Formality of Floer complex of the ideal boundary of hyperbolic knot complement. (arXiv:1901.02258v1 [math.SG])

This is a sequel to the authors' article [BKO]. We consider a hyperbolic knot $K$ in a closed 3-manifold $M$ and the cotangent bundle of its complement $M \setminus K$. We equip a hyperbolic metric $h$ with $M \setminus K$ and the induced kinetic energy Hamiltonian $H_h = \frac{1}{2} |p|_h^2$ and Sasakian almost complex structure $J_h$ with the cotangent bundle $T^*(M \setminus K)$. We consider the conormal $\nu^*T$ of a horo-torus $T$, i.e., the cusp cross-section given by a level set of the Busemann function in the cusp end and maps $u: (\Sigma, \partial \Sigma) \to (T^*(M \setminus K), \nu^*T)$ converging to a \emph{non-constant} Hamiltonian chord of $H_h$ at each puncture of $\Sigma$, a boundary-punctured open Riemann surface of genus zero with boundary. We prove that all non-constant Hamiltonian chords are transversal and of Morse index 0 relative to the horo-torus $T$. As a consequence, we prove that $\widetilde{\mathfrak m}^k = 0$ unless $k \neq 2$ and an $A_\infty$-algebra asso 查看全文>>