New proof associates partitions to isotopic pseudo-Anosov homeomorphisms.
problem Stable and unstable foliations for pseudo-Anosov homeomorphisms.
method Geometric realization of Fathi's result for isotopic homeomorphisms.
result Associated stable and unstable partitions for isotopic pseudo-Anosov homeomorphisms.
Generates special homeomorphisms for complex surfaces.
problem Creating specific homeomorphisms for infinite-type surfaces.
method General conditions for producing endperiodic loxodromics.
result Produces homeomorphisms acting loxodromically on arc graphs.
The study of periodic subgroups in homeomorphism groups of manifolds.
problem Burnside problem for homeomorphism groups of manifolds.
method Analyzing surface and circle homeomorphism groups, extending Tits alternative.
result Every finitely generated periodic subgroup is finite for most manifolds.
Study homeomorphism groups of ordinals, proving strong distortion and normal generators.
problem Understanding algebraic and geometric properties of homeomorphism groups of ordinals.
method Analyzing successor ordinals with connections to permutation groups and manifolds.
result Proves strong distortion and normal generators for homeomorphism groups of ordinals.
Uniform interpretation of group theory in manifold homeomorphisms.
problem Understanding group properties in manifold homeomorphisms.
method First order theory interpretation of second order group theory.
result Many group theory problems encoded in homeomorphism groups.
Study shows similar result to Margulis for Cantor set homeomorphisms.
problem Understanding groups of homeomorphisms of Cantor sets.
method Analogous to Margulis's proof for linear groups.
result Groups of homeomorphisms either preserve a measure or contain a free subgroup.
This thesis classifies pseudo-Anosov homeomorphisms using geometric Markov partitions.
problem Classifying pseudo-Anosov homeomorphisms up to topological conjugacy.
method Algorithmic approach using geometric Markov partitions.
result Geometric type is a complete invariant of conjugation.
Positive factorization for pseudoperiodic homeomorphisms on surfaces.
problem Factorization of pseudoperiodic homeomorphisms on surfaces.
method Generalization of classical results on smooth germs of surfaces, topological characterization of monodromies, and use of positive factorization criteria.
result Pseudoperiodic homeomorphisms on surfaces with positive fractional Dehn twist coefficients and screw numbers admit a positive factorization.
Homeomorphisms of hyperbolic 3-manifolds have invariant sets under certain conditions.
problem Understanding invariant sets for homeomorphisms on hyperbolic 3-manifolds.
method Shadowing techniques and analysis of invariant sets under positive speed with respect to a uniform foliation.
result Homeomorphisms of hyperbolic 3-manifolds are forced to have invariant sets, preventing them from being minimal.
We show that the Gromov boundary of the free product of two infinite hyperbolic groups is uniquely determined up to homeomorphism by the homeomorphism types of the boundaries of its factors. We generalize this result to graphs of hyperbolic groups over finite subgroups. Finally, we give a necessary and sufficient condi…
We show that the topological groups Diff+1(I) and Diff+1(S1) of orientation-preserving C1-diffeomorphisms of the interval and the circle, respectively, admit finitely generated dense subgroups. We also investigate the question of genericity (in the sense of Baire category) of such finite to…
The study explores homeomorphism groups of self-similar 2-manifolds, including the 2-sphere and Cantor set.
problem Understanding the structure and properties of homeomorphism groups of self-similar 2-manifolds.
method Survey of recent results, exposition of classical results, treatment of stable sets, and proof of new theorems.
result Characterization of homeomorphisms of perfectly self-similar 2-manifolds and extensions of existing results.
The notion of a locally continuously perfect group is introduced and studied. This notion generalizes locally smoothly perfect groups introduced by Haller and Teichmann. Next, we prove that the path connected identity component of the group of all homeomorphisms of a manifold is locally continuously perfect. The case o…
Extends Palais' result on diffeomorphisms to homeomorphisms and bi-Lipschitz mappings.
problem Extending diffeomorphisms to global mappings of manifolds.
method Elementary argument for diffeomorphisms, deep results for homeomorphisms and bi-Lipschitz mappings.
result Extension of Palais' result to homeomorphisms and bi-Lipschitz mappings.
The paper creates a deformation retraction for homeomorphisms of the projective plane.
problem Deformation retraction of homeomorphisms of the projective plane.
method Equivariant strong deformation retraction from homeomorphism group to special orthogonal group.
result Induces a SO(3)-equivariant strong deformation retraction from projective plane homeomorphisms to SO(3).
Generic homeos on complex manifolds have full metric mean dimension.
problem Understanding the metric mean dimension of generic homeomorphisms.
method Analyzing C0-generic homeomorphisms on compact manifolds. result The metric mean dimension equals the manifold's dimension.
Classifies when homeomorphism groups of stable surfaces have automatic continuity.
problem Determining when homeomorphism groups of stable surfaces are continuous.
method Developed a general framework to prove automatic continuity for homeomorphism groups, applied to stable surfaces and Stone spaces.
result Classification of stable surfaces with respect to automatic continuity of their homeomorphism groups.
We describe a circle of ideas relating the dynamics of 2-dimensional homeomorphisms to that of 1-dimensional endomorphisms. This is used to introduce a new class of maps generalizing that of Thurston's pseudo-Anosov homeomorphisms.
Study on homeomorphism groups of manifolds using set theory.
problem Relationship between set theory and homeomorphism groups of manifolds.
method First-order rigidity, type versus conjugacy, axiom of constructibility, projective determinacy.
result Under V=L, homeomorphism groups of manifolds are first-order rigid and conjugacy class is determined by type.
Locally approximating groups of homeomorphisms reveal manifold properties.
problem Understanding the structure and properties of homeomorphism groups on manifolds.
method Analyzing dense subgroups in Euclidean charts and interpreting first-order arithmetic.
result Locally approximating groups of homeomorphisms uniquely determine manifold properties.
Study on homeomorphism groups of telescoping 2-manifolds showing strong distortion.
problem Characterizing the homeomorphism group of telescoping 2-manifolds.
method Introduced telescoping 2-manifolds, studied homeomorphism groups, and used commutator subgroup properties.
result Homeomorphism group of telescoping 2-manifolds is strongly distorted.
Proves existence of sentences to identify homeomorphic manifolds.
problem Identifying homeomorphic manifolds using group properties.
method Defines sentences in group language to match homeomorphic manifolds.
result Existence of sentences to distinguish homeomorphic manifolds.
Study cohomology of homeomorphisms and diffeomorphisms of manifolds.
problem Determine bounded and unbounded cohomology of homeomorphism and diffeomorphism groups.
method Analyzing specific manifolds like the circle, 2-disc, and spheres.
result Identify the bounded cohomology of homeomorphisms and diffeomorphisms groups of certain manifolds.
We give some general criteria of being a homeomorphism for continuous mappings of topological manifolds, as well as criteria of being a diffeomorphism for smooth mappings of smooth manifolds. As an illustration, we apply these criteria to the problems arising in two- and three-dimensional grid generation.
New homeomorphism found in Klein bottle group.
problem Understanding homeomorphisms of Klein bottle.
method Using recent results on commutator length.
result Existence of homeomorphism with positive stable commutator length.
New connection found between complex polynomials and surface homeomorphisms.
problem Investigating the existence of generalized pseudo-Anosov maps from quadratic polynomials.
method Developed a new connection between dynamics of quadratic polynomials and surface homeomorphisms, focusing on Hubbard trees.
result Identified conditions for constructing generalized pseudo-Anosov maps from quadratic polynomials.
Study weak conjugacy in surface homeomorphisms.
problem Understanding weak conjugacy in homeomorphisms of surfaces.
method Exploring the group of homeomorphisms isotopic to the identity.
result New insights into weak conjugacy relations.
Paper explains dynamics of homeomorphisms to mapping tori geometry.
problem Understanding dynamics of end-periodic homeomorphisms.
method Illustration-driven overview of recent results.
result Analogue of Brock's theorem for infinite-type surfaces.
Bestvina and Handel have found an effective algorithm that determines whether a given homeomorphism of an orientable, possibly punctured surface is pseudo-Anosov. We present a software package in Java that realizes this algorithm for surfaces with one puncture. Moreover, the package allows the user to define homeomorph…
Two subset germs of Euclidean spaces are called blow-spherically equivalent, if their spherical modifications are homeomorphic and the homeomorphism induces homeomorphic tangent links. Blow-spherical equivalence is stronger than the topological equivalence but weaker than the Lipschitz equivalence. We introduce the thi…
In this paper we introduce a general notion of weak extension property for embeddings induced by a group actions. As an example, for the group H(M, m) of measure-preserving homeomorphisms of a noncompact manifold M, we deduce weak type extension theorems, and as an application we exhibit the local contractibility of th…
Unified framework recovers and improves classical Brouwer homeomorphism results.
problem Classical Brouwer homeomorphism theory and its dynamics.
method Unified foliated framework combining Le Calvez's and Handel's methods.
result Recovery and improvement of classical results in Brouwer homeomorphism theory.
Every homeomorphism of Euclidean space is a commutator of two homeomorphisms.
problem Understanding the structure of homeomorphisms in Euclidean space.
method Proving every orientation-preserving homeomorphism can be written as a commutator of two such homeomorphisms.
result Every orientation-preserving homeomorphism of Euclidean space is a commutator of two homeomorphisms.
Paper improves proof of theorem on convex 2-disk homeomorphisms.
problem Space of SL homeomorphisms of a convex 2-disk. method Improved proof of main lemma.
result Major improvement in understanding homeomorphisms of convex 2-disk.
We generalize the "hamiltonian topology" on hamiltonian isotopies to an intrinsic "symplectic topology" on the space of symplectic isotopies. We use it to define the group SSympeo(M,ω) of strong symplectic homeomorphisms, which generalizes the group Hameo(M,ω) of hamiltonian homeomorphisms introduced by Oh and Mull…
Adapts pivoting technique to circle homeomorphisms for proofs.
problem Probabilistic Tits alternative and exponential synchronization.
method Adapts Gou{ë}zel's pivoting technique.
result Different proofs of probabilistic Tits alternative and exponential synchronization.
No algorithm exists to decide 4-manifold homeomorphism.
problem Deciding homeomorphism for 4-manifolds.
method Demonstrated through a specific example of a connected sum of 12 copies of S^2 × S^2.
result No algorithm exists for deciding homeomorphism of 4-manifolds.
Proof shows homeomorphism problem is unsolvable.
problem Unsolvable homeomorphism problem in topology.
method Detailed proof of Markov's theorem.
result Existence of unrecognizable manifolds in higher dimensions.
Study covers of sphere with homeomorphisms lifting property.
problem Finite abelian covers of sphere with lifting homeomorphisms.
method Completely determined covers with specific lifting property.
result Properties of finite abelian covers with lifting homeomorphisms.
We construct a finitely presented group with infinitely many non-homeomorphic asymptotic cones. We also show that the existence of cut points in asymptotic cones of finitely presented groups does, in general, depend on the choice of scaling constants and ultrafilters.
Goldman bracket distinguishes surface homeomorphisms.
problem Characterizing homeomorphisms between non-compact surfaces.
method Using the Goldman bracket to distinguish homeomorphisms.
result A homotopy equivalence is a homeomorphism if it preserves the Goldman bracket.
Any two knots admit orientation preserving homeomorphic Seifert surfaces, as can be seen by stabilizing. There is a generalization of a Seifert surface to the setting of links called a C-complex. In this paper, we ask when two links will admit orientation preserving homeomorphic C-complexes. In the case of 2-component …
Study actions of homeomorphism groups on manifolds with specific properties.
problem Actions of homeomorphism groups on manifolds with nontrivial finite free actions.
method Analyzes the action of extHomeo0(M) on another manifold N under certain conditions. result If M is not an Fp-homology sphere, then N≅MimesK for a homology manifold K. Extends knot surgery to exotic four-manifolds.
problem Constructing exotic definite four-manifolds.
method Generalizes RBG construction to n-surgery. result Produces pairs of knots with same n-surgery. This paper determines all possible topological symmetry groups of generalized Petersen graphs.
problem Identifying all topological symmetry groups of generalized Petersen graphs.
method Analyzing embeddings of generalized Petersen graphs in S3 and considering homeomorphisms. result All groups that can be topological symmetry groups of generalized Petersen graphs are identified.
Let X be a path-connected topological space admitting a universal cover. Let Homeo(X,a) denote the group of homeomorphisms of X preserving degree one cohomology class a. We investigate the distortion in Homeo(X,a). Let g be an element of Homeo(X,a). We define a Nielsen-type equivalence relation on the space of g-invari…
Study shows certain surface homeomorphisms groups can't be precompact.
problem Understanding precompactness of groups of surface homeomorphisms.
method Analyzing subgroup properties and transitivity of homeomorphisms.
result No subgroup of surface homeomorphisms can be Roelcke precompact.
We prove that the set of non-properness of a polynomial mapping of the three dimensional space which is a local homeomorphism cannot be homeomorphic to the real line R.