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