The group of symplectomorphisms

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Symplectic Manifold ◽  
Universal Cover ◽  
Topological Properties ◽  
Basic Properties ◽  
Group Of Diffeomorphisms ◽  
Path Connected ◽  
Flux Homomorphism ◽  
Calabi Homomorphism

This chapter discusses the basic properties of the group of symplectomorphisms of a compact connected symplectic manifold and its subgroup of Hamiltonian symplectomorphisms. It begins by showing that the group of symplectomorphisms is locally path-connected and then moves on to the flux homomorphism. The main result here is a theorem of Banyaga that characterizes the Hamiltonian symplectomorphisms in terms of the flux homomorphism. In the noncompact case there is another interesting homomorphism, called the Calabi homomorphism, that takes values in the reals and may be defined on the universal cover of the group of Hamiltonian symplectomorphisms. The chapter ends with a brief comparison of the topological properties of the group of symplectomorphisms with those of the group of diffeomorphisms.

scholarly journals Onθ-regular spaces

1994 ◽  
Vol 17 (4) ◽  
pp. 687-692 ◽  
Author(s):  
Martin M. Kovár
Keyword(s):  
Regular Space ◽  
Topological Properties ◽  
Locally Compact ◽  
Basic Properties ◽  
Paracompact Space ◽  
Paracompact Spaces

In this paper we studyθ-regularity and its relations to other topological properties. We show that the concepts ofθ-regularity (Janković, 1985) and point paracompactness (Boyte, 1973) coincide. Regular, strongly locally compact or paracompact spaces areθ-regular. We discuss the problem when a (countably)θ-regular space is regular, strongly locally compact, compact, or paracompact. We also study some basic properties of subspaces of aθ-regular space. Some applications: A space is paracompact iff the space is countablyθ-regular and semiparacompact. A generalizedFσ-subspace of a paracompact space is paracompact iff the subspace is countablyθ-regular.

scholarly journals Topological convexity spaces

1974 ◽  
Vol 19 (2) ◽  
pp. 125-132 ◽  
Author(s):  
Victor Bryant
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Topological Properties ◽  
Convexity Space ◽  
Basic Properties

We shall start by recalling the definition and some basic properties of a convexity space; a topological convexity space (tcs) will then be a convexity space together with an admissible topology, and will be a generalisation of a topological vector space (tvs). After showing that the usual tvs results connecting the linear and topological properties extend to this new setting we then prove a form of the Krein-Milman theorem in a tcs.

scholarly journals EQUIVARIANT ALGEBRAIC INDEX THEOREM

2019 ◽  
pp. 1-27 ◽  
Author(s):  
Alexander Gorokhovsky ◽  
Niek de Kleijn ◽  
Ryszard Nest
Keyword(s):  
Symplectic Manifold ◽  
Discrete Group ◽  
Pseudodifferential Operators ◽  
Index Theorem ◽  
Group Of Automorphisms ◽  
Algebraic Version ◽  
Group Of Diffeomorphisms ◽  
Formal Deformation ◽  
Underlying Manifold ◽  
Equivariant Version

We prove a $\unicode[STIX]{x1D6E4}$ -equivariant version of the algebraic index theorem, where $\unicode[STIX]{x1D6E4}$ is a discrete group of automorphisms of a formal deformation of a symplectic manifold. The particular cases of this result are the algebraic version of the transversal index theorem related to the theorem of A. Connes and H. Moscovici for hypo-elliptic operators and the index theorem for the extension of the algebra of pseudodifferential operators by a group of diffeomorphisms of the underlying manifold due to A. Savin, B. Sternin, E. Schrohe and D. Perrot.

scholarly journals THE NONLINEAR MASLOV INDEX AND THE CALABI HOMOMORPHISM

2007 ◽  
Vol 09 (06) ◽  
pp. 769-780 ◽  
Author(s):  
GABI BEN SIMON
Keyword(s):  
Projective Space ◽  
Open Subset ◽  
Complex Projective Space ◽  
Maslov Index ◽  
Universal Cover ◽  
Hamiltonian Diffeomorphisms ◽  
Calabi Invariant ◽  
Nonlinear Maslov Index ◽  
Complex Projective ◽  
Calabi Homomorphism

In this paper, we find that the asymptotic nonlinear Maslov index defined on the universal cover of the group of all contact Hamiltonian diffeomorphisms of the standard (2n - 1)-dimensional contact sphere is a quasimorphism. Then we show our main result: Let M be standard (n - 1)-dimensional complex projective space. We prove that the value of the pullback of the asymptotic nonlinear Maslov index to the universal cover of the group of Hamiltonian diffeomorphisms of M, when evaluated on a diffeomorphism supported in a sufficiently small open subset of M, equals [Formula: see text] times the Calabi invariant of this diffeomorphism.

COVER OF THE BROWNIAN BRIDGE AND STOCHASTIC SYMPLECTIC ACTION

2000 ◽  
Vol 12 (01) ◽  
pp. 91-137 ◽  
Author(s):  
R. LÉANDRE
Keyword(s):  
Symplectic Manifold ◽  
Sobolev Spaces ◽  
Morse Theory ◽  
Loop Space ◽  
Brownian Bridge ◽  
Universal Cover ◽  
Symplectic Action

We define the universal cover of the Brownian bridge of a symplectic manifold. This allows us to define a non-trivial functional over it called the stochastic symplectic action, and to define local Sobolev spaces over the universal cover, such that the symplectic action belong to them. This is done in the purpose of an analytical Morse theory in the sense of Witten over the loop space associated to this symplectic action. In this purpose, a stochastic Witten complex is constructed over the universal cover of the loop space, and modulo some weights, it is shown that its cohomology is equal to the stochastic cohomology of the universal cover.

scholarly journals Universal covers of topological modules and a monodromy principle

2003 ◽  
Vol 34 (4) ◽  
pp. 299-308
Author(s):  
Osman Mucuk ◽  
Mehmet Ozdemir
Keyword(s):  
Universal Cover ◽  
Topological Ring ◽  
Universal Covers ◽  
Topological Modules ◽  
Simply Connected ◽  
Path Connected

Let $R$ be a simply connected topological ring and $M$ be a topological left $R$-module in which the underling topology is path connected and has a universal cover. In this paper, we prove that a simply connected cover of $M$ admits the structure of a topological left $R$-module, and prove a Monodromy Principle, that a local morphism on $M$ of topological left $R$-modules extends to a morphism of topological left $R$-modules.

scholarly journals The fundamental group of quotients of products of some topological spaces by a finite group - A generalization of a Theorem of Bauer-Catanese-Grunewald-Pignatelli: Le groupe fondamental de quotients de produits de certains espaces topologiques par ungroupe fini — Généralisation d’un théorème de Bauer–Catanese–Grunewald–Pignatelli

2021 ◽  
Vol Volume 5 ◽  
Author(s):  
Rodolfo Aguilar
Keyword(s):  
Finite Group ◽  
Finite Number ◽  
Fundamental Group ◽  
Universal Cover ◽  
Connected Components ◽  
Topological Spaces ◽  
Group A ◽  
A Finite Group ◽  
Path Connected

We provide a description of the fundamental group of the quotient of a product of topological spaces X i, each admitting a universal cover, by a finite group G, provided that there is only a finite number of path-connected components in X g i for every g ∈ G. This generalizes previous work of Bauer-Catanese-Grunewald-Pignatelli and Dedieu-Perroni. Nous fournissons une description du groupe fondamental du quotient d’un produitd’espaces topologiques Xi , chacun admettant un revêtement universel, par un groupe fini G,pourvu qu’il n’existe qu’un nombre ni de composantes connexes par arcs dans Xgi pour chaque g ∈ G. Cela généralise des résultats antérieurs de Bauer–Catanese–Grunewald–Pignatelli et deDedieu–Perroni.

scholarly journals Covering groups of non-connected topological groups revisited

1994 ◽  
Vol 115 (1) ◽  
pp. 97-110 ◽  
Author(s):  
Ronald Brown ◽  
Osman Mucuk
Keyword(s):  
Topological Group ◽  
Universal Cover ◽  
Topological Groups ◽  
Covering Spaces ◽  
Unique Structure ◽  
Underlying Space ◽  
Path Connected ◽  
Covering Groups

All spaces are assumed to be locally path connected and semi-locally 1-connected. Let X be a connected topological group with identity e, and let be the universal cover of the underlying space of X. It follows easily from classical properties of lifting maps to covering spaces that for any point ẽ in with pẽ = e there is a unique structure of topological group on such that ẽ is the identity and is a morphism of groups. We say that the structure of topological group on X lifts to .

Thicknesses of knots

1999 ◽  
Vol 126 (2) ◽  
pp. 293-310 ◽  
Author(s):  
Y. DIAO ◽  
C. ERNST ◽  
E. J. JANSE VAN RENSBURG
Keyword(s):  
Smooth Surface ◽  
Solid Torus ◽  
Great Care ◽  
Topological Properties ◽  
Closed Curves ◽  
Large Curvature ◽  
Basic Properties ◽  
The Core ◽  
Strong Deformation ◽  
The Relationship

In this paper we define a set of radii called thickness for simple closed curves denoted by K, which are assumed to be differentiable. These radii capture a balanced view between the geometric and the topological properties of these curves. One can think of these radii as representing the thickness of a rope in space and of K as the core of the rope. Great care is taken to define our radii in order to gain freedom from small pieces with large curvature in the curve. Intuitively, this means that we tend to allow the surface of the ropes that represent the knots to deform into a non smooth surface. But as long as the radius of the rope is less than the thickness so defined, the surface of the rope will remain a two manifold and the rope (as a solid torus) can be deformed onto K via strong deformation retract. In this paper we explore basic properties of these thicknesses and discuss the relationship amongst them.

Sign in / Sign up

Export Citation Format

Share Document