The Identity Component of the Leaf Preserving Diffeomorphism Group is Perfect.
Commutators of diffeomorphisms of a manifold with boundary
A well known theorem of Herman-Thurston states that the identity component of the group of diffeomorphisms of a boundaryless manifold is perfect and simple. We generalize this result to manifolds with boundary. Remarks on -diffeomorphisms are included.
A note on generalized flag structures
Generalized flag structures occur naturally in modern geometry. By extending Stefan's well-known statement on generalized foliations we show that such structures admit distinguished charts. Several examples are included.
On Lie algebras of vector fields related to Riemannian foliations
Riemannian foliations constitute an important type of foliated structures. In this note we prove two theorems connecting the algebraic structure of Lie algebras of foliated vector fields with the smooth structure of a Riemannian foliation.
Lie algebras of vector fields and codimension one foliations.
The main result is a Pursell-Shanks type theorem for codimension one foliations. This theorem can be viewed as a partial solution of a hypothetical general version of the theorem of Pursell-Shanks. Several propositions and lemmas on foliations are contained in the proof.
Correspondence between diffeomorphism groups and singular foliations
It is well-known that any isotopically connected diffeomorphism group G of a manifold determines a unique singular foliation . A one-to-one correspondence between the class of singular foliations and a subclass of diffeomorphism groups is established. As an illustration of this correspondence it is shown that the commutator subgroup [G,G] of an isotopically connected, factorizable and non-fixing diffeomorphism group G is simple iff the foliation defined by [G,G] admits no proper minimal sets....
On the uniform perfectness of groups of bundle homeomorphisms
Groups of homeomorphisms related to locally trivial bundles are studied. It is shown that these groups are perfect. Moreover if the homeomorphism isotopy group of the base is bounded then the bundle homeomorphism group of the total space is uniformly perfect.
On the homeomorphism groups of manifolds and their universal coverings
Let H c(M) stand for the path connected identity component of the group of all compactly supported homeomorphisms of a manifold M. It is shown that H c(M) is perfect and simple under mild assumptions on M. Next, conjugation-invariant norms on Hc(M) are considered and the boundedness of Hc(M) and its subgroups is investigated. Finally, the structure of the universal covering group of Hc(M) is studied.
Local and nice structures of the groupoid of an equivalence relation
Groups of -diffeomorphisms related to a foliation
The notion of a -diffeomorphism related to a foliation is introduced. A perfectness theorem for the group of -diffeomorphisms is proved. A remark on -diffeomorphisms is given.
Preface
On the flux homomorphism for regular Poisson manifolds
Author’s abstract: “We introduce the concept of the flux homomorphism for regular Poisson manifolds. First we establish a one-to-one correspondence between Poisson diffeomorphisms close to and closed foliated 1-forms close to 0. This allows to show that the group of Poisson automorphisms is locally contractible and to define the flux locally. Then, by means of the foliated cohomology, we extend this local homomorphism to a global one”.
On admissible groups of diffeomorphisms
The phenomenon of determining a geometric structure on a manifold by the group of its automorphisms is a modern analogue of the basic ideas of the Erlangen Program of F. Klein. The author calls such diffeomorphism groups admissible and he describes them by imposing some axioms. The main result is the followingTheorem. Let , , be a geometric structure such that its group of automorphisms satisfies either axioms 1, 2, 3 and 4, or axioms 1, 2, 3’, 4, 5, 6 and 7, and is compact, or axioms 1, 2,...
An infinite dimensional version of the third Lie theorem
The concept of evolution operator is used to introduce a weak Lie subgroup of a regular Lie group, and to give a new version of the third Lie theorem. This enables the author to formulate and to study the problem of integrability of infinite-dimensional Lie algebras. Several interesting examples are presented.
Integrability of the Poisson algebra on a locally conformal symplectic manifold
Summary: It is proven that the Poisson algebra of a locally conformal symplectic manifold is integrable by making use of a convenient setting in global analysis. It is also observed that, contrary to the symplectic case, a unified approach to the compact and non-compact case is possible.
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