Non-loose Legendrian spheres with trivial contact homology DGA (Q2826647)

If \((Y^{2n+1},\xi)\) is a contact manifold, then a closed, connected, embedded submanifold \(L^n\to Y\) so that \(TL\subset\xi\) is a Legendrian knot. Knots with topology \(S^n\), and knots embedded in \((\mathbb R^{2n+1},\xi_{\mathrm{std}})\) are Legendrian knots, where \(\xi=\ker(dz-\sum y_idx_i)\). If \(f:L^n\hookrightarrow (Y^{2n+1},\xi)\) is a smooth embedding, \(F_s=df\) is a homotopy of bundle monomorphisms, covering \(f\) for all \(s\), so that \(F_0=df\) and \(F_1(TL)\) is a Lagrangian subspace of \(\xi\), then the pair \((f,F_s)\) is called a formal Legendrian knot. A Legendrian knot can be viewed as a formal Legendrian knot by letting \(F_s=df\) for all \(s\). Two Legendrian knots are said to be formally isotopic if there exists a smooth isotopy \(f_t:L\to Y\) between them, and \(df_t\) is homotopic through paths of monomorphisms, fixed at the endpoints, to a path of Lagrangian monomorphisms. If \(B\subset\mathbb R^3_{\mathrm{std}}\) is an open ball containing a stabilization of action \(a\), \(V_\rho=\{|p|,|q|\leq\rho\}\subset T^*\mathbb R^{n-1}\), and \(\Lambda\) is the Cartesian product of the stabilization and the zero section, which is Legendrian in the open convex set \(B\times V_\rho\) in \(\mathbb R^{2n+1}_{\mathrm{std}}\), then the pair \((B\times V_\rho,\Lambda)\) is called a Legendrian twist. A Legendrian twist satisfying \(\frac{a}{\rho^2}<2\) is called a loose chart. If \(L\) is a Legendrian knot in a contact manifold \((Y,\xi)\) and if there is a Darboux chart \(U\subset Y\) so that \((U,U\cap L)\) is a loose chart, then \(L\) is called loose. If \(L\) is an \(n\)-manifold, \(Y\) is a manifold of larger dimension, and \(A\subset \mathrm{Gr}_n(Y)\), where \(\mathrm{Gr}_n(Y)\) denotes the bundle of \(n\)-planes in \(TY\) with fiber \(\mathrm{Gr}_{\dim(Y),n}\), then an \(A\)-directed embedding is an embedding \(L\to Y\) so that \(TL\subset A\), and a formal \(A\)-directed embedding is a smooth embedding \(f:L\to Y\), together with a path of bundle monomorphisms \(F_s:TL\to TY\) covering \(f\), so that \(F_0=df\) and \(\mathrm{Im}(F_1)\subset A\). To say an \(h\)-principle holds for \(A\)-directed embeddings is to say the inclusion of \(A\)-directed embeddings into formal \(A\)-directed embeddings is a weak homotopy equivalence. In [``Loose Legendrian embeddings in high dimensional contact manifolds'', Preprint, \url{arXiv:1201.2245}], \textit{E.~Murphy} gave an \(h\)-principle type result for a class of Legendrian embeddings in contact manifolds of dimension at least \(5\), and showed that loose Legendrian \(n\)-submanifolds are flexible in the \(h\)-principle sense, that is, any two loose Legendrian submanifolds that are formally Legendrian isotopic are also actually Legendrian isotopic. Legendrian contact homology is a Floer theoretic invariant that associates a differential graded algebra \({\mathcal A}_\Lambda\) to a Legendrian submanifold \(\Lambda\subset Y\).NEWLINENEWLINEIn this paper, the author considers differential graded algebras for Legendrian submanifolds of \(\mathbb R^{2n+1}_{\mathrm{std}}\). It is known that the differential graded algebra of a loose Legendrian submanifold is trivial. The author shows that the converse is not true. It is shown that there is a Legendrian \(n\)-sphere \(\Lambda\subset \mathbb R^{2n+1}_{\mathrm{std}}\) such that the Legendrian differential graded algebra of \(\Lambda\) is trivial but \(\Lambda\) is not loose.