Let $D$ be a 2-dimensional closed unit disk and $\rm{Symp}(D,0)_{\rm{rel}}$ the group of symplectomorphisms preserving the origin and the boundary $\partial D$ pointwise. We consider the $\mathbb{R}$-valued flux homomorphism on $\rm{Symp}(D,0)_{\rm{rel}}$ and define the central $\mathbb{R}$-extension called the $\mathbb{R}$-valued flux extension. We determine the Euler class of this extension and investigate the relation between the extension, the group $2$-cocycle defined by Ismagilov, Losik, and Michor, and the Calabi invariant of $D$.
Let $M$ be a closed connected smooth manifold and $G=\textmd{Diff}_0(M)$ denote the connected component of the diffeomorphism group of $M$ containing the identity. The natural action of $G$ on $M$ induces the trace homomorphism on homology. We show that the image of trace homomorphism is annihilated by the subalgebra of the cohomology ring of $M$, generated by the characteristic classes of $M$. Analogously, if $J$ is an almost complex structure on $M$ and $G$ denotes the identity component of the group of diffeomorphisms of $M$ preserving $J$ then the image of the corresponding trace homomorph…
Let $M$ and $N$ be two closed $C^{\infty}$ manifolds and let $\text{Diff}_c(M)$ denote the group of $C^{\infty}$ diffeomorphisms isotopic to the identity. We prove that any (discrete) group homomorphism between $\text{Diff}_c(M)$ and $\text{Diff}_c(N)$ is continuous. We also show that a non-trivial group homomorphism $Φ: \text{Diff}_c(M) \to \text{Diff}_c(N)$ implies that $\dim(M) \leq \dim(N)$ and give a classification of such homomorphisms when $\dim(M) = \dim(N)$.
Let $M$ be a closed manifold. Polterovich constructed a linear map from the vector space of quasi-morphisms on the fundamental group $π_{1}(M)$ of $M$ to the space of quasi-morphisms on the identity component ${\rm Diff}_Ω^{\infty} (M)_{0}$ of the group of volume-preserving diffeomorphisms of $M$. In this paper, the restriction $H^{1}(π_{1}(M); {\mathbb R})\to H^{1}({\rm Diff}_Ω^{\infty} (M)_{0} ; {\mathbb R})$ of the linear map is studied and its relationship with the flux homomorphism is described.
For r at least 3, p at least 2, we classify all actions of the groups Diff^r_c(R) and Diff^r_+(S1) by C^p -diffeomorphisms on the line and on the circle. This is the same as describing all nontrivial group homomorphisms between groups of compactly supported diffeomorphisms on 1- manifolds. We show that all such actions have an elementary form, which we call topologically diagonal. As an application, we answer a question of Ghys in the 1-manifold case: if M is any closed manifold, and Diff(M)_0 injects into the diffeomorphism group of a 1-manifold, must M be 1 dimensional? We show that the answ…
Given a smooth spacelike surface $Σ$ of negative curvature in Anti-de Sitter space of dimension 3, invariant by a representation $ρ:π_1(S)\to\mathrm{PSL}_2\mathbb{R}\times\mathrm{PSL}_2\mathbb{R}$ where $S$ is a closed oriented surface of genus $\geq 2$, a canonical construction associates to $Σ$ a diffeomorphism $ϕ_Σ$ of $S$. It turns out that $ϕ_Σ$ is a symplectomorphism for the area forms of the two hyperbolic metrics $h$ and $h'$ on $S$ induced by the action of $ρ$ on $\mathbb{H}^2\times\mathbb{H}^2$. Using an algebraic construction related to the flux homomorphism, we give a new proof of …
On a closed symplectic surface Sigma of genus two or more, we give a new construction of an extended flux map (a crossed homomorphism from the symplectomorphism group Symp(Sigma) to the cohomology group H^1(Sigma;R) that extends the flux homomorphism). This construction uses the topology of the Jacobian of the surface and a correction factor related to the Johnson homomorphism. For surfaces of genus three or more, we give another new construction of an extended flux map using hyperbolic geometry.
Let $M$ be the circle or a compact interval, and let $α=k+τ\ge1$ be a real number such that $k=\lfloor α\rfloor$. We write $\mathrm{Diff}_+^α(M)$ for the group of $C^k$ diffeomorphisms of $M$ whose $k^{th}$ derivatives are Hölder continuous with exponent $τ$. If $α\ge1$, we prove that there exists a finitely generated subgroup $G_α\le\mathrm{Diff}_+^α(M)$ with the property that $G_α$ admits no injective homomorphisms into $\mathrm{Diff}_+^β(M)$ for all $β>α$. If $α>1$, we also show the dual result: there exists a finitely generated group $H_α\le\bigcap_{β<α}\mathrm{Diff}_+^β(M)$ with the prope…
We study the flux homomorphism for closed forms of arbitrary degree, with special emphasis on volume forms and on symplectic forms. The volume flux group is an invariant of the underlying manifold, whose non-vanishing implies that the manifold resembles one with a circle action with homologically essential orbits.
Denote by $\DC(M)_0$ the identity component of the group of compactly supported $C^\infty$ diffeomorphisms of a connected $C^\infty$ manifold $M$, and by $\HR$ the group of the homeomorphisms of $\R$. We show that if $M$ is a closed manifold which fibers over $S^m$ ($m\geq 2$), then any homomorphism from $\DC(M)_0$ to $\HR$ is trivial.
We show that the group cohomology of the diffeomorphisms of the disk with $n$ punctures has the cohomology of the braid group of $n$ strands as the summand. As an application of this method, we also prove that there is no cohomological obstruction to lifting the "standard" embedding $\mathrm{Br}_{2g+2}\hookrightarrow \mathrm{Mod}_{g,2}$ to a group homomorphism between diffeomorphism groups.
Let $D^2$ be the open unit disc in the Euclidean plane and let $G:= Diff(D2; area)$ be the group of smooth compactly supported area-preserving diffeomorphisms of $D^2$. We investigate the properties of G endowed with the autonomous metric. In particular, we construct a bi-Lipschitz homomorphism $Z^k \rightarrow G$ of a finitely generated free abelian group of an arbitrary rank. We also show that the space of homogeneous quasi-morphisms vanishing on all autonomous diffeomorphisms in the above group is infinite dimensional.
In this paper we discuss the relationship between groups of diffeomorphisms of spheres and balls. We survey results of a topological nature and then address the relationship as abstract (discrete) groups. We prove that the identity component Diff_0(S^{2n-1}) of the group of smooth diffeomorphisms of S^{2n+1} admits no nontrivial homomorphisms to the group of C^1 diffeomorphisms of the ball B^m for any n and m. This result generalizes theorems of Ghys and Herman. We also examine finitely generated subgroups of Diff_0(S^n) and produce an example of a finitely generated torsion free group Gamma w…
We investigate the representation theory of the polynomial core of the quantum Teichmuller space of a punctured surface S. This is a purely algebraic object, closely related to the combinatorics of the simplicial complex of ideal cell decompositions of S. Our main result is that irreducible finite-dimensional representations of this polynomial core are classified, up to finitely many choices, by group homomorphisms from the fundamental group of the surface to the isometry group of the hyperbolic 3--space. We exploit this connection between algebra and hyperbolic geometry to exhibit new invaria…
Let Σ_g be a closed orientable surface let Diff_0(Σ_g; area) be the identity component of the group of area-preserving diffeomorphisms of Σ_g. In this work we present an extension of Gambaudo-Ghys construction to the case of a closed hyperbolic surface Σ_g, i.e. we show that every non-trivial homogeneous quasi-morphism on the braid group on n strings of Σ_g defines a non-trivial homogeneous quasi-morphism on the group Diff_0(Σ_g; area). As a consequence we give another proof of the fact that the space of homogeneous quasi-morphisms on Diff_0(Σ_g; area) is infinite dimensional. Let Ham(Σ_g) be …
Let $M$ be a manifold, $N$ a 1-dimensional manifold. Assuming $r \neq \dim(M)+1$, we show that any nontrivial homomorphism $ρ: \text{Diff}^r_c(M)\to \text{Homeo}(N)$ has a standard form: necessarily $M$ is $1$-dimensional, and there are countably many embeddings $ϕ_i: M\to N$ with disjoint images such that the action of $ρ$ is conjugate (via the product of the $ϕ_i$) to the diagonal action of $\text{Diff}^r_c(M)$ on $M \times M \times ...$ on $\bigcup_i ϕ_i(M)$, and trivial elsewhere. This solves a conjecture of Matsumoto. We also show that the groups $\text{Diff}^r_c(M)$ have no countable ind…
Suppose M is a noncompact connected oriented C^infty n-manifold and omega is a positive volume form on M. Let D^+(M) denote the group of orientation preserving diffeomorphisms of M endowed with the compact-open C^infty topology and D(M; omega) denote the subgroup of omega-preserving diffeomorphisms of M. In this paper we propose a unified approach for realization of mass transfer toward ends by diffeomorphisms of M. This argument together with Moser's theorem enables us to deduce two selection theorems for the groups D^+(M) and D(M; omega). The first one is the extension of Moser's theorem to …
We give general classification and structure theorems for actions of groups of homeomorphisms and diffeomorphisms on manifolds, reminiscent of classical results for actions of (locally) compact groups. This gives a negative answer to Ghys' "extension problem" for diffeomorphisms of manifolds with boundary, as well as a classification of all homomorphisma $\mathrm{Homeo}_0(M) \to \mathrm{Homeo}_0(N)$ when dim(M) = dim(N) (and related results for diffeomorphisms), and a complete classification of actions of $\mathrm{Homeo}_0(S^1)$ on surfaces. This resolves many problems in a program initiated b…
For any closed oriented surface F of genus at least three, we prove the existence of foliated F-bundles over surfaces such that the signatures of the total spaces are non-zero. We can arrange that the total holonomy of the horizontal foliations preserve a prescribed symplectic form on the fiber. We relate the cohomology class represented by the transverse symplectic form to a crossed homomorphism from the symplectomophism group of the fiber to its first real cohomology. This crossed homomorphism extends the flux homomorphism defined on the identity component of the symplectomorphism group.
Let S be a compact surface - or the interior of a compact surface - and let V be the manifold of cooriented contact elements of S equiped with its canonical contact structure. A diffeomorphism of V that preserves the contact structure and its coorientation is called a contact transformation over S. We prove the following results. 1) If S is neither a sphere nor a torus then the inclusion of the diffeomorphism group of S into the contact transformation group is 0-connected. 2) If S is a sphere then the contact transformation group is connected. 3) if S is a torus then the homomorphism from the …
We give a sufficient and necessary condition of the fundamental group homomorphism of a map between manifolds to induce homology equivalences. Moreover, a classification of one-sided h-cobordism of manifolds up to diffeomorphisms is obtained, based on Quillen's plus construction with Whitehead torsions.
The Nielsen Realization problem asks when the group homomorphism from Diff(M) to pi_0 Diff(M) admits a section. For M a closed surface, Kerckhoff proved that a section exists over any finite subgroup, but Morita proved that if the genus is large enough then no section exists over the entire mapping class group. We prove the first nonexistence theorem of this type in dimension 4: if M is a smooth closed oriented 4-manifold which contains a K3 surface as a connected summand then no section exists over the whole of the mapping class group. This is done by showing that certain obstructions lying i…
By Torelli topology the author understands aspects of the topology of surfaces (potentially) relevant to the study of Torelli groups. The extension problem in Torelli topology is the problem of determining when a diffeomorphism of compact connected subsurface of a closed surface can be extended to a diffeomorphism of the whole surface acting trivially on its homology. The special case of extensions by the identity diffeomorphism was considered by A. Putman, whose work was one of the main sources of inspiration for the present paper. It turns out that the answer depends on the position of the s…
Let $f:M\to \mathbb{R}$ be a Morse function on a smooth closed surface, $V$ be a connected component of some critical level of $f$, and $\mathcal{E}_V$ be its atom. Let also $\mathcal{S}(f)$ be a stabilizer of the function $f$ under the right action of the group of diffeomorphisms $\mathrm{Diff}(M)$ on the space of smooth functions on $M,$ and $\mathcal{S}_V(f) = \{h\in\mathcal{S}(f)\,| h(V) = V\}.$ The group $\mathcal{S}_V(f)$ acts on the set $π_0\partial \mathcal{E}_V$ of connected components of the boundary of $\mathcal{E}_V.$ Therefore we have a homomorphism $ϕ:\mathcal{S}(f)\to \mathrm{Au…
Given a manifold $M$ and a point in its interior, the point-pushing map describes a diffeomorphism that pushes the point along a closed path. This defines a homomorphism from the fundamental group of $M$ to the group of isotopy classes of diffeomorphisms of $M$ that fix the basepoint. This map is well-studied in dimension $d = 2$ and is part of the Birman exact sequence. Here we study, for any $d \geqslant 3$ and $k \geqslant 1$, the map from the $k$-th braid group of $M$ to the group of homotopy classes of homotopy equivalences of the $k$-punctured manifold $M \smallsetminus z$, and analyse i…
Let $\hat{G}$ be a group and $G$ its normal subgroup. In this paper, we study $\hat{G}$-invariant quasimorphisms on $G$ which appear in symplectic geometry and low dimensional topology. As its application, we prove the non-existence of a section of the flux homomorphism on closed surfaces of higher genus. We also prove that Py's Calabi quasimorphism and Entov-Polterovich's partial Calabi quasimorphism are non-extendable to the group of symplectomorphisms. We show that Py's Calabi quasimorphism is the unique non-extendable quasimorphism to some group.
Let $G$ be a group. A function $l:G\rightarrow \lbrack 0,\infty )$ is called a length function if (1) $l(g^{n})=|n|l(g)$ for any $g\in G$ and $n\in \mathbb{Z};$ (2) $l(hgh^{-1})=l(g)$ for any $h,g\in G;$ and (3) $l(ab)\leq l(a)+l(b)$ for commuting elements $a,b.$ Such length functions exist in many branches of mathematics, mainly as stable word lengths, stable norms, smooth measure-theoretic entropy, translation lengths on $\mathrm{CAT}(0)$ spaces and Gromov $δ$% -hyperbolic spaces, stable norms of quasi-cocycles, rotation numbers of circle homeomorphisms, dynamical degrees of birational maps …
