arXiv subjects
Publications and source records attributed to Andrés Pedroza.
We introduce a generalization of Weinstein's morphism, defined on π_{2k-1}(Ham(M,ω)) for 1 < k \leq n, where (M,ω) is a 2n-dimensional symplectic manifold. Using this morphism, we show that for n > 1 and 1 < k \leq n, the homotopy groups π_{2k-1}(Ham(CP^n,ω_{FS})) and π_{2k-1}(Ham(\tilde CP^n,\tildeω_ρ)) are nontrivial. Here, (\tilde CP^n,\tildeω_ρ) denotes the symplectic one-point blow-up of (CP^n,ω_FS) of weigh ρ.
We present a generalization of Weinstein's morphism defined on $π_{2k-1} ( \textup{Ham} (M,ω)) $. We use this morphism to show that for $n\geq 2$ the Lie group $SU(2)$ induces an element in $π_3(\textup{Ham} (\mathbb{C}P^n,ω_\textup{FS}))$ of infinite order.
We prove that the fundamental group of the group of Hamiltonian diffeomorphisms of the symplectic manifold that is obtain by blowing up a submanifold contains an element of infinite order. We prove this using Weinstein's morphism and by constructing explicitly such loop of Hamiltonian diffeomorphisms.
We show that for any positive integer $k$ there exists a closed symplectic $4$-manifold $(M,ω)$, such that $H^1_\mathrm{dR}(M;\mathbb{R})$ is a $k$-dimensional real vector space and its Flux group is equal to $H^1_\mathrm{dR}(M_1;\mathbb{Z})$.
We show that for any positive integer $k$ there exists a closed symplectic $4$-manifold, such that the rank of the fundamental group of the group of Hamiltonian diffeomorphisms is at least $k.$
We show that two Hamiltonian isotopic Lagrangians in (CP^2,ω_\textup{FS}) induce two Lagrangian submanifolds in the one-point blow-up (\widetilde{CP}^2,\widetildeω_ρ) that are not Hamiltonian isotopic. Furthermore, we show that for any integer k>1 there are k Hamiltonian isotopic Lagrangians in (CP^2,ω_\textup{FS}) that induce k Lagrangian submanifolds in the one-point blow-up such that no two of them are Hamiltonian isotopic.
We show how to compute the Lagrangian Floer homology in the one-point blow up of the proper transform of Lagrangians submanifolds, solely in terms of information of the base manifold. As an example we present an alternative computation of the Lagrangian quantum homology in the one-point blow up of (CP^2,ω) of the proper transform of the Clifford torus.
We present an algebraic framework for the computation of low-degree cohomology of a class of bigraded complexes which arise in Poisson geometry around (pre)symplectic leaves. We also show that this framework can be applied to the more general context of Lie algebroids. Finally, we apply our results to compute the low-degree cohomology in some particular cases.
We study the behavior of the modular class of an orientable Poisson manifold and formulate some unimodularity criteria in the semilocal context, around a (singular) symplectic leaf. Our results generalize some known unimodularity criteria for regular Poisson manifolds related to the notion of the Reeb class. In particular, we show that the unimodularity of the transverse Poisson structure of the leaf is a necessary condition for the semilocal unimodular property. Our main tool is an explicit formula for a bigraded decomposition of modular vector fields of a coupling Poisson structure on a foliated manifold. Moreover, we also exploit the notion of the modular class of a Poisson foliation and its relationship with the Reeb class.
In this note we present a brief introduction to Lagrangian Floer homology and its relation with the solution of Arnol'd conjecture, on the minimal number of non-degenerate fixed points of a Hamiltonian diffeomorphism. We start with the basic definition of critical point on smooth manifolds, in oder to sketch some aspects of Morse theory. Introduction to the basics concepts of symplectic geometry are also included, with the idea of understanding the statement of Arnol'd Conjecture and how is related to the intersection of Lagrangian submanifolds.
We prove the bounded isometry conjecture of F. Lalonde and L. Polterovich for a special class of closed symplectic manifolds. As a byproduct, it is shown that the flux group of a product of these special symplectic manifold is isomorphic to the direct sum of the flux group of each symplectic manifold.