Homotopical Dynamics in Symplectic Topology (original) (raw)
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2021
We construct a Floer type boundary operator for generalised Morse-Smale dynamical systems on compact smooth manifolds by counting the number of suitable flow lines between closed (both homoclinic and periodic) orbits and isolated critical points. The same principle works for the discrete situation of general combinatorial vector fields, defined by Forman, on CW complexes. We can thus recover the Z2 homology of both smooth and discrete structures directly from the flow lines (V-paths) of our vector field.
African Diaspora Journal of Mathematics , 2017
This paper continues to carry out a foundational study of Banyaga's topologies of a closed symplectic manifold (M,ω) [4]. Our intention in writing this paper is to work out several “symplectic analogues” of some results found in the study of Hamiltonian dynamics. By symplectic analogue, we mean if the first de Rham's group (with real coefficients) of the manifold is trivial, then the results of this paper reduce to some results found in the study of Hamiltonian dynamics. Especially, without appealing to the positivity of the symplectic displacement energy, we point out an impact of the L∞−version of Hofer-like length in the investigation of the symplectic nature of the C0−limit of a sequence of symplectic maps. This yields a symplectic analogue of a result that was proved by Hofer-Zehnder [10] (for compactly supported Hamiltonian diffeomorphisms on R2n); then reformulated by Oh-Müller [14] for Hamiltonian diffeomorphisms in general. Furthermore, we show that Polterovich's regularization process for Hamiltonian paths extends over the whole group of symplectic isotopies, and then use it to prove the equality between the two versions of Hofer-like norms. This yields the symplectic analogue of the uniqueness result of Hofer's geometry proved by Polterovich [13]. Our results also include the symplectic analogues of some approximation lemmas found by Oh-Müller [14]. As a consequence of a result of this paper, we prove by other method a result found by McDuff-Salamon.
$C^0-$Symplectic Geometry Under Displacement
JDSGT, 2019
This paper studies the group of all symplectic homeomorphisms of a closed symplectic manifold (M,ω). After given a direct proof of the positivity result of the symplectic displacement energy, we show that the uniqueness theorem of generators of strong symplectic isotopies extends to any closed symplectic manifold: This generalizes a uniqueness result due to BuhovskySeyfaddini, and extends a uniqueness result found by Banyaga-Tchuiaga to any closed symplectic manifolds. An explicit formula for the mass flow of any strong symplectic isotopy with respect to its generator is given, and we prove that the mass flows of two strong symplectic isotopies with the same endpoints are equal whenever both paths have homotopic orbits of a certain point on M. It follows that on a Lefschetz type closed symplectic manifold, any strong symplectic isotopy with a trivial mass flow is homotopic relatively to fix endpoint to a continuous Hamiltonian flow, and some conjectures are also formulated.
Asian Journal of Mathematics, 2012
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The Beginnings of Symplectic Topology in Bochum in the Early Eighties
Jahresbericht der Deutschen Mathematiker-Vereinigung
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Symplectic homology for symplectomorphism and symplectic isotopy problem
2014
We introduce a combination of fixed point Floer homology and symplectic homology for Liouville domains. As an application, we detect non-trivial elements in the symplectic mapping class group of a Liouville domain. 2. Floer homology for symplectomorphism of a Liouville domain 2.1. Floer data and Admissible data. Definition 2.1. A symplectomorphism φ : W → W is called exact if the 1-form φ * λ − λ is exact and compactly supported. Under these assumptions there exists a unique compactly supported function F φ : W → R such that φ * λ − λ = dF φ. The exact symplectomorphisms form a group, denoted by Symp(W , λ/d). The functions associated to composition and inverse are given by
On certain geometric and homotopy properties of closed symplectic manifolds
2000
The paper deals with relations between the Hard Lefschetz property, (non)vanishing of Massey products and the evenness of odd-degree Betti numbers of closed symplectic manifolds. It is known that closed symplectic manifolds can violate all these properties (in contrast with the case of Kaehler manifolds). However, the relations between such homotopy properties seem to be not analyzed. This analysis may shed a new light on topology of symplectic manifolds. In the paper, we summarize our knowledge in tables (different in the simply-connected and in symplectically aspherical cases). Also, we discuss the variation of symplectically harmonic Betti numbers on some 6-dimensional manifolds.