arXiv research

Study on homeomorphism groups of manifolds using set theory.

Generates special homeomorphisms for complex surfaces.

Classifies Morse boundaries of 3-manifold groups.

Paper explains dynamics of homeomorphisms to mapping tori geometry.

This thesis classifies pseudo-Anosov homeomorphisms using geometric Markov partitions.

Study covers of surfaces, showing types and properties.

Study finds infinitely many surface bundle types with zero signature.

In this paper, based upon the basic theory for glued manifolds in M.W. Hirsch (1976) \cite[Chapter 8, §2 Gluing Manifolds Together]{h}, we give a method of constructing homeomorphisms between two small covers over simple convex polytopes. As a result we classify, up to homeomorphism, all small covers over a 3-dimension…

Countable modular groups found on surfaces with infinite type.

Finite-type surfaces have a topological Hopf property.

Study of homeomorphisms on infinite type surfaces with a classification theorem.

We consider closed topological 4-manifolds $M$ with universal cover ${S^2\times{S^2}}$ and Euler characteristic $χ(M) = 1$. All such manifolds with $π=π_1(M)\cong {\mathbb Z}/4$ are homotopy equivalent. In this case, we show that there are four homeomorphism types, and propose a candidate for a smooth example which is …

Every weak Perron number is realized as a stretch factor of a homeomorphism on a surface.

We give three sufficient criteria for two quasitoric manifolds (M,M') to be (weakly) equivariantly homeomorphic. We apply these criteria to count the weakly equivariant homeomorphism types of quasitoric manifolds with a given cohomology ring.

We shall show that for a given homeomorphism type and a set of end invariants (including the parabolic locus) with necessary topological conditions which a topologically tame Kleinian group with that homeomorphism type must satisfy, there is an algebraic limit of minimally parabolic, geometrically finite Kleinian group…

In this paper we investigate the relationship between isotopy classes of knots and links in S^3 and the diffeomorphism types of homeomorphic smooth 4-manifolds. As a corollary of this initial investigation, we begin to uncover the surprisingly rich structure of diffeomorphism types of manifolds homeomorphic to the K3 s…

Let $φ:\mathcal{C}(S)\to\mathcal{C}(S')$ be a simplicial isomorphism between the curve graphs of two infinite-type surfaces. In this paper we show that in this situation $S$ and $S'$ are homeomorphic and $φ$ is induced by a homeomorphism $h:S\to S'$.

For each $n > 0$ there is a one complex parameter family of homeomorphisms of the circle consisting of linear fractional transformations `conjugated by $z \to z^n$'. We show that these families are free of relations, which determines the structure of `the group of homeomorphisms of finite type'. We also discuss a numbe…

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…

Classifies specific types of Lorentzian manifolds with unipotent holonomy.

We prove that for any orientable connected surface of finite type which is not a a sphere with at most four punctures or a torus with at most two punctures, any homeomorphism of the space of geodesic laminations of this surface, equipped with the Thurston topology, is induced by a homeomorphism of the surface.

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…

We give a proof of the Neilsen-Thurston classification theorem of a homeomorphism f of a standard surface of finite type as either periodic, pseudo-Anosov, or reducible. In the periodic case, we show that there exists an integer n>0 such that f is isotopic to h with h^n isotopic to the identity. This is the weaker vers…

In this note we prove that equivariantly homeomorphic quasitoric manifolds are diffeomorphic. As a consequence we show that up to finite ambiguity the diffeomorphism type of certain quasitoric manifolds $M$ is determined by their cohomology rings and first Pontrjagin classes.

We show that any collection of n-dimensional orbifolds with sectional curvature and volume uniformly bounded below, diameter bounded above, and with only isolated singular points contains orbifolds of only finitely many orbifold homeomorphism types. This is a generalization to the orbifold category of a similar result …

For Gamma a finite, connected metric graph, we consider the space of configurations of n points in Gamma with a restraint parameter r dictating the minimum distance allowed between each pair of points. These restricted configuration spaces come up naturally in topological robotics. In this paper, we study the homotopy,…

Suppose M is a noncompact connected PL 2-manifold and let H(M)_0 denote the identity component of the homeomorphism group of M with the compact-open topology. In this paper we classify the homotopy type of H(M)_0 by showing that {\cal H}(M)_0 has the homotopy type of the circle if M is the plane, an open or half open a…

We show that any isomorphism between mapping class groups of orientable infinite-type surfaces is induced by a homeomorphism between the surfaces. Our argument additionally applies to automorphisms between finite-index subgroups of these `big' mapping class groups and shows that each finite-index subgroup has finite ou…

New examples of manifolds in tangent cones of non-collapsed Ricci limit spaces.

The paper finds linked periodic orbits in disc homeomorphisms using braids.

We prove that (apart from dimension $n=4$), each Riemannian solenoidal lamination with transitive homeomorphism group and leaves isometric to a symmetric space $X$ of noncompact type, is homeomorphic to the inverse limit of the system of finite covers of a compact locally-symmetric $n$-manifold.

We introduce a general procedure called `reverse engineering' that can be used to construct infinite families of smooth 4-manifolds in a given homeomorphism type. As one of the applications of this technique, we produce an infinite family of pairwise nondiffeomorphic 4-manifolds homeomorphic to CP^2#3(-CP^2).

Upper bound on 3-manifold volumes from surface homeomorphisms.

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…

Study constraints on diffeomorphisms and homeomorphisms of 4-manifolds with boundary.

Construct symplectic Lefschetz fibrations with any signature and spin type.

Study arcs on surfaces, focusing on topological aspects and group actions.

We prove that a class of asymptotically nonnegatively curved manifolds (in the sense of Abresch) satisfying some uniform Euclidean type volume growth conditions contains only finitely many homeomorphism types.

Suppose M is a noncompact connected PL 2-manifold. In this paper we study the topological property of the triple (H(M)_0, H^PL(M)_0, H^PL, c(M)_0), where H(M)_0 is the identity component of the homeomorphism group {\cal H}(M) of M with the compact-open topology, and H^PL(M)_0 and H^PL, c(M)_0 are the identity component…

Suppose $X$ and $Y$ are compact connected topological 4-manifolds with fundamental group $π$. For any $r \geqslant 0$, $Y$ is $r$-stably homeomorphic to $X$ if $Y \# r(S^2 \times S^2)$ is homeomorphic to $X \# r(S^2\times S^2)$. How close is stable homeomorphism to homeomorphism? When the common fundamental group $π$ i…

Any quasi-isometry of the complex of curves is bounded distance from a simplicial automorphism. As a consequence, the quasi-isometry type of the curve complex determines the homeomorphism type of the surface.

Proofs show finite subgroups of homeomorphism groups are almost nilpotent.

We define a new finite type invariant for stably homeomorphic class of curves on compact oriented surfaces without boundaries and extend to a regular homotopy invariant for spherical curves.

The long-standing problem of the perfectness of the compactly supported equivariant homeomorphism group on a $G$-manifold (with one orbit type) is solved in the affirmative. The proof is based on an argument different than that for the case of diffeomorphisms. The theorem is a starting point for computing $H_1(\mathcal…

The study explores homeomorphism groups of self-similar 2-manifolds, including the 2-sphere and Cantor set.

The goal of this paper is to demonstrate that, at least for nonsimply connected 4-manifolds, the Seiberg-Witten invariant alone does not determine diffeomorphism type within the same homeomorphism type.

Suppose M is a noncompact connected 2-manifold and m is a good Radon measure of M with m(partial M) = 0. Let H(M)_0 denote the identity component of the group of homeomorphisms of M equipped with the compact-open topology and let H(M; m)_0 denote the identity component of the subgroup consisting of m-preserving homeomo…

We determine the homeomorphism type of the space of smooth complete nonnegatively curved metrics on surfaces of positive Euler characteristic equipped with the topology of $C^γ$ uniform convergence on compact sets, when $γ$ is infinite or is not an integer. If $γ=\infty$, the space of metrics is homeomorphic to the sep…