Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0705.3450

The symplectic geometry of cotangent bundles from a categorical viewpoint

Kenji Fukaya·Paul Seidel·Ivan Smith
View original source ↗Read source document ↗Export metadata ↓

Abstract

We describe various approaches to understanding Fukaya categories of cotangent bundles. All of the approaches rely on introducing a suitable class of noncompact Lagrangian submanifolds. We review the work of Nadler-Zaslow (math/0604379, math/0612399) and the authors (math/0701783), before discussing a new approach using family Floer cohomology and the ``wrapped Fukaya category''. The latter, inspired by Viterbo's symplectic homology, emphasises the connection to loop spaces, hence seems particularly suitable when trying to extend the existing theory beyond the simply-connected case.

Explore related subjects

math.SGmath.AT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Kenji Fukaya, Paul Seidel, Ivan Smith. 2007-09-26. The symplectic geometry of cotangent bundles from a categorical viewpoint. https://arxiv.org/abs/0705.3450

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The local Floer cohomology of indicator functions

Let $K$ be a compact domain with contact-type boundary in a symplectic manifold $M$. We associate an unweighted local Floer cohomology group to each isolated Reeb component of $\partial K$ and show that, in sufficiently small Novikov windows, these groups together with $H^*(K)$ describe the symplectic cohomology of $M$ with support on $K$. The main analytic input is an energy-barrier construction which remains uniform under finite-slope approximations of the indicator Hamiltonian of $K$. We also study functoriality. For inclusions of contact-type domains we describe the local blocks of the restriction map without requiring the Liouville primitives on the two boundaries to agree. For a monotone isotopy carrying an isolated family of Reeb components, the corresponding block is the local Floer transport map multiplied by the Novikov weight determined by the source-minus-target action difference. These results provide the local Floer-theoretic input for companion work on relative symplectic cohomology near singularities of SYZ fibrations.

math.SG↗
arXiv2026

Classical cylindrical contact homology is not invariant over $\bZ$

We show that classical cylindrical contact homology over $\mathbb{Z}$, equipped with its decomposition by free homotopy classes, is not invariant under changes of hypertight contact form. Specifically, we construct two nondegenerate hypertight perturbations of the standard Seifert contact form on $Σ(2,4,5)$ whose cylindrical contact homologies in the regular-fiber class are respectively torsion-free and contain $\mathbb{Z}/5$-torsion.

math.SG↗
arXiv2026

Bi-contact plugs and dynamical hyperbolicity

We develop a theory of bi-contact plugs in dimension three to construct Anosov flows or, more generally, structurally stable nonsingular flows. For a transversely orientable hyperbolic plug with orientable filling Morse-Smale boundary laminations, we prove that every strongly transverse gluing map can be isotoped through strongly transverse maps to identify a strongly adapted contact form up to sign. The resulting flow is hyperbolic and is Anosov when the quotient is closed. This answers a question of Béguin-Bonatti-Yu under the stated orientability assumptions. The result also extends to gluing nonsingular partially hyperbolic flows. Anosov completion by reflection embeds each such hyperbolic plug in an Anosov flow on its oriented double. Further applications include a classification of structurally generic transversely oriented projectively Anosov flows without saddle periodic orbits, Morse-Smale examples with prescribed saddle count, a generalization of the construction of Bonatti-Bowden-Potrie to give embeddings of (attracting) hyperbolic plugs into (partially hyperbolic) projectively Anosov flows with the same topological entropy, and constructions on doubles and torus bundles with controlled invariant torus dynamics. Finally, every cooriented partially hyperbolic bi-contact plug admits a Liouville structure on its thickening, with Liouville trajectories projecting to positively reparametrized flow lines. Repelling plugs yield four-dimensional Liouville domains, often with chaotic skeletons.

math.SG↗