Mapping Class Research Articles

Nielsen fixed point theory for split n-valued maps on the Klein bottle

In this work we begin the study of n-valued maps on the Klein bottle, denoted by K, where we focus on the split ones. We provide an explicit description of Pn(K) (the n-th pure braid group of K) as an iterated semi-direct product of the form Fn⋊θn−1(⋯⋊(F2⋊θ1π1(K))), where the Fi′s are free groups on i letters. Given a split n-valued map Φ={f1,…,fn} its pointed homotopy class is determined by a pair of braids in Pn(K). We also provide a formula for N(Φ), the Nielsen number of Φ, which is completely determined by two braids, which in turn also determine the homotopy classes of the functions fi′s. If Φ={f1,f2} is a 2-valued map with N(Φ)=0, we show that there exists at least one 2-valued map Φ′={f1′,f2′}, such that Φ′ is fixed point free and for i=1,2 it holds that [fi]=[fi′], where [] denotes a pointed homotopy class of maps. Finally, we display an infinite family of pointed homotopy classes in [K,F2(K)], such that N(ϕ)=0, for any ϕ in the family. Furthermore, the map from [K,F2(K)] to [K,K×K], induced by the inclusion F2(K)→K×K, takes this family to one single element in [K,K×K]. We do not know if these 2-valued maps of the family can be deformed to fixed point free maps.

CFT Correlators and Mapping Class Group Averages

Mapping class group averages appear in the study of 3D gravity partition functions. In this paper, we work with 3D topological field theories to establish a bulk-boundary correspondence between such averages and correlators of 2D rational CFTs whose chiral mapping class group representations are irreducible and satisfy a finiteness property. We show that Ising-type modular fusion categories satisfy these properties on surfaces with or without field insertions, extending results in [1], and we comment on the absence of invertible global symmetries in the examples we consider.

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.

Workers Reconstituting the Factory

Abstract This comment on Moritz Altenried's The Digital Factory discusses how the book offers four interrelated theoretical contributions to the study of labour in the digital economy – redefining the factory, specifying digital Taylorism, materializing its infrastructure, and mapping class relations – through four sites of investigation. The piece discusses the implications of the resulting multiplication of labour and labour relations for reconfigured class relations and resistance and argues that the differentiated social relations across spatial and material contexts ask for a theorization of the conjunctural nature of these relations.

Asymptotic mapping class groups of Cantor manifolds and their finiteness properties (with an appendix by Oscar Randal-Williams)

We prove that the infinite family of asymptotic mapping class groups of surfaces defined by Funar–Kapoudjian and Aramayona–Funar are of type F_{\infty} , thus answering a problem of Funar–Kapoudjian–Sergiescu and a question of Aramayona–Funar. This result is a specific case of a more general theorem which allows us to deduce that asymptotic mapping class groups of certain Cantor manifolds , also introduced in this paper, are of type F_{\infty} . As important examples, we obtain type F_{\infty} asymptotic mapping class groups that contain, respectively, the mapping class group of every compact surface with non-empty boundary, the automorphism group of every free group of finite rank, or infinite families of arithmetic groups.In addition, for certain types of manifolds, the homology of our asymptotic mapping class groups coincides with the stable homology of the relevant mapping class groups, as studied by Harer and Hatcher–Wahl.

Constraining mapping class group homomorphisms using finite subgroups

We classify homomorphisms from mapping class groups by using arguments involving finite subgroups. First, we give a new proof of a result of Aramayona–Souto that all homomorphisms between certain mapping class groups of closed surfaces are trivial. Second, we show that only finitely many mapping class groups of closed surfaces have nontrivial homomorphisms to Homeo(Sn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {Homeo}(\\mathbb {S}^n)$$\\end{document} for any n, where Sn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {S}^n$$\\end{document} is the n-sphere. We also effectivize this result for small values of n; for instance, we prove that every homomorphism from Mod(Sg)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {Mod}(S_g)$$\\end{document} to Homeo(S2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {Homeo}(\\mathbb {S}^2)$$\\end{document} or Homeo(S3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {Homeo}(\\mathbb {S}^3)$$\\end{document} is trivial if g≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g\\ge 3$$\\end{document}, extending a result of Franks–Handel.

Correction: Monodromy and mapping class groups of 3-dimensional hypersurfaces

Correction: Monodromy and mapping class groups of 3-dimensional hypersurfaces

On a subclass of alpha-quasi convex mappings

In this paper, we introduce the class of α-quasi convex mappings on the unit ball in a complex Banach space, with the aim of providing a unified approach to the Fekete-Szegö problem for subclasses of normalized starlike mappings and normalized quasi-convex mappings of type B on the open unit ball B of a complex Banach space. In particular, using some restrictive assumptions, we investigate inclusion relations among the classes of alpha-quasi convex mappings, starlike mappings and quasi-convex mappings of type B.

A class of Schwarz qubit maps with diagonal unitary and orthogonal symmetries

Abstract A class of unital qubit maps displaying diagonal unitary and orthogonal symmetries is analyzed. Such maps have already found a lot applications in quantum information theory. We provide a complete characterization of this class of maps showing intricate relation between positivity, operator Schwarz inequality, and complete positivity. Finally, it is shown how to generalize the entire picture beyond unital case (so called generalized Schwarz maps). Interestingly, the first example of Schwarz but not completely positive map found by Choi belongs to our class. As a case study we provide a full characterization of Pauli maps. Our analysis leads to generalization of seminal Fujiwara–Algoet conditions for Pauli quantum channels.

Extreme fractal dimension at periodicity cascades in parameter spaces.

In the parameter spaces of nonlinear dynamical systems, we investigate the boundaries between periodicity and chaos and unveil the existence of fractal sets characterized by a singular fractal dimension that deviates greatly from the fractal sets in their vicinity. This extreme fractal dimension stands out from the typical value previously considered universal for these parameter boundaries. We show that such singular fractal sets dwell along parameter curves, called extreme curves, that intersect periodicity cascades at their centers of stability across all scales of parameter spaces. The results reported here are generally demonstrated for the class of one-dimensional maps with at least two control parameters. Generalizations to other classes of systems are possible.

Structures Associated with the Borromean Rings’ Complement in the Poincaré Ball

Guided by physical needs, we deal with the rotationally isotropic Poincar´e ball, when considering the complement of Borromean rings embedded in it. We consistently describe the geometry of the complement and realize the fundamental group as isometry subgroup in three dimensions. Applying this realization, we reveal normal stochastization and multifractal behavior within the examined model of directed random walks on the rooted Cayley tree, whose sixbranch graphs are associated with dendritic polymers. According to Penner, we construct the Teichm¨uller space of the decorated ideal octahedral surface related to the quotient space of the fundamental group action. Using the conformality of decoration, we define six moduli and the mapping class group generated by cyclic permutations of the ideal vertices. Intending to quantize the geometric area, we state the connection between the induced geometry and the sine-Gordon model. Due to such a correspondence we obtain the differential two-form in the cotangent bundle of the moduli space.

Stable twisted cohomology of the mapping class groups in the unit tangent bundle homology

AbstractWe compute the stable cohomology groups of the mapping class groups of compact orientable surfaces with one boundary, with twisted coefficients given by the rational homology of the unit tangent bundles of the surfaces. These coefficients define a covariant functor over the classical category associated to mapping class groups, rather than a contravariant one, and are thus out of the scope of the traditional framework to study twisted cohomological stability. A remarkable property is that the computed stable twisted cohomology is not free as a module over the stable cohomology algebra with constant rational coefficients. For comparison, we also compute the stable cohomology group with coefficients in the first rational cohomology of the unit tangent bundle of the surface, which fits into the traditional framework.

Morse subsets of injective spaces are strongly contracting

We show that a quasi-geodesic in an injective metric space is Morse if and only if it is strongly contracting. Since mapping class groups and, more generally, hierarchically hyperbolic groups act properly and coboundedly on injective metric spaces, we deduce various consequences relating, for example, to growth tightness and genericity of pseudo-Anosovs/Morse elements. Moreover, we show that injective metric spaces have the Morse local-to-global property and that a non-virtually cyclic group acting properly and coboundedly on an injective metric space is acylindrically hyperbolic if and only if contains a Morse ray. We also show that strongly contracting geodesics of a space stay strongly contracting in the injective hull of that space.

Non-realizability of some big mapping class groups

In this note, we prove that the compactly supported mapping class group of a surface containing a genus 3 3 subsurface has no realization as a subgroup of the homeomorphism group. We also prove that for certain surfaces with order 6 6 symmetries, their mapping class groups have no realization as a subgroup of the homeomorphism group. Examples of such surfaces include the plane minus a Cantor set and the sphere minus a Cantor set.

Fixed Point Results for New Classes of k-Strictly Asymptotically Demicontractive and Hemicontractive Type Multivalued Mappings in Symmetric Spaces

Fixed point theory is a significant area of mathematical analysis with applications across various fields such as differential equations, optimization, and dynamical systems. Recently, multivalued mappings have gained attention due to their ability to model more complex and realistic problems. ln this work, novel classes of nonlinear mappings called k-strictly asymptotically demicontractive-type and asymptotically hemicontractive-type multivalued mappings are introduced in real Hilbert spaces that are symmetric spaces. In addition, we discuss the weak and strong convergence results by considered modified algorithms, and a demiclosedness property, for these classes of mappings are proved. Several non-trivial examples are demonstrated to validate the newly defined mappings. Consequently, the results and iterative methods obtained in this study improve and extend several known outcomes in the literature.

Snapback Repellers, Computational Chaos and Extreme Multistability in Discrete-Time Memristor Murali–Lakshmanan–Chua Circuit

Discretization schemes such as Euler method and Runge–Kutta techniques are extensively used to find approximate solutions of Continuous-Time (CT) dynamical system. While the approximation is good for small discretization step sizes, as pointed out by Lorenz, when the step size increases, computational chaos and computational instability are frequently observed, the former phenomenon being a precursor to the latter. By computational instability, it is meant that there is a blow up of trajectories for the Discrete-Time (DT) system. Computational chaos instead means that for certain step sizes, the DT system displays chaos while the CT counterpart is not chaotic. This paper studies the dynamics of a class of second-order maps obtained via the discretization of a Memristor Murali–Lakshmanan–Chua Circuit (MMLCC). The discretization, which is based on the DT Flux–Charge Analysis Method (FCAM), guarantees that the first integrals of a CT-MMLCC are preserved exactly for the DT system. Hence the dynamics of DT-MMLCC evolves on invariant manifolds and it is characterized by the coexistence of infinitely many different attractors (extreme multistability). The paper uses analytic techniques introduced by Marotto, based on the concept of snapback repellers and transverse homoclinic orbits, to study the chaotic behaviors of the maps. Regions of the parameter space are singled out where there exist snapback repellers for DT-MMLCC, thus implying that the maps display chaos in the Marotto and in the Li–Yorke sense. Since the corresponding CT-MMLCC does not display chaos, the observed chaos of DT-MMLCC is genuinely a consequence of the discretization scheme used in the paper, i.e. it can be actually considered as computational chaos. It is also verified that computational chaos is a precursor to computational instability of the DT-MMLCC maps. Finally, the paper analyzes the effect on chaos obtained by changing the invariant manifold where the dynamics evolves.

Dynamic properties of the dynamical system [formula omitted]

Let (X,f) be a dynamical system, where X is a nondegenerate continuum and f is a map. For any positive integer n, we consider the hyperspace Fn(X) with the Vietoris topology. For n>1 and K∈Fn(X) the subset Fn(K,X) of Fn(X) is defined as the collection of elements of Fn(X) containing K. We consider the quotient hyperspace FnK(X)=Fn(X)⧸Fn(K,X), which is obtained from Fn(X) by shrinking Fn(K,X) to one point set. Furthermore, we consider the induced maps Fn(f):Fn(X)→Fn(X) and FnK(f):FnK(X)→FnK(X). In this paper, we introduce the dynamical system (FnK(X),FnK(f)) and we study relationships between the conditions f∈M, Fn(f)∈M and FnK(f)∈M, where M is one of the following classes of maps: transitive, mixing, weakly mixing, totally transitive, exact, exact in the sense of Akin-Auslander-Nagar, strongly transitive in the sense of Akin-Auslander-Nagar, exact transitive, fully exact, strongly exact transitive, strongly product transitive, orbit-transitive, Devaney chaotic, irreducible, TT++, strongly transitive and very strongly transitive.

Monodromy and mapping class groups of 3-dimensional hypersurfaces

AbstractWe describe the subgroup of the mapping class group of a hypersurface in $$\mathbb{C}\mathbb{P}^4$$ C P 4 consisting of those diffeomorphisms which can be realised by monodromy.

The Generalized Toeplitz Determinants for a Class of Holomorphic Mappings in Several Complex Variables

The Generalized Toeplitz Determinants for a Class of Holomorphic Mappings in Several Complex Variables

Induced mappings on the hyperspace of totally disconnected sets

Abstract The symbol TD(X) denotes the hyperspace of all nonempty totally disconnected compact subsets of a Hausdorff space X. This hyperspace is endowed with the Vietoris topology. For a mapping between Hausdorff spaces f : X → Y, define the induced mapping TD(f) : TD(X) → TD(Y) by TD(f)(A) = f(A) (the image of A under f). In the current paper, we study the relationships between the condition f belongs to a class of mappings between Hausdorff spaces 𝕄 and the condition TD(f) belongs to 𝕄.