ON THE TRANSITIVITY OF THE GROUP OF ORBIFOLD DIFFEOMORPHISMS

Consider a connected manifold of dimension at least two and the group of compactly supported diffeomorphisms that are isotopic to the identity through a compactly supported isotopy. This group acts n-transitively: any n-tuple of points can be moved to any other n-tuple by an element of this group. The group of diffeomorphisms of an orbifold is typically not n-transitive: simple obstructions are given by isomorphism classes of isotropy groups of points. In this paper we investigate the transitivity properties of the group of compactly supported diffeomorphisms of orbifolds that are isotopic to the identity through a compactly supported isotopy. We also study an example in the category of area preserving mappings.

Deformations of circle-valued Morse functions on 2-torus

In this paper we give an algebraic description of fundamental groups of orbits of circle-valued Morse functions on T2 with respect to the action of the group of diffeomorphisms of T2

Diffeomorphism cocycles over partially hyperbolic systems

We consider Hölder continuous cocycles over an accessible partially hyperbolic system with values in the group of diffeomorphisms of a compact manifold $\mathcal {M}$ . We obtain several results for this setting. If a cocycle is bounded in $C^{1+\gamma }$ , we show that it has a continuous invariant family of $\gamma $ -Hölder Riemannian metrics on $\mathcal {M}$ . We establish continuity of a measurable conjugacy between two cocycles assuming bunching or existence of holonomies for both and pre-compactness in $C^0$ for one of them. We give conditions for existence of a continuous conjugacy between two cocycles in terms of their cycle weights. We also study the relation between the conjugacy and holonomies of the cocycles. Our results give arbitrarily small loss of regularity of the conjugacy along the fiber compared to that of the holonomies and of the cocycle.

On the group of diffeomorphisms of foliated manifolds

Now the foliations theory is intensively developing branch of modern differential geometry, there are numerous researches on the foliation theory. The purpose of our paper is study the structure of the group DiffF(M) of diffeomorphisms and the group IsoF(M) of isometries of foliated manifold (M,F). It is shown the group DiffF(M) is closed subgroup of the group Diff(M) of diffeomorphisms of the manifold M in compact-open topology and also it is proven the group IsoF(M) is Lie group. It is introduced new topology on DiffF(M) which depends on foliation F and called F- compact open topology. It's proven that some subgroups of the group DiffF(M) are topological groups with F-compact open topology.

The cohomology of Torelli groups is algebraic

The Torelli group of $W_g = \#^g S^n \times S^n$ is the group of diffeomorphisms of $W_g$ fixing a disc that act trivially on $H_n(W_g;\mathbb{Z} )$ . The rational cohomology groups of the Torelli group are representations of an arithmetic subgroup of $\text{Sp}_{2g}(\mathbb{Z} )$ or $\text{O}_{g,g}(\mathbb{Z} )$ . In this article we prove that for $2n \geq 6$ and $g \geq 2$ , they are in fact algebraic representations. Combined with previous work, this determines the rational cohomology of the Torelli group in a stable range. We further prove that the classifying space of the Torelli group is nilpotent.

Homotopy properties of smooth functions on the Möbius band

Let $B$ be a M\"obius band and $f:B \to \mathbb{R}$ be a Morse map taking a constant value on $\partial B$, and $\mathcal{S}(f,\partial B)$ be the group of diffeomorphisms $h$ of $B$ fixed on $\partial B$ and preserving $f$ in the sense that $f\circ h = f$. Under certain assumptions on $f$ we compute the group $\pi_0\mathcal{S}(f,\partial B)$ of isotopy classes of such diffeomorphisms. In fact, those computations hold for functions $f:B\to\mathbb{R}$ whose germs at critical points are smoothly equivalent to homogeneous polynomials $\mathbb{R}^2\to\mathbb{R}$ without multiple factors. Together with previous results of the second author this allows to compute similar groups for certain classes of smooth functions $f:N\to\mathbb{R}$ on non-orientable compact surfaces $N$.

EQUIVARIANT ALGEBRAIC INDEX THEOREM

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.

Minimizing acceleration on the group of diffeomorphisms and its relaxation

We study a second-order variational problem on the group of diffeomorphisms of the interval [0, 1] endowed with a right-invariant Sobolev metric of order 2, which consists in the minimization of the acceleration. We compute the relaxation of the problem which involves the so-called Fisher–Rao functional, a convex functional on the space of measures. This relaxation enables the derivation of several optimality conditions and, in particular, a sufficient condition which guarantees that a given path of the initial problem is also a minimizer of the relaxed one. Based on these sufficient conditions, the main result is that, when the value of the (minimized) functional is small enough, the minimizers are classical, that is the defect measure vanishes.

Unusual view of the Schwarzian theory

A polar decomposition of the Wiener measure based on its quasi-invariance under the group of diffeomorphisms is proposed. As a result, functional integrals in the Schwarzian theory can be written as the Fourier transform of the functional integrals in the quantum oscillator model with the Calogero potential.