arXiv research

Universal spaces for finite topological spaces simplify shape descriptions.

Distance function to a finite set is a topological Morse function.

In this paper, we study the critical case of the Allard regularity theorem. Combining with Reifenberg's topological disk theorem, we get a critical Allard-Reifenberg type regularity theorem. As a main result, we get the topological finiteness for a class of properly immersed surfaces in $\mathbb{R}^n$ with finite Willm…

Let G be a lattice in PSL(2,C). The pro-normal topology on G is defined by taking all cosets of non-trivial normal subgroups as a basis. This topology is finer than the pro-finite topology, but it is not discrete. We prove that every finitely generated subgroup H<G is closed in the pro-normal topology. As a corollary w…

3-manifolds study Hasse norm principle, akin to number fields.

Finite-type surfaces have a topological Hopf property.

Study uses equivariant topology to measure distances between G metric spaces.

We prove:(1) the existence, for every integer n > 3, of a noncompact smooth n-dimensional topological manifold whose diffeomorphism group contains an isomorphic copy of every finitely presented group; (2) a finiteness theorem on finite simple subgroups of diffeomorphism groups of compact smooth topological manifolds.

This thesis introduces big mapping class groups and their structure.

Locally flat submanifolds have finite CW complex complements.

Generic smooth plane-to-plane map germs are topologically equivalent to cones of mappings of the circle. We carry out a complete topological classification of smooth stable mappings of the circle and show how this classification leads, via the result mentioned above, to a topological classification of finitely determin…

Minimal topology on surface homeomorphisms proven.

We prove there exists a compact embedded minimal surface in a complete finite volume hyperbolic $3$-manifold $\mathcal{N}$. We also obtain a least area, incompressible, properly embedded, finite topology, $2$-sided surface. We prove a properly embedded minimal surface of bounded curvature has finite topology. This dete…

The sinh-Gordon equation is solved on finite, symmetric graphs.

The paper proves finite topological type theorems for open manifolds with non-negative Ricci curvature and almost maximal local rewinding volume.

Let $S$ be an orientable, connected surface with infinitely-generated fundamental group. The main theorem states that if the genus of $S$ is finite and at least 4, then the isomorphism type of the pure mapping class group associated to $S$, denoted $\mathrm{PMap}(S)$, detects the homeomorphism type of $S$. As a corolla…

We present sufficient conditions for topological stability of continuous functions $f:\mathbb{R}\to\mathbb{R}$ having finitely many local extrema with respect to averagings by discrete measures with finite supports.

Finite presentation for a specific group in 3D handlebody topology.

Finite type invariants separate PL links in 3D space.

No CMC surfaces exist in certain hyperbolic 3-manifolds.

The study classifies translating and self-expanding solitons in 3D space.

We show that a complete Riemannian manifold has finite topological type (i.e., homeomorphic to the interior of a compact manifold with boundary), provided its Bakry-Émery Ricci tensor has a positive lower bound, and either of the following conditions: (i) the Ricci curvature is bounded from above; (ii) the Ricci curvat…

Proof of genus formula for 3-manifolds using arithmetic topology.

Magnitude is not continuous but may be stable for most finite metric spaces.

The standard actions of finite groups on spheres S^d are linear actions, i.e. by finite subgroups of the orthogonal group O(d+1). We prove that, in each dimension d>5, there is a finite group G which admits a faithful, topological action on a sphere S^d but is not isomorphic to a subgroup of O(d+1). The situation remai…

The paper proves UV finiteness and vanishing anomalies for hybrid topological-holomorphic field theories.

Finite simplicial complexes dominate certain manifolds with a bounded number of simplices.

Abstract commensurators linked to topological models of solenoids.

Only vertical planes are asymptotic to other planes in 3D space.

We present and prove a topological characterization of geodesic laminations on hyperbolic surfaces of finite type.

The paper studies strong topologies for complex Monge-Ampère equations on Kähler manifolds.

The study examines conditions for minimal volume entropy of simplicial complexes.

We prove that compact topological 4-manifolds can be effectively presented by a finite amount of data.

Minimal volume entropy vanishes or is positive under certain fiber growth conditions.

We consider orientation-preserving actions of a finite group G on the 3-sphere S^3 (and also on Euclidean space R^3). By the geometrization of finite group actions on 3-manifolds, if such an action is smooth then it is conjugate to an orthogonal action, and in particular G is isomorphic to a subgroup of the orthogonal …

A new topology improves decentralized learning efficiency and accuracy.

We prove that for every nonnegative integer $g$, there exists a bound on the number of ends of a complete, embedded minimal surface $M$ in $\mathbb{R}^3$ of genus $g$ and finite topology. This bound on the finite number of ends when $M$ has at least two ends implies that $M$ has finite stability index which is bounded …

In this paper we prove that a complete, embedded minimal surface $M$ in $\mathbb{R}^3$ with finite topology and compact boundary (possibly empty) is conformally a compact Riemann surface $\overline{M}$ with boundary punctured in a finite number of interior points and that $M$ can be represented in terms of meromorphic …

The paper explores mapping class groups and their quantum field theory representations.

Two flows are topologically almost commensurable if, up to removing finitely many periodic orbits and taking finite coverings, they are topologically equivalent. We prove that all suspensions of automorphisms of the 2-dimensional torus and all geodesic flows on unit tangent bundles to hyperbolic 2-orbifolds are pairwis…

Category theory generalizes finite type invariants using diagrams systems.

For a compact smooth manifold $M$ (with boundary) we prove that the topological rank of the diffeomorphism group Diff$_0^k(M)$ is finite for all $k\geq 1$. This extends a result from [2] where the same claim is proved in the special case of dim M = k = 1.

We show that the topological complexity of a finitely generated torsion free hyperbolic group $π$ with $\cdπ=n$ equals $2n$.

Study shows automorphism groups of certain hyperbolic manifolds are infinitely generated.

We construct a simple finite-dimensional topological quantum field theory for compact 3-manifolds with triangulated boundary.

This survey paper concerns mainly with some asymptotic topological properties of finitely presented discrete groups: quasi-simple filtration (QSF), geometric simple connectivity (GSC), topological inverse-representations, and the notion of easy groups. As we will explain, these properties are central in the theory of d…

In this paper we shall establish that properly embedded constant mean curvature one surfaces in H^3 of finite topology are of finite total curvature and each end is regular. In particular, this implies the horosphere is the only simply connected such example, and the catenoid cousins the only annular examples of this n…

Proofs show finite subgroups of homeomorphism groups are almost nilpotent.