We consider the Fukaya-Floer $A_\infty$-structures arising from a basis of thimbles in a Lefschetz fibration. Particular attention is paid to the relation between these structures in the total space of the fibration, and their counterparts in the fibre.

In this paper, we prove that the “quantization commutes with reduction” phenomenon of Guillemin and Sternberg applies in the context of the metaplectic correction.

This paper has two purposes. First, as a continuation of "Perturbed Floer homology of some fibered three manifolds," we apply a similar method to compute the perturbed $HF^+$ for some special classes of fibered three-manifolds in the second highest $spin^c$-structures, including the mapping tori of Dehn twists along a single non-separating curve and along a transverse pair of curves. Second, we establish an adjunction inequality for the perturbed Heegaard Floer homology, which indicates a potential connection between the $U$-action on the homology group and the Thurston norm of a three-manifold. As an application, we find the $U$-action on the perturbed $HF^+$ of the above classes of fibered three-manifolds is trivial.

A Bott manifold is a closed smooth manifold obtained as the total space of an iterated $\mathbb{C}P^1$-bundle starting with a point, where each $\mathbb{C}P^1$-bundle is the projectivization of a Whitney sum of two complex line bundles. A $\mathbb{Q}$-trivial Bott manifold of dimension $2n$ is a Bott manifold whose cohomology ring is isomorphic to that of $(\mathbb{C}P^1)^n$ with $\mathbb{Q}$-coefficients. We find all diffeomorphism types of $\mathbb{Q}$-trivial Bott manifolds and show that they are distinguished by their cohomology rings with $\mathbb{Z}$-coefficients. As a consequence, the number of diffeomorphism classes of $\mathbb{Q}$-trivial Bott manifolds of dimension $2n$ is equal to the number of partitions of $n$. We even show that any cohomology ring isomorphism between two $\mathbb{Q}$-trivial Bott manifolds is induced by a diffeomorphism.

In 1970, Moser showed that the Hamiltonian flow of the Kepler problem in $\mathbb{R}^n$ for a fixed negative energy level is regularized via stereographic projection to the geodesic flow on the punctured cotangent bundle of the unit sphere in $\mathbb{R}^{n+1}$, in such a way that the time parameter in the Kepler problem and the arc length for the geodesic flow are related by the Kepler equation. Ligon and Schaaf gave an alternative regularization of the Kepler problem, treating the whole negative energy part of the phase space at once, such that the Kepler flow and the Delaunay flow on the punctured cotangent bundle of the sphere become related by a canonical transformation. The rather elaborate calculations of Ligon and Schaaf were simplified by Cushman and Duistermaat.

In this paper, we derive the Ligon–Schaaf regularization as an almost trivial adaptation of the Moser regularization. As a consequence, the hidden symmetry of the Kepler problem becomes naturally visible.

We prove a conjecture of Barraud and Cornea in the monotone setting, refining a result of Chekanov on the Hofer distance between two Hamiltonian isotopic Lagrangian submanifolds.