Class Of Homomorphisms Research Articles

Presentations of mapping class groups and an application to cluster algebras from surfaces

In this paper, we give presentations of the mapping class groups of marked surfaces stabilizing boundaries for any genus. Note that in the existing works, the mapping class groups of marked surfaces were the isotopy classes of homeomorphisms fixing boundaries pointwise. The condition for stabilizing boundaries of mapping class groups makes the requirement for mapping class groups to fix boundaries pointwise to be unnecessary.As an application of presentations of the mapping class groups of marked surfaces stabilizing boundaries, we obtain the presentation of the cluster automorphism group of a cluster algebra from a feasible surface (S,M).Lastly, for the case (1) 4-punctured sphere, the cluster automorphism group of a cluster algebra from the surface is characterized. Since cluster automorphism groups of cluster algebras from those surfaces were given in [1] in the cases (2) the once-punctured 4-gon and (3) the twice-punctured digon, we indeed give presentations of cluster automorphism groups of cluster algebras from surfaces which are not feasible.

On Matrix Representations of Homeomorphism Classes

In this study, we investigate the matrix representations of homeomorphism classes. Considering well-known concepts such as the matrix of ones, column-sum, row-sum, one's complement, Hadamard product, and regular addition for matrices, we explore binary matrices' relationships with subsets of a finite set. The main results establish connections between matrix operations and set operations, providing insights into the structure of homeomorphism classes. The paper concludes with the formulation of a topology on a set based on specific matrix conditions.

The logarithmic Dirichlet Laplacian on Ahlfors regular spaces

We introduce the logarithmic analogue of the Laplace-Beltrami operator on Ahlfors regular metric-measure spaces. This operator is intrinsically defined with spectral properties analogous to those of elliptic pseudo-differential operators on Riemannian manifolds. Specifically, its heat semigroup consists of compact operators which are trace-class after some critical point in time. Moreover, its domain is a Banach module over the Dini continuous functions and every Hölder continuous function is a smooth vector. Finally, the operator is compatible, in the sense of noncommutative geometry, with the action of a large class of non-isometric homeomorphisms.

Circle Homeomorphisms with Square Summable Diamond Shears

Abstract We introduce and study the space of homeomorphisms of the circle (up to Möbius transformations), which are in $\ell ^{2}$ with respect to modular coordinates called diamond shears along the edges of the Farey tessellation. Diamond shears are related combinatorially to shear coordinates and are also closely related to the $\log \Lambda $-lengths of decorated Teichmüller space introduced by Penner. We obtain sharp results comparing this new class to the Weil–Petersson class and Hölder classes of circle homeomorphisms. We also express the Weil–Petersson metric tensor and symplectic form in terms of infinitesimal shears and diamond shears.

On exponential asymptotics of one class of homeomorphisms at a point of the complex plane

The presented paper investigates the exponential asymptotics at a point of the complex plane of the ring Q-homeomorphisms with respect to p-modulus for p>2. Examples showing sharpness of the obtained results have been constructed.

-Homemorphism In Topological Spaces

The perception of an -Generalized -closed sets in topological spaces and also derived their homeomorphism. Relationships are established between αGµ- homeomorphism in topological spaces containing the class of homeomorphisms.

Motion Groupoids and Mapping Class Groupoids

Here {underline{M}} denotes a pair (M, A) of a manifold and a subset (e.g. A=partial M or A=varnothing ). We construct for each {underline{M}} its motion groupoidtextrm{Mot}_{{underline{M}}}, whose object set is the power set {{mathcal {P}}}M of M, and whose morphisms are certain equivalence classes of continuous flows of the ‘ambient space’ M, that fix A, acting on {{mathcal {P}}}M. These groupoids generalise the classical definition of a motion group associated to a manifold M and a submanifold N, which can be recovered by considering the automorphisms in textrm{Mot}_{{underline{M}}} of Nin {{mathcal {P}}}M. We also construct the mapping class groupoidtextrm{MCG}_{{underline{M}}} associated to a pair {underline{M}} with the same object class, whose morphisms are now equivalence classes of homeomorphisms of M, that fix A. We recover the classical definition of the mapping class group of a pair by taking automorphisms at the appropriate object. For each pair {underline{M}} we explicitly construct a functor {textsf{F}}:textrm{Mot}_{{underline{M}}} rightarrow textrm{MCG}_{{underline{M}}}, which is the identity on objects, and prove that this is full and faithful, and hence an isomorphism, if pi _0 and pi _1 of the appropriate space of self-homeomorphisms of M are trivial. In particular, we have an isomorphism in the physically important case {underline{M}}=([0,1]^n, partial [0,1]^n), for any nin {mathbb {N}}. We show that the congruence relation used in the construction textrm{Mot}_{{underline{M}}} can be formulated entirely in terms of a level preserving isotopy relation on the trajectories of objects under flows—worldlines (e.g. monotonic ‘tangles’). We examine several explicit examples of textrm{Mot}_{{underline{M}}} and textrm{MCG}_{{underline{M}}} demonstrating the utility of the constructions.

On the Behavior of One Class of Homeomorphisms at Infinity

On the Behavior of One Class of Homeomorphisms at Infinity

Simplifying indefinite fibrations on 4-manifolds

The main goal of this article is to connect some recent perspectives in the study of 4 4 -manifolds from the vantage point of singularity theory. We present explicit algorithms for simplifying the topology of various maps on 4 4 -manifolds, which include broken Lefschetz fibrations and indefinite Morse 2 2 -functions. The algorithms consist of sequences of moves, which modify indefinite fibrations in smooth 1 1 -parameter families. These algorithms allow us to give purely topological and constructive proofs of the existence of simplified broken Lefschetz fibrations and Morse 2 2 -functions on general 4 4 -manifolds, and a theorem of Auroux–Donaldson–Katzarkov on the existence of certain broken Lefschetz pencils on near-symplectic 4 4 -manifolds. We moreover establish a correspondence between broken Lefschetz fibrations and Gay–Kirby trisections of 4 4 -manifolds, and show the existence and stable uniqueness of simplified trisections on all 4 4 -manifolds. Building on this correspondence, we also provide several new constructions of trisections, including infinite families of genus- 3 3 trisections with homotopy inequivalent total spaces, and exotic same genera trisections of 4 4 -manifolds in the homeomorphism classes of complex rational surfaces.

Geodesic flow on an intersection of several confocal quadrics in $\mathbb{R}^n$

By the Jacobi-Chasles theorem, for each geodesic on an $n$-axial ellipsoid in $n$-dimensional Euclidean space, apart from it there exist also $n-2$ quadrics confocal with it that are tangent to all the tangent lines of this geodesic. It is shown that this result also holds for the geodesic flow on the intersection of several nondegenerate confocal quadrics. As in the case of the Jacobi-Chasles theorem, this fact ensures the integrability of the corresponding geodesic flow. For each compact intersection of several nondegenerate confocal quadrics its homeomorphism class is determined, and it turns out that such an intersection is always homeomorphic to a product of several spheres. Also, a sufficient condition for a potential is presented which ensures that the addition of this potential preserved the integrability of the corresponding dynamical system on the intersection of an arbitrary number of confocal quadrics. Bibliography: 16 titles.

On a Classification of Periodic Maps on the 2-Torus

In this paper, following J. Nielsen, we introduce a complete characteristic of orientation-preserving periodic maps on the two-dimensional torus. All admissible complete characteristics were found and realized. In particular, each of the classes of orientation-preserving periodic homeomorphisms on the 2-torus that are nonhomotopic to the identity is realized by an algebraic automorphism. Moreover, it is shown that the number of such classes is finite. According to V. Z. Grines and A. Bezdenezhnykh, any gradient-like orientation-preserving diffeomorphism of an orientable surface is represented as a superposition of the time-1 map of a gradient-like flow and some periodic homeomorphism. Thus, the results of this work are directly related to the complete topological classification of gradient-like diffeomorphisms on surfaces.

Topological analysis of pseudo-Euclidean Euler top for special values of the parameters

An analogue of the Euler top is considered for a pseudo-Euclidean space is under consideration. In the cases when the geometric integral or area integral vanishes the bifurcation diagrams of the moment map are constructed and the homeomorphism class of each leaf of the Liouville foliation is determined. For each arc of the bifurcation diagram, for one of the two possible cases of the mutual arrangement of the moments of inertia, the types of singularities in the preimage of a small neighbourhood of this arc (analogues of Fomenko 3-atoms) are determined, and for nonsingular isoenergy and isointegral surfaces an invariant of rough Liouville equivalence (an analogue of a rough molecule) is constructed. The pseudo-Euclidean Euler system turns out to have noncompact noncritical bifurcations. Bibliography: 23 titles.

New Homeomorphism in topological spaces

In this paper a new class of homeomorphism called minimal Regular weakly homeomorphism and maximal Regular weakly homeomorphism are introduced and investigated and during this process some properties of the new concepts have been studied.

On harmonic homeomorphisms from the punctured unit disc onto its exterior

The purpose of this paper is to study the class of harmonic sense-preserving homeomorphisms from the punctured unit disc onto its planar exterior which extend homeomorphically between the unit circle and itself and admit a simple pole at the origin.

TWO-WEIGHTED COMPOSITION OPERATORS ON SOBOLEV SPACES AND QUASICONFORMAL ANALYSIS

We establish some equivalent conditions for a homeomorphism \(\varphi : D\rightarrow D'\) of Euclidean domains in \(\mathbb R^n\), \(n\ge 2\), to induce a bounded composition operator \(\varphi ^*: {L}^1_p(D';\omega ) \cap \text {Lip}_l(D')\rightarrow L^1_q(D;\theta )\), where \(1< q \le p<\infty\), by the composition rule: \(\varphi ^*(f)=f\circ \varphi\). Here \(\omega :D'\rightarrow (0,\infty )\) is an arbitrary weight function on the domain \(D'\), and \(\theta :D\rightarrow (0,\infty )\) is some weight function in Muckenhoupt’s \(A_{q}\)-class on the domain D. Moreover, we prove that the class of homeomorphisms under consideration is completely determined by the controlled variation of the weighted capacity of cubical condensers whose shells are concentric cubes.

Minimal direct products

A space is called minimal if it admits a minimal continuous selfmap. We give examples of metrizable continua $X$ admitting both minimal homeomorphisms and minimal noninvertible maps, whose squares $X\times X$ are not minimal, i.e., they admit neither minimal homeomorphisms nor minimal noninvertible maps, thus providing a definitive answer to a question posed by Bruin, Kolyada and the second author in 2003. (In 2018, Boronski, Clark and Oprocha provided an answer in the case when only homeomorphisms were considered.) Then we introduce and study the notion of product-minimality. We call a compact metrizable space $Y$ product-minimal if, for every minimal system $(X,T)$ given by a metrizable space $X$ and a continuous selfmap $T$, there is a continuous map $S\colon Y\to Y$ such that the product $(X\times Y,T\times S)$ is minimal. If such a map $S$ always exists in the class of homeomorphisms, we say that $Y$ is a homeo-product-minimal space. We show that many classical examples of minimal spaces, including compact connected metrizable abelian groups, compact connected manifolds without boundary admitting a free action of a nontrivial compact connected Lie group, and many others, are in fact homeo-product-minimal.

Distortion theorems for homeomorphic Sobolev mappings of integrable p-dilatations

"We study the distortion features of homeomorphisms of Sobolev class $W^{1,1}_{\rm loc}$ admitting integrability for $p$-outer dilatation. We show that such mappings belong to $W^{1,n-1}_{\rm loc},$ are differentiable almost everywhere and possess absolute continuity in measure. In addition, such mappings are both ring and lower $Q$-homeomorphisms with appropriate measurable functions $Q.$ This allows us to derive various distortion results like Lipschitz, H\""older, logarithmic H\""older continuity, etc. We also establish a weak bounded variation property for such class of homeomorphisms."

Stable diffeomorphism classification of some unorientable 4‐manifolds

Kreck's modified surgery theory reduces the classification of closed, connected 4-manifolds, up to connect sum with some number of copies of $S^2\times S^2$, to a series of bordism questions. We implement this in the case of unorientable 4-manifolds M and show that for some choices of fundamental groups, the computations simplify considerably. We use this to solve some cases in which $\pi_1(M)$ is finite of order 2 mod 4: under an assumption on cohomology, there are nine stable diffeomorphism classes for which M is pin$^+$, one stable diffeomorphism class for which M is pin$^-$, and four stable diffeomorphism classes for which M is neither. We also determine the corresponding stable homeomorphism classes.

Torus Actions on Alexandrov 4-Spaces

We obtain an equivariant classification for orientable, closed, four-dimensional Alexandrov spaces admitting an isometric torus action. This generalizes the equivariant classification of Orlik and Raymond of closed four-dimensional manifolds with torus actions. Moreover, we show that such Alexandrov spaces are equivariantly homeomorphic to 4-dimensional Riemannian orbifolds with isometric {T^{2}}-actions. We also obtain a partial homeomorphism classification.

Topological classification of billiards bounded by confocal quadrics in three-dimensional Euclidean space

We study billiards on compact connected domains in bounded by a finite number of confocal quadrics meeting in dihedral angles equal to . Billiards in such domains are integrable due to having three first integrals in involution inside the domain. We introduce two equivalence relations: combinatorial equivalence of billiard domains determined by the structure of their boundaries, and weak equivalence of the corresponding billiard systems on them. Billiard domains in are classified with respect to combinatorial equivalence, resulting in 35 pairwise nonequivalent classes. For each of these obtained classes, we look for the homeomorphism class of the nonsingular isoenergy 5-manifold, and we show this to be one of three types: either , or , or . We obtain 24 classes of pairwise nonequivalent (with respect to weak equivalence) Liouville foliations of billiards on these domains restricted to a nonsingular energy level. We also define bifurcation atoms of three-dimensional tori corresponding to the arcs of the bifurcation diagram. Bibliography: 59 titles.