Chekanov-Eliashberg dg-algebras for singular Legendrians (Q2687307)

The Chekanov-Eliashberg dg-algebra(or Legendrian contact homology) is an invariant of a pair \((\Lambda; Y=\partial W)\) of a Legendrian submanifold \(\Lambda\) of a contact manifold \(Y\). Here, we consider the contact manifold \(Y=\partial W\) that is the (positive) contact boundary of an Weinstein domain(cobordism) \(W\) (in the case that the negative contact boundary has no Reeb orbits). The Reeb chords of the contact structure, with boundaries on the Legendrian submanifold, generate the dg-algebra and counting pseudo-holomorphic curves, bounding Reeb chords, give the differential maps. The authors of the paper under review, extend the definition of Chekanov-Eliashberg dg-algebra for singular Legendrian submanifolds, which come from the Weinstein-skeleton of the Weinstein submanifold, Legendrian-embedded into the contact boundary of a Weinstein domain. The definition is as follows: Let \(W\) be a Weinstein domain of dimension \(2n\) and \((V,h)\subset \partial W\) be a Legendrian embedding of the Weinstein \((2n-2)\)-subdomain \(V\) with the Weinstein handle decomposition \(h\). Denote the subcritical part of \(V\) by \(V_{0}\) and the core-disks of the critical handles by \(l_{j}, j=1,\cdots, m\) and the attaching spheres for the corresponding handle by \(\partial l_{j}\). Then a small neighborhood of \(V\) in \(\partial W\) can be identified with \(V\times (-\epsilon,\epsilon)\), where the second factor is along the Reeb flow. Define \(V\)-handle to be an \(\epsilon\)-neighborhood \(V\times T^{*}_{\leq \epsilon}[-1,1]\) of \(V\times [-1.1]\) in \(V\times T^{*}[-1,1]\), where \(T^{*}[-1,1]\) is the cotangent bundle of the interval \([-1,1]\subset \mathbb{R}\). Attaching \(V\)-handle to \(\partial W\sqcup (\mathbb{R}\times V)\) along \(V\times (-\epsilon,\epsilon)\), we get an Weinstein cobordism \(W_{V}\) with negative end \(V\times \mathbb{R}\). Attaching \(V_{0}\)-handlle to \(W\sqcup (\mathbb{R}\times V)\), we get a subset \(W_{V}^{0}\) of the Weinstein cobordism \(W_{V}\). Then consider the attaching link \(\Sigma(h)\) in the sub-Weinstein cobordism \(W_{V}^{0}\) as a Legendrian \((n-1)\)-sphere of \(\partial_{+}W_{V}^{0}\), where \(\partial_{+}W_{V}^{0}\) denotes the positivity boundary of \(W_{V}^{0}\). When we attach Weinstein \(n\)-handles to \(W_{V}^{0}\) along \(\Sigma(h_{j})\) for each top-dimensional handle \(h_{j}\) of \(V\), we get \(W_{V}\). One of important observation is Lemma 3.1, which says that the negative end of the cobordism \(W_{V}\) have no closed Reeb orbits and the Weinstein handle decomposition of \(W_{V}\) is the union of handles of \(W\) and handles of \(V\) with dimension shifted up by \(1\). Therefore, we can define the ordinary Chekanov-Eliashberg dg-algebra for \(\Sigma(h)\subset \partial W_{V}^{0}\). The Checkanov-Eliashberg dg-algebra of \((V,h)\subset \partial W\) is defined to be the usual one of \(\Sigma(h)\subset \partial W_{V}^{0}\), \(CE^{*}((V,h);\partial W):=CE^{*}(\Sigma(h);\partial W_{V}^{0})\). We remark that the original notation in the paper uses \(W\) instead of \(\partial W\), for example, \(CH^{*}((V,h);W)\). For smooth Legendrians, where the Weinstein subdomain \(V\) is a smooth neighborhood of the zero section \(T^{*}\Lambda\), the new definition recovers the usual Chekanov-Eliashberg dg-algebra \(CE^{*}(\Lambda, C_{-*}(\Omega \Lambda);\partial W)\) with the loop space coefficients \(C_{-*}(\Omega \Lambda)\). Newly, the extended definition provides dg-algebras for non-closed Legendrian submanifolds \(\Lambda\subset \partial W\) with Legendrian boundary in \(\partial V\). It is considered as a sister-concept of the partially wrapped Fukaya categories, [\textit{Z. Sylvan}, J. Topol. 12, No. 2, 372--441 (2019; Zbl 1430.53097)]. Important properties of the extended Checkanov-Eliashberg dg-algebra are as follows: First, there is a natural push-out diagram coming from the joining \(W\#_{V}W'\) of two Weinstein manifolds \(W, W'\) via attaching \(V\)-handle along the Legendrian embeding of \(V\) in their boundary, respectively, Theorem 1.3. It is also proven that there is an isomorphism between partially wrapped Floer cohomology in \(W\) stopped at \(V\) and Chekanov-Eliashberg dg-algebras with coefficients in chains on the based loop space. In sum, their cut-and-paste formulas for holomorphic curve invariants parallels results on sectorial descent for wrapped Fukaya categories in [\textit{S. Ganatra} et al., ``Sectorial descent for wrapped Fukaya categories'', Preprint, \url{arXiv:1809.03427}]. The interested readers may find a sequel paper, [``Simplicial descent for Chekanov-Eliashberg DG-algebras'', Preprint, \url{arXiv:2112.01915}], by the first author.