In one of his papers, C. Viterbo defined a distance on the set of Hamiltonian diffeomorphisms of ${ℝ}^{2n}$ endowed with the standard symplectic form ${\omega }_0=dp\wedge dq$. We study the completions of this space for the topology induced by Viterbo’s distance and some others derived from it, we study their different inclusions and give some of their properties. In particular, we give a convergence criterion for these distances that allows us to prove that the completions contain non-ordinary elements, as for example, discontinuous...

Hermann and Thurston proved that the group of diffeomorphisms with compact support of a smooth manifold M which are isotopic to the identity is a perfect group. We consider the case where M has a geometric structure. In this paper we shall survey on the recent results of the first homology of the diffeomorphism groups which preserve a smooth G-action or a foliated structure on M. We also work in Lipschitz category.

The group of real analytic diffeomorphisms of a real analytic manifold is a rich group. It is dense in the group of smooth diffeomorphisms. Herman showed that for the $n$-dimensional torus, its identity component is a simple group. For $U\left(1\right)$ fibered manifolds, for manifolds admitting special semi-free $U\left(1\right)$ actions and for 2- or 3-dimensional manifolds with nontrivial $U\left(1\right)$ actions, we show that the identity component of the group of real analytic diffeomorphisms is a perfect group.

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.

This paper reports on some results concerning:a) The homotopy type of the group of diffeomorphisms $\mathrm{Diff}\left(M^n\right)$ of a differentiable compact manifold $M^n$ (with $C^{\infty }$-topology).b) the result of the homotopy comparison of this space with the group of all homeomorphisms Homeo $M^n$ (with $C^o$-topology). As a biproduct, one gets new facts about the homotopy groups of $\mathrm{Diff}(D^n,\partial D^n),{\mathrm{Top}}_n$, ${\mathrm{Top}}_n/O_n$ and about the number of connected components of the space of topological and combinatorial pseudoisotopies.The results are contained in Section 1 and Section...

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.