Smooth perfectness for the group of diffeomorphisms
Stefan Haller, Tomasz Rybicki, Josef Teichmann
备注：
Fully revised and extended second version of our paper math.DG/0409605 with improved results. Tomasz Rybicki contributed substantially to this second version and joined us as third author
Given a result of Herman, we provide a new elementary proof of the fact that the connected component of the group of compactly supported diffeomorphisms is perfect and hence simple. Moreover, we show that every diffeomorphism $g$, which is sufficiently close to the identity, can be represented as a product of four commutators, $g=[h_1,k_1]\circ...\circ[h_4,k_4]$, where the factors $h_i$ and $k_i$ can be chosen to depend smoothly on $g$.
It is proven that the identity component of the group preserving the leaves of a generalized foliation is perfect. This shows that a well-known simplicity theorem on the diffeomorphism group extends to the nontransitive case.
In this paper we study the uniform perfectness, boundedness and uniform simplicity of diffeomorphism groups of compact manifolds with boundary and open manifolds and obtain some upper bounds of their diameters with respect to commutator length, those with support in balls and conjugation-generated norm.
We show that on a closed smooth manifold $M$ equipped with $k$ fiber bundle structures whose vertical distributions span the tangent bundle, every smooth diffeomorphism $f$ of $M$ sufficiently close to the identity can be written as a product $f=f_1... f_k$, where $f_i$ preserves the $i^{\text{th}}$-fiber. The factors $f_i$ can be chosen smoothly in $f$. We apply this result to show that on a certain class of closed smooth manifolds every diffeomorphism sufficiently close to the identity can be written as product of commutators and the factors can be chosen smoothly. Furthermore we get concret…
For a compact, smooth C^r orbifold (without boundary), we show that the topological structure of the orbifold diffeomorphism group is a Banach manifold for finite r \ge 1 and a Frechet manifold if r=infty. In each case, the local model is the separable Banach (Frechet) space of C^r (C^infty, resp.) orbisections of the tangent orbibundle.
We study the action of the diffeomorphism group $\Diff(M)$ on the space of proper immersions $\Imm_{\text{prop}}(M,N)$ by composition from the right. We show that smooth transversal slices exist through each orbit, that the quotient space is Hausdorff and is stratified into smooth manifolds, one for each conjugacy class of isotropy groups.
In this paper we introduce the notion of tangent space TG of a (not necessary smooth) subgroup G of the diffeomorphism group Diff(M) of a compact manifold M. We prove that TG is a Lie subalgebra of the Lie algebra of smooth vector fields on M. The construction can be generalized to subgroups of any (finite or infinite dimensional) Lie groups. The tangent Lie algebra TG introduced this way is a generalization of the classical Lie algebra in the smooth cases. As a working example, we discuss in detail the tangent structure of the holonomy group and fibered holonomy group of Finsler manifolds.
Smale proved that the orientation-preserving diffeomorphism group of S^2 has a continuous strong deformation retraction to SO(3). In this paper, we construct such a strong deformation retraction which is diffeologically smooth.
For a compact contact manifold it is shown that the anisotropic Folland-Stein function spaces form an algebra. The notion of anisotropic regularity is extended to define the space of Folland-Stein contact diffeomorphisms, which is shown to be a topological group under composition and a smooth Hilbert manifold. These results are used in a subsequent paper to analyze the action of the group of contact diffeomorphisms on the space of CR structures on a compact, three dimensional manifold.
Let $M$ be a manifold and $T^*M$ be the cotangent bundle. We introduce a 1-cocycle on the group of diffeomorphisms of $M$ with values in the space of linear differential operators acting on $C^{\infty} (T^*M).$ When $M$ is the $n$-dimensional sphere, $S^n$, we use this 1-cocycle to compute the first-cohomology group of the group of diffeomorphisms of $S^n$, with coefficients in the space of linear differential operators acting on contravariant tensor fields.
Techniques of gauge theory are used to define and compute an invariant of certain diffeomorphisms of 4-manifolds. The invariant vanishes for any diffeomorphism which is smoothly isotopic to the identity. As an application, we give the first example of a diffeomorphism of a simply-connected 4-manifold which is homotopic to the identity map, but not smoothly isotopic to the identity.
We study perturbations of a partially hyperbolic toral automorphism L which is diagonalizable over C and has a dense center foliation. For a small perturbation of L with a smooth center foliation we establish existence of a smooth leaf conjugacy to L. We also show that if a small perturbation of an ergodic irreducible L has smooth center foliation and is bi-Holder conjugate to L, then the conjugacy is smooth. As a corollary, we show that for any symplectic perturbation of such an L any bi-Holder conjugacy must be smooth. For a totally irreducible L with two-dimensional center, we establish a n…
Let $X$ be a smooth, compact, oriented $4$-manifold. Building upon work of Li-Liu, Ruberman, Nakamura and Konno, we consider a families version of Seiberg-Witten theory and obtain obstructions to the existence of certain group actions on $X$ by diffeomorphisms. The obstructions show that certain group actions on $H^2(X , \mathbb{Z})$ preserving the intersection form can not be lifted an action of the same group on $X$ by diffeomorphisms. Using our obstructions, we construct numerous examples of group actions which can be realised continuously but can not be realised smoothly for any differenti…
We consider the groups $\operatorname{Diff}_{\mathcal B}(\mathbb R^n)$, $\operatorname{Diff}_{H^\infty}(\mathbb R^n)$, and $\operatorname{Diff}_{\mathcal S}(\mathbb R^n)$ of smooth diffeomorphisms on $\mathbb R^n$ which differ from the identity by a function which is in either $\mathcal B$ (bounded in all derivatives), $H^\infty = \bigcap_{k\ge 0}H^k$, or $\mathcal S$ (rapidly decreasing). We show that all these groups are smooth regular Lie groups.
In this paper, we are concerned with interactions between isoparametric theory and differential topology. Two foliations are called equivalent if there exists a diffeomorphism between the foliated manifolds mapping leaves to leaves. Using differential topology, we obtain several results towards the classification problem of isoparametric foliations up to equivalence. In particular, we show that each homotopy $n$-sphere has the "same" isoparametric foliations as the standard sphere $S^n$ has except for $n=4$, reducing the classification problem on homotopy spheres to that on the standard sphere…
In a previous work it is shown that every finite group $G$ of diffeomorphisms of a connected smooth manifold $M$ of dimension $\geq 2$ equals, up to quotient by the flow, the centralizer of the group of smooth automorphisms of a $G$-invariant complete vector field $X$ (shortly $X$ describes $G$). Here the foregoing result is extended to show that every finite group of diffeomorphisms of $M$ is described, within the group of all homeomorphisms of $M$, by a vector field. As a consequence, it is proved that a finite group of homeomorphisms of a compact connected topological $4$-manifold, whose ac…
Given a compact manifold M, we prove that any bracket generating and invariant under multiplication on smooth functions family of vector fields on M generates the connected component of unit of the group Diff(M).
We give constraints on smooth families of 4-manifolds with boundary using Manolescu's Seiberg-Witten Floer stable homotopy type, provided that the fiberwise restrictions of the families to the boundaries are trivial families of 3-manifolds. As an application, we show that, for a simply-connected oriented compact smooth 4-manifold $X$ with boundary with an assumption on the Frøyshov invariant or the Manolescu invariants $α, β, γ$ of $\partial X$, the inclusion map $\mathrm{Diff}(X,\partial) \hookrightarrow \mathrm{Homeo}(X,\partial)$ between the groups of diffeomorphisms and homeomorphisms whic…
We consider a variety of geodesic equations on Sobolev diffeomorphism groups, including the equations of ideal hydrodynamics. We prove that solutions of the corresponding two-point boundary value problems are precisely as smooth as their boundary conditions. We further utilise this regularity property to construct continuously differentiable exponential maps in the Frechét setting.
A smooth diffeomorphism is said to be distributionally uniquely ergodic (DUE for short) when it is uniquely ergodic and its unique invariant probability measure is the only invariant distribution (up to multiplication by a constant). Ergodic translations on tori are classical examples of DUE diffeomorphisms. In this article we construct DUE diffeomorphisms supported on closed manifolds different from tori, providing some counterexamples to a conjecture proposed by Forni in [For08].
We construct a smooth Lie group structure on the group of real analytic diffeomorphisms of a compact analytic manifold with corners. This generalises the known analogous results in the situation where the real analytic manifold has no corners. Additionally our approach uses a different construction.
We show that if $M$ is a compact smooth manifold diffeomorphic to the total space of an orientable $S^2$ bundle over the torus $T^2$, then its diffeomorphism group does not have the Jordan property, i.e., Diff$(M)$ contains a finite subgroup $G_n$ for any natural number $n$ such that every abelian subgroup of $G_n$ has index at leat $n$. This gives a counterexample to an old conjecture of Ghys.
Let G be a compact Lie group and X be a compact smooth G-manifold with finitely many G-fixed points. We show that if X admits a G-equivariant hyperbolic diffeomorphism having a certain convergence property, there exists an open covering of X indexed by the G-fixed points so that each open set is G-stable and G-equivariantly diffeomorphic to the tangential G-representation at the corresponding G-fixed point. We also show that the converse is also true in case of holomorphic torus actions
Let M be a smooth compact connected oriented manifold of dimension at least two endowed with a volume form. Assuming certain conditions on the fundamental group $π_1(M)$ we construct quasi-isometric embeddings of either free Abelian or direct products of non-Abelian free groups into the group of volume preserving diffeomorphisms of M equipped with the L^p metric induced by a Riemannian metric on M.
