Symplectic homology and the Eilenberg-Steenrod axioms
type: Item
instance of: scholarly article
title: Symplectic homology and the Eilenberg-Steenrod axioms
zbMATH Open document ID: 1392.53093
author: Alexandru Oancea
author: Peter Albers
author: K. Cieliebak
published in: Algebraic \& Geometric Topology
publication date: 2018-05-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/1511.00485
review text: The authors say that there are too many Floer (co)homologies for noncompact objects, e.g., wrapped Floer homology, symplectic homology, Rabinowitz-Floer homology and linearized contact homology. In this paper, based on the notion Liouville cobordism with filling, six types of symplectic homologies are introduced in a uniform way (\S2). They are shown to satisfy the Eilenberg-Steenrod axioms except the dimension axiom (Theorem 1.4. proved in \S3--\S7), and the Rabinowitz-Floer homology and the wrapped Floer homology are derived from them (\S8). A Liouville domain is an exact symplectic manifold \((W^{2n},\omega,\lambda)\), \(\omega=d\lambda\) such that \(\alpha=\lambda|_{\partial W}\) is a positive contact form. A Liouville cobordism is defined similarly, but so that \(\alpha\wedge(d\alpha)^{n-1}\) defines the opposite of the boundary orientation on some of components of \(\partial W\), which is the negative boundary \(\partial^-W\) of \(W\). The positive boundary \(\partial^+W\) of \(W\) is defined by \(\partial W\setminus\partial^-W\). A filling \(F\) of the Liouville cobordism is a Liouville domain whose positive boundary is isomorphic to the negative boundary of \(W\) so that the concatenation \(F\circ W\) is a Liouville domain with positive boundary \(\partial^+W\). On a Liouville cobordism with filling \(F\) the following (smooth time-dependent 1-periodic approximations of a) Hamiltonian \(H_{\mu,\tau}\), such that \[ \begin{aligned} H_{\mu.\tau}=&\tau r-\tau \;\mathrm{on} \;[1,\infty)\times \partial^+W, \quad H_{\mu,\tau}=0 \;\mathrm{on} \;W, \\ H_{\mu,\tau}=&-\mu r+\mu \mathrm{on} \;(\delta,1]\times\partial^-W\subset F,\end{aligned} \] and \(H_{\mu,\tau}\) is constant on the remaining part of \(F\) is constructed. The Hamiltonian Floer homology defined by \(H_{\mu,\tau}\), truncated in the finite action window \((a,b)\) is denoted by \(\mathrm{HF}_\ast^{(a,b)}(H_{\mu,\tau})\). Taking limits with respect to \((a,b)\), six variants of symplectic homologies of a…
reviewed by: Akira Asada
Mathematics Subject Classification ID: 53D40
Mathematics Subject Classification ID: 55N40
Mathematics Subject Classification ID: 57R17
Mathematics Subject Classification ID: 57R90
zbMATH DE Number: 6867653
zbMATH Keywords: Floer homology
zbMATH Keywords: symplectic homology
zbMATH Keywords: Rabinowitz-Floer homology
zbMATH Keywords: Eilenberg-Steenrod axioms for a homology theory
zbMATH Keywords: Liouville cobordism
zbMATH Keywords: filling of the Liouville cobordism
MaRDI profile type: Publication
OpenAlex ID: W3098527749
arXiv ID: 1511.00485
cites work: Symplectic cohomology and Viterbo's theorem
cites work: An open string analogue of Viterbo functoriality
cites work: Symplectic Tate homology
cites work: LEAF-WISE INTERSECTIONS AND RABINOWITZ FLOER HOMOLOGY
cites work: Orderability and the Weinstein conjecture
cites work: Factorization homology of topological manifolds
cites work: Lagrangian cobordism. I
cites work: Existence and classification of overtwisted contact structures in all dimensions
cites work: Symplectic homology product via Legendrian surgery
cites work: Effect of Legendrian surgery
cites work: Compactness results in symplectic field theory
cites work: An exact sequence for contact- and symplectic homology
cites work: Symplectic homology, autonomous Hamiltonians, and Morse-Bott moduli spaces
cites work: The Gysin exact sequence for \(S^1\)-equivariant symplectic homology
cites work: S1-Equivariant Symplectic Homology and Linearized Contact Homology
cites work: A remark on equivariant Poincaré duality
cites work: Handle attaching in symplectic homology and the chord conjecture
cites work: Symplectic homology. II: A general construction
cites work: A Floer homology for exact contact embeddings
cites work: MORSE HOMOLOGY ON NONCOMPACT MANIFOLDS
cites work: Rabinowitz Floer homology and symplectic homology
cites work: Compactness for punctured holomorphic curves
cites work: Lifting pseudo-holomorphic polygons to the symplectisation of \(P \times \mathbb{R}\) and applications
cites work: On homological rigidity and flexibility of exact Lagrangian endocobordisms
cites work: On the groups \(H(\Pi,n)\). I
cites work: Rational SFT, Linearized Legendrian Contact Homology, and Lagrangian Floer Cohomology
cites work: A duality exact sequence for Legendrian contact homology
cites work: The contact homology of Legendrian submanifolds in \(\mathbb R^{2n+1}\)
cites work: Legendrian knots and exact Lagrangian cobordisms
cites work: Symplectic and contact differential graded algebras
cites work: Symplectic homology. I: Open sets in \(\mathbb{C}^ n\)
cites work: The Symplectic Geometry of Cotangent Bundles from a Categorical Viewpoint
cites work: A note on Lagrangian cobordisms between Legendrian submanifolds of \(\mathbb R^{2n+1}\)
cites work: Cyclic homology, derivations, and the free loopspace
cites work: Loop products and closed geodesics
cites work: The positive equivariant symplectic homology as an invariant for some contact manifolds
cites work: On the minimal number of periodic orbits on some hypersurfaces in \(\mathbb{R}^{2n}\)
cites work: Multiplicative structure in equivariant cohomology
cites work: Some algebraic properties of cyclic homology groups
cites work: Cyclic homology and equivariant homology
cites work: The Fixed Point Theorem in Equivariant Cohomology
cites work: Symplectic homology of displaceable Liouville domains and leafwise intersection points
cites work: Brieskorn manifolds in contact topology
cites work: Weak and strong fillability of higher dimensional contact manifolds
cites work: The additivity of traces in triangulated categories
cites work: Lagrangian Rabinowitz Floer homology and twisted cotangent bundles
cites work: Autonomous Hamiltonian flows, Hofer's geometry and persistence modules
cites work: Novikov-symplectic cohomology and exact Lagrangian embeddings
cites work: Topological quantum field theory structure on symplectic cohomology
cites work: Morse theory for periodic solutions of hamiltonian systems and the maslov index
cites work: A biased view of symplectic cohomology
cites work: Functors and computations in Floer homology with applications. I
cites work: Cylindrical contact homology of subcritical Stein-fillable contact manifolds
Wikidata QID: Q129880755

Symplectic homology and the Eilenberg-Steenrod axioms (DOI)
DOI: 10.2140/AGT.2018.18.1953