X_0\), and \(A_{\sigma_k}\) is a dg-algebra associated to each \(k\)-face \(\sigma_k\) of \(C\) so that an inclusion of faces in \(C\) induces a reverse inclusion of corresponding dg-algebras.
As an application, given a plumbing of copies of cotangent bundles of spheres of dimension at least three according to any plumbing quiver, the author produces a simplicial decomposition. Via this he computes the Chekanov-Eliashberg dg-algebra of the Legendrian attaching spheres of the given plumbing. Moreover he shows that this algebra is quasi-isomorphic to the Ginzburg dg-algebra of the plumbing quiver [V. Ginzburg, “Calabi-Yau algebras”, Preprint, arXiv:math/0612139], extending previous results of Y. Lekili and K. Ueda [Algebr. Geom. 8, No. 5, 562–586 (2021; Zbl 1476.53104)] and T. Etgü and Y. Lekili [Quantum Topol. 10, No. 4, 777–813 (2019; Zbl 1442.57013)].