Finite-type surfaces have a topological Hopf property.
problem Characterizing surfaces with a topological Hopf property.
method Using topological analogs of the Hopf property.
result Infinite-type surfaces do not have the Hopf property.
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48 results for “finite degree property”
Large and iterated finite group actions on manifolds admitting non-zero degree maps to nilmanifoldsmath.GT
Study finite group actions on manifolds with non-zero degree maps to nilmanifolds.
problem Understanding finite group actions on manifolds with specific properties.
method Analyzes effective actions, bounds discrete degrees, studies toral rank conjecture, and introduces iterated actions.
result Proves Homeo(M) is Jordan and bounds the discrete degree of symmetry.
New computations show various properties of bounded cohomology in finitely presented groups.
problem Understanding bounded cohomology properties in finitely presented groups.
method Computational and theoretical analysis of bounded cohomology.
result Existence of finitely presented non-amenable boundedly acyclic groups and groups with uncountable bounded cohomology.
Injectivity result for light ray transform on Lorentzian manifolds.
problem Injectivity of light ray transform on Lorentzian manifolds.
method Explicit relationship between geodesic and magnetic vector fields.
result Injectivity up to natural obstruction under certain conditions.
The paper finds new graph covers with exceptionally low degree.
problem Graph covers with primitive homology not spanned by lifts.
method Focused on finite -group deck groups, developed a character table-based algorithm.
result Found the smallest known nontrivial examples with covering degree 128.
Two new polynomial invariants for long virtual knots.
problem Defining new polynomial invariants for long virtual knots.
method Introducing and , establishing properties, and showing realizability.
result First derivatives of and at define finite type invariants of degree three.
In this paper we study infinitesimal and finite flexibility for generic semidiscrete surfaces. We prove that generic 2-ribbon semidiscrete surfaces have one degree of infinitesimal and finite flexibility. In particular we write down a system of differential equations describing isometric deformations in the case of exi…
The paper studies random covers of torus knot complements and their statistical properties.
problem Understanding the statistical behavior of finite covers of torus knot complements.
method Asymptotic subgroup growth analysis and Benjamini-Schramm limit theorems.
result Determination of the linear growth rate of Betti numbers for random covers of torus knot complements.
Graphons of Line Graphsstat.ML
The paper explores graphons of line graphs from sparse finite graphs.
problem Estimating graph limits from sparse finite graphs.
method Mapping finite graphs to their line graphs and analyzing graphs with the square-degree property.
result Graphons of line graphs can distinguish between sparse graphs like star graphs and superlinear preferential attachment graphs.
The study explores mapping degree sets and their properties for manifolds.
problem Understanding the structure and properties of mapping degree sets for manifolds.
method Analyzes the properties of mapping degree sets and their relationships with self-mapping degree sets.
result Not every multiplicative set containing 0,1 is a self-mapping degree set.
Every closed orientable surface S has the following property: any two connected covers of S of the same degree are homeomorphic (as spaces). In this, paper we give a complete classification of compact 3-manifolds with empty or toroidal boundary which have the above property. We also discuss related group-theoretic ques…
Study Kazdan-Warner equations on graphs using Brouwer degree theory.
problem Proving existence of solutions to Kazdan-Warner equations on finite graphs.
method Degree theory approach to uniformly bound and compute Brouwer degree.
result New proofs of existence results for Kazdan-Warner equations.
The aim of this paper is twofold. On the one hand, it provides a review of the links between random tensor models, seen as quantum gravity theories, and the PL-manifolds representation by means of edge-colored graphs (crystallization theory). On the other hand, the core of the paper is to establish results about the to…
We introduce the class of perturbed right-angled Artin groups. These are constructed by gluing Bieri double groups into standard right-angled Artin groups. As a first application of this construction we obtain families of CAT(0) groups containing finitely presented subgroups which are not of type , and h…
Homotopy classes of nanowords and nanophrases are combinatorial generalizations of virtual knots and links. Goussarov, Polyak and Viro defined finite type invariants for virtual knots and links via semi-virtual crossings. We extend their definition to nanowords and nanophrases. We study finite type invariants of low de…
In this paper, it is shown that every orientable closed 3-manifold maps with nonzero degree onto at most finitely many homeomorphically distinct irreducible non-geometric orientable closed 3-manifolds. Moreover, given any nonzero integer, as a mapping degree up to sign, every orientable closed 3-manifold maps with that…
Homology growth of specific mapping tori vanishes for certain groups.
problem Homology growth of polynomially growing mapping tori in various groups.
method Proof of the cheap rebuilding property for specific groups.
result Torsion homology growth vanishes for Farber sequences in every degree.
Paper solves whether zero sets are mapping degree sets.
problem Whether finite sets containing zero are mapping degree sets.
method Examined oriented closed connected manifolds of the same dimension.
result Affirmative answer given for both integer and rational settings.
In this paper, we give a complete set of finite type string link invariants of degree <5. In addition to Milnor invariants, these include several string link invariants constructed by evaluating knot invariants on certain closure of (cabled) string links. We show that finite type invariants classify string links up to …
For a single cusped hyperbolic 3-manifold, Hodgson proved that there are only finitely many Dehn fillings of it whose trace fields have bounded degree. In this paper, we conjecture the same for manifolds with more cusps, and give the first positive results in this direction. For example, in the 2-cusped case, if a mani…
The paper studies connectivity properties of Morse complexes as simplicial complexes grow.
problem Understanding connectivity of Morse complexes as simplicial complexes evolve.
method Bestvina-Brady Morse theory applied to a generalized Morse complex.
result Proves becomes arbitrarily highly connected as grows.
A new topology improves decentralized learning efficiency and accuracy.
problem Finding efficient decentralized learning topologies with fast consensus and low maximum degree.
method Proposed the Base- Graph topology for decentralized learning.
result The Base- Graph enables faster convergence and better communication efficiency than the exponential graph.
Lower bounds for cover degrees of hyperbolic 3-manifolds.
problem Finding lower bounds for cover degrees of hyperbolic 3-manifolds.
method Using volume, diameter, and new invariants to establish lower bounds.
result Established lower bounds for cover degrees in terms of given parameters.
In this paper, we define finite type invariants for cyclic equivalence classes of nanophrases and construct the universal ones. Also, we identify the universal finite type invariant of degree 1 essentially with the linking matrix. It is known that extended Arnold's basic invariants to signed words are finite type invar…
Open manifolds can be covered by with finite or infinite degree.
problem Covering open manifolds with .
method Proving simple branched covers for open manifolds of dimensions 2 and 3, and investigating universal bases.
result Open manifolds can be covered by with finite or infinite degree.
New polynomial invariants defined for long virtual knots.
problem Defining and studying polynomial invariants for long virtual knots.
method Intersection numbers of cycles on a closed surface, considering crossing order.
result Intersection polynomials are finite-type invariants of degree two under crossing changes, but not under virtualizations.
Study finite group actions on aspherical manifolds, proving rigidity and symmetry bounds.
problem Understanding actions of finite groups on aspherical manifolds.
method Analyzing the outer automorphism group and homeomorphism group of the fundamental group.
result Proves the homeomorphism group is Jordan and bounds the discrete degree of symmetry.
Properties of glued knotsmath.GT
Study on knots formed by gluing ellipses, defining gluing degree.
problem Properties of glued knots.
method Defined gluing degree to relate to knot properties.
result Classified all knots up to gluing degree 6.
Groups with specific properties have vanishing -Betti numbers.
problem Understanding -Betti numbers for certain groups.
method Introduced cheap 1-rebuilding property and used structure theorem of Tucker-Drob.
result First -Betti numbers vanish for specified groups.
Paper proves edge-connectivity equals minimum degree for graphs with non-negative curvature.
problem Edge-connectivity vs. minimum degree in graphs with non-negative curvature.
method Analyzes finite connected graphs with non-negative Lin-Lu-Yau curvature.
result Edge-connectivity equals minimum degree for graphs with non-negative curvature.
In this paper, we investigate twist sequences for Kauffman finite-type invariants and Goussarov-Polyak-Viro finite-type invariants. It is shown that one obtains a Kauffman or GPV type of degree if and only if an invariant is a polynomial of degree on every twist lattice of the right form. The main resul…
The sinh-Gordon equation is solved on finite, symmetric graphs.
problem Solving the sinh-Gordon equation with nonzero prescribed functions on finite graphs.
method Uniform a priori estimate to define topological degree, case-by-case calculation of degree, classical sinh-Gordon equation analysis.
result The classical sinh-Gordon equation with nonzero prescribed function is always solvable on finite, symmetric graphs.
Finite type and finitely generated homotopy groups for manifold automorphisms.
problem Finite type and homotopy group properties of manifold automorphism spaces.
method Analyzing the classifying space of diffeomorphism groups and using simple homotopy theory.
result The classifying space of diffeomorphism groups has finitely generated homotopy groups.
Maps between circle bundles: Fiber-preserving, Finiteness and Realization of mapping degree setsmath.GT
Maps between circle bundles are studied, proving fiber-preserving and finiteness results for mapping degrees.
problem Understanding the properties of maps between circle bundles and their degrees.
method Analyzing the structure of circle bundles over aspherical manifolds and using homotopy and homology properties.
result The mapping degree set of fiber-preserving maps from to is determined and finite under certain conditions.
Study of harmonic functions on infinite penny graphs.
problem Characterizing harmonic functions on infinite penny graphs.
method Proving volume doubling and Poincaré inequalities, analyzing polynomial growth harmonic functions.
result Finite dimensional property of ancient solutions of the heat equation.
In the previous three papers in this series, [WKO1]-[WKO3] (arXiv:1405.1956, arXiv:1405.1955, and to appear), Z. Dancso and I studied a certain theory of "homomorphic expansions" of "w-knotted objects", a certain class of knotted objects in 4-dimensional space. When all layers of interpretation are stripped off, what r…
By constructing certain maps, this note completes the answer of the Question: For which closed orientable 3-manifold , the set of mapping degrees is finite for any closed orientable 3-manifold ?
The abstract proves the existence and regularity of Brakke flows starting from a given set.
problem Existence and regularity of Brakke flows starting from a given set.
method Proves the existence and regularity of Brakke flows using a closed countably 1-rectifiable set in R^2.
result For almost all time, the flow locally consists of a finite number of embedded curves of class W^{2,2} whose endpoints meet at junctions with angles of 0, 60, or 120 degrees.
3-manifolds can be virtually dominated by maps of degree 8.
problem Understanding virtual domination of 3-manifolds.
method Proving existence of finite covers with specific map properties.
result Virtual 8-dominance of 3-manifolds.
We study generalizations of finite-type knot invariants obtained by replacing the crossing change in the Vassiliev skein relation by some other local move, analyzing in detail the band-pass and doubled-delta moves. Using braid-theoretic techniques, we show that, for a large class of local moves, generalized Goussarov's…
The main goal of this note is to provide evidence that the first rational homology of the Johnson subgroup of the mapping class group of a genus g surface with one marked point is finite-dimensional. Building on work of Dimca-Papadima, we use symplectic representation theory to show that, for all , the…
We compute the cohomology with group ring coefficients of the complement of a finite collection of affine hyperplanes in a finite dimensional complex vector space. It is nonzero in exactly one degree, namely the degree equal to the rank of the hyperplane arrangement.
Degree theory for orbifoldsmath.GT
In [3] Borzellino and Brunsden started to develop an elementary differential topology theory for orbifolds. In this paper we carry on their project by defining a mapping degree for proper maps between orbifolds, which counts preimages of regular values with appropriate weights. We show that the mapping degree satisfies…
The perturbative Chern-Simons theory is studied in a finite-dimensional version or assuming that the propagator satisfies certain properties (as is the case, e.g., with the propagator defined by Axelrod and Singer). It turns out that the effective BV action is a function on cohomology (with shifted degrees) that solves…
Uniformly finite homology is a coarse homology theory, defined via chains that satisfy a uniform boundedness condition. By construction, uniformly finite homology carries a canonical -semi-norm. We show that, for uniformly discrete spaces of bounded geometry, this semi-norm on uniformly finite homology in …
The paper introduces transposed Poisson superalgebras and their properties.
problem Developing algebraic structures for quantum physics.
method Defining transposed Poisson superalgebras using derivations and proving their identities.
result Six identities are crucial for understanding transposed Poisson superalgebras.
First the title could be also understood as ``3-manifolds related by non-zero degree maps" or "Degrees of maps between 3-manifolds" for some aspects in this survey talk. The topology of surfaces was completely understood at the end of 19th century, but maps between surfaces kept to be an active topic in the 20th centur…
We introduce a class of probability measure-valued diffusions, coined polynomial, of which the well-known Fleming--Viot process is a particular example. The defining property of finite dimensional polynomial processes considered by Cuchiero et al. (2012) and Filipovic and Larsson (2016) is transferred to this infinite …