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Jarek Kedra

Publications and source records attributed to Jarek Kedra.

14 recordsLinked to original sources
arXiv2026

On quasimorphisms and distortion in homeomorphism groups

Let $M$ be a smooth compact oriented connected manifold, and ${\rm Homeo}_0(M,μ)$ the group of homeomorphisms of $M$ supported away from $\partial M,$ which preserve a Borel probability measure $μ$ induced by a volume form on $M$, and are isotopic to the identity. In this paper, we identify those Gambaudo-Ghys and Polterovich quasimorphisms $Ψ\colon {\rm Diff}_0(M,μ)\to R$ which extend $C^0$-continuously to ${\rm Homeo}_0(M,μ)$ as quasimorphisms, and to ${\rm Homeo}_0(M)$ as group cochains whose differentials are semi-bounded cocycles. We present several applications of this result which include unboundedness of certain bi-invariant metric on the commutator subgroup of ${\rm Homeo}_0(M,μ)$, and conditions under which a homeomorphism in ${\rm Homeo}_0(M)$ is undistorted.

math.GT↗
arXiv2013

On the autonomous metric on the group of area-preserving diffeomorphisms of the 2-disc

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.

math.GT↗
arXiv2012

Quasi-Isometric Embeddings into Diffeomorphism Groups

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.

math.GT↗
arXiv2008

A cocycle on the group of symplectic diffeomorphisms

We define a cocycle on the group of symplectic diffeomorphisms of a symplectic manifold and investigate its properties. The main applications are concerned with symplectic actions of discrete groups. For example, we give an alternative proof of the Polterovich theorem about the distortion of cyclic subgroups in finitely generated groups of Hamiltonian diffeomorphisms.

math.SG↗
arXiv2007

Symplectically aspherical manifolds

This is a survey article on symplectically aspherical manifolds. The paper contains a discussion on constructions of symplectically aspherical manifolds, their topological properties and the role of this class in symplectic topology. Research perspectives are discussed.

math.SG↗
arXiv2007

Symplectically hyperbolic manifolds

A symplectic form is called hyperbolic if its pull-back to the universal cover is a differential of a bounded one-form. The present paper is concerned with the properties and constructions of manifolds admitting hyperbolic symplectic forms. The main results are: * If a symplectic form represents a bounded cohomology class then it is hyperbolic. * The symplectic hyperbolicity is equivalent to a certain isoperimetric inequality. * The fundamental group of symplectically hyperbolic manifold is non-amenable. We also construct hyperbolic symplectic forms on certain bundles and Lefschetz fibrations, discuss the dependenc of the symplectic hyperbolicity on the fundamental group and discuss some properties of the group of symplectic diffeomorphisms of a symplectically hyperbolic manifold.

math.SG↗
arXiv2006

Symplectic configurations

We define a class of symplectic fibrations called symplectic configurations. They are natural generalization of Hamiltonian fibrations. Their geometric and topological properties are investigated. We are mainly concentrated on integral symplectic manifolds. We construct the classifyng space \B of symplectic integral configurations. The properties of the classifying map \B --> BSymp(M,w) are examined. The universal symplectic bundle over \B has a natural connection whose holonomy group is isomorphic to the enlarged Hamiltonian group recently defined by McDuff. The space \B is identified with the classifying space of an extension of certain subgroup of the symplectomorphism group.

math.SG↗
arXiv2005

Homotopy properties of Hamiltonian group actions

Consider a Hamiltonian action of a compact Lie group H on a compact symplectic manifold (M,w) and let G be a subgroup of the diffeomorphism group Diff(M). We develop techniques to decide when the maps on rational homotopy and rational homology induced by the classifying map BH --> BG are injective. For example, we extend Reznikov's result for complex projective space CP^n to show that both in this case and the case of generalized flag manifolds the natural map H_*(BSU(n+1)) --> H_*(BG) is injective, where G denotes the group of all diffeomorphisms that act trivially on cohomology. We also show that if lambda is a Hamiltonian circle action that contracts in G = Ham(M,w) then there is an associated nonzero element in pi_3(G) that deloops to a nonzero element of H_4(BG). This result (as well as many others) extends to c-symplectic manifolds (M,a), ie, 2n-manifolds with a class a in H^2(M) such that a^n is nonzero. The proofs are based on calculations of certain characteristic classes and elementary homotopy theory.

math.SG↗
arXiv2003

Evaluation fibrations and topology of symplectomorphisms

There are two main results. The first states that isotropy subgroups of groups acting transitively on a rationally hyperbolic spaces have infinitely generated rational cohomology algebra. Using this fact, we prove that the analogous statement holds for groups of symplectomorphisms of certain blow-ups.

math.AT↗
arXiv2002

Characteristic classes of smooth fibrations

We construct characteristic classes of smooth (Hamiltonian) fibrations as as fiber integrals of products of Pontriagin (or Chern) classes of vertical vector bundles over the total space of the universal fibration. We give explicit formulae of these fiber integrals for toric manifolds and get estimates of the dimension of the cohomology groups of classifying spaces.

math.SG↗