New class of maps restricts manifolds strongly in algebraic topology.
problem Restricting manifolds in algebraic topology.
method Proposed a class of generalized special generic maps.
result Extended fundamental results on structures and algebraic topological properties.
This paper constructs real algebraic maps that are topologically special generic maps.
problem Constructing smooth maps in differential topology and real algebraic geometry.
method Constructs real algebraic maps that are topologically special generic maps.
result Real algebraic maps are topologically special generic maps.
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…
Topological normal generation proved for mapping class groups of certain surfaces.
problem Proving topological normal generation for mapping class groups of surfaces.
method Analyzing the end space of surfaces and using topological normal closure properties.
result Topological normal generation is equivalent to uniquely self-similar for surfaces with countable end space.
Computes mapping class groups of 4-manifolds with boundary.
problem Computing mapping class groups for 4-manifolds with boundary.
method Topological and smooth methods applied to compact, simply connected 4-manifolds.
result Description of topological and stable smooth mapping class groups.
Stability of mapping spaces is shown to be related to the D-topology.
problem Understanding the relationship between stability and the D-topology of mapping spaces.
method Reformulation and proof of stability theorems in diffeological étale manifolds.
result Stable classes of mapping spaces are D-open.
The study examines the topology of map germs and their images.
problem Understanding the topology of map germs and their images.
method Using the topology of the link to analyze the normal and non-normal images.
result Normal images of map germs are quotient singularities.
Study numerical invariants under retraction maps between topological spaces.
problem Understand behavior of invariants like cohomological dimensions under retractions.
method Introduced a notion of retraction and studied several numerical invariants.
result Proved inequalities between invariants hold under retractions.
This thesis introduces big mapping class groups and their structure.
problem Understanding mapping class groups of infinite-type surfaces.
method Systematic introduction and analysis of structure and topological generation.
result Key differences from finite-type mapping class groups.
Study on minimal torsion topological generators for mapping class groups of infinite-type surfaces.
problem Minimal topological generating sets of mapping class groups consisting of torsion elements.
method Investigation of minimal topological generating sets for Map(S(n)) consisting entirely of torsion elements, with special attention to involutions. result Minimal topological generating sets for Map(S(n)) consisting of torsion elements are found for various n. This paper characterizes stable polynomial mappings in a specific set.
problem Characterizing stable polynomial mappings in a given set.
method Analyzing polynomial mappings with specific degrees and determining topological equivalence.
result Effective determination of mappings with generic topology.
Overview of infinite surface mapping class groups.
problem Understanding mapping class groups of infinite surfaces.
method Survey of recent research findings.
result Recent developments in mapping class groups of infinite surfaces.
We investigate how Viro's integral calculus applies for the study of the topology of stable maps. We also discuss several applications to Morin maps and complex maps.
Study bounds topological entropy of maps on surfaces with punctures based on mapping torus homology.
problem Relating topological entropy of pseudo-Anosov maps to homology of mapping tori.
method Analyzing the topological entropy of pseudo-Anosov maps on surfaces with punctures and relating it to the rank of the first homology of their mapping tori.
result Entropy of a pseudo-Anosov map is bounded by a formula involving the genus, number of punctures, and homology rank.
New EZ-structure maps mapping class group actions.
problem Mapping class group actions on surfaces.
method Constructing a boundary with minimal, strongly proximal, and topologically free action.
result Boundary acts on mapping class group with desired properties.
Involutions generate mapping class groups of infinite surfaces.
problem Generating involutions for mapping class groups of infinite surfaces.
method Analyzing infinite surfaces with n ends, showing involutions generate groups for n ≥ 6 and n ≥ 3.
result Involutions generate mapping class groups for n ≥ 6 and n ≥ 3.
The construction of topological index maps for equivariant families of Dirac operators requires factoring a general smooth map through maps of a very simple type: zero sections of vector bundles, open embeddings, and vector bundle projections. Roughly speaking, a normally non-singular map is a map together with such a …
Study topological quantum mechanics on orbifolds with geometric interpretation.
problem Quantum mechanical models on symplectic orbifolds.
method Explicit orbifold version of quantum HKR map and exact semi-classical approximation.
result Geometric and quantum field theoretic interpretation of orbifold algebraic index.
This paper finds minimal sets of generators for mapping class groups of specific surfaces.
problem Finding minimal sets of generators for mapping class groups of infinite-type surfaces.
method Analyzing specific surfaces S(n) to determine minimal sets of generators. result Minimal sets of generators for Map(S(n)) are identified for n≥8 (3 elements), n≥3 (4 elements), and S(1) (2 elements). There is a concept in digital topology of a shy map. We define an analogous concept for topological spaces: We say a function is shy if it is continuous and the inverse image of every path-connected subset of its image is path-connected. Some basic properties of such maps are presented. For example, every shy map onto …
Maps can be embedded in higher dimensions if they lift to embeddings in product spaces.
problem Embedding maps in higher dimensions without self-intersections.
method Lifting maps to embeddings in product spaces.
result Maps can be embedded in higher dimensions if they lift to embeddings in product spaces.
Topology on mapping class group leads to generic results on surface properties.
problem Defining and analyzing a topology on mapping class groups.
method Introduced a twist-cofinite topology on mapping class groups of surfaces.
result Generic properties of subsets, such as pseudo-Anosov maps and Heegaard manifolds.
The existence theorem for mapping cylinder neighborhoods is discussed as a prototypical example of controlled topology and its applications. The first of a projected series developed from lectures at the Summer School on High-Dimensional Topology, Trieste Italy 2001
Study approximates top Lyapunov exponents for surface mapping classes.
problem Approximating topological Lyapunov exponents for surface mapping classes.
method Periodic approximation and joint spectral radius extension.
result Top Lyapunov exponents can be approximated by periodic orbits.
The study restricts manifolds with certain explicit SGL maps and constructs them.
problem Restrictions on manifolds admitting specific SGL maps.
method Generalization of Morse functions and canonical projections to construct SGL maps.
result Manifolds admitting certain explicit SGL maps are strongly topologically restricted.
We prove implicit function theorems for mappings on topological vector spaces over valued fields. In the real and complex cases, we obtain implicit function theorems for mappings from arbitrary (not necessarily locally convex) topological vector spaces to Banach spaces.
The bending map of a hyperbolic 3-manifold maps a convex cocompact hyperbolic metric on a hyperbolic 3-manifold with boundary to its bending measured geodesic lamination. In the present paper we study the extension of this map to the space of geometrically finite hyperbolic metrics. We introduce a relationship on the s…
Maps with boundary definite fold points restrict manifold structure.
problem Restricting the global structure of manifolds with boundary.
method Introducing boundary special generic maps and deriving differential-topological restrictions.
result New results on non-singular extensions of special generic maps.
The paper explores mapping class groups and their quantum field theory representations.
problem Understanding finite dimensional representations of mapping class groups.
method Survey of topological quantum field theory aspects.
result Discussion of finite dimensional representations in quantum field theory.
We analyze the signature type of a cascade of periodic orbits associated to period doubling renormalizable maps of the two dimensional disk. The signature is a sequence of rational numbers which describes how periodic orbits turn each other and is invariant by topological conjugacies that preserve orientation. We prove…
In this article, we introduce the notion of a functor on coarse spaces being coarsely excisive- a coarse analogue of the notion of a functor on topological spaces being excisive. Further, taking cones, a coarsely excisive functor yields a topologically excisive functor, and for coarse topological spaces there is an ass…
Study cohomology rings of 3D manifolds with round fold maps into the plane.
problem Understanding cohomology rings of 3D manifolds with round fold maps.
method Analyzing cohomology rings of 3D manifolds admitting round fold maps into the plane.
result Explicit new study showing relation between coefficient rings and topological types of round fold maps.
We construct pairs of non-isometric hyperbolic 3-orbifolds with the same topological type and volume. Topologically these orbifolds are mapping tori of pseudo-Anosov maps of the surface of genus 2, with singular locus a fibred (hyperbolic) link with five components.
New framework models high-Hopf-index hopfions using generalized fold maps.
problem Modeling and understanding high-Hopf-index hopfions.
method Proposes a new mathematical framework based on generalized Hopf maps and fold maps.
result Establishes a robust bridge between singular maps and experimentally observed hopfion topologies.
The paper studies the topological properties of convex sets and their polar mappings.
problem Investigating the topological nature of polar mappings on convex sets.
method Proving homeomorphism and topological conjugacy between polar mappings and involution.
result Inclusion-reversing involutions on convex sets are topologically conjugate to polar mappings.
We study the topology of a complete asymptotically hyperbolic Einstein manifold such that its conformal boundary has positive Yamabe invariant. We proved that all maps from such manifold into any nonpositively curved manifold are homotopically trivial. Our proof is based on a Bochner type argument on harmonic maps.
Topological degrees of continuous mappings between manifolds of even dimension are studied in terms of index theory of pseudo-differential operators. The index formalism of non-commutative geometry is used to derive analytic integral formulas for the index of a 0:th order pseudo-differential operator twisted by a Hölde…
The topological fundamental group π1top is a topological invariant that assigns to each space a quasi-topological group and is discrete on spaces which are well behaved locally. For a totally path-disconnected, Hausdorff, unbased space X, we compute the topological fundamental group of the "hoop earring" spac…
In this article we determine, for an infinite family of maps on the plane, the topology of the surface on which the minimal regular covering occurs. This infinite family includes all Archimedean maps.
The paper explores the dynamics of composite symplectic Dehn twists with nonuniform hyperbolicity.
problem Understanding the dynamics and properties of composite symplectic Dehn twists.
method Analyzing the form of nonuniform hyperbolicity, growth of Floer cohomology, and classification of symplectic mapping classes.
result Composite symplectic Dehn twists exhibit positive topological entropy and exponential growth in Floer cohomology.
We study properties of shy maps in digital topology.
We give a necessary and sufficient condition for the mapping class group of the pair of the 3-sphere and a graph embedded in it to be isomorphic to the topological symmetry group of the embedded graph.
New algorithm constrains SOMs to create supervised low-dimensional mappings.
problem Creating supervised mappings in neural networks with known internal topology.
method Developed Supervised Topological Maps (STMs) by modifying SOMs to incorporate target distances.
result STMs allow for supervised generation of new data with known internal structure.
Develops topological concepts for Morrey-Sobolev bundles in high dimensions.
problem Lack of continuity in transition maps for Morrey-Sobolev bundles.
method Introduces topological isomorphism classes and uses connection-oriented approach.
result Derives approximability results for bundles and connections in Morrey-Sobolev setting.
The paper studies branched surfaces and their properties.
problem Global topologies of branched surfaces and their maps.
method Explicit construction of maps and study of topological properties.
result New insights into the geography of branched surfaces and their maps.
Minimal simplicial maps constructed for spheres and manifolds.
problem Constructing minimal simplicial maps of specific degrees.
method Triangulations and degree constructions for manifolds and spheres.
result Minimal triangulations for degree d self-maps of Sn−1imesS1. The paper extends properties of smooth functions to closed sets and maps.
problem Properties of smooth functions on closed sets and maps.
method Extending properties of smooth functions to closed sets and maps, proving isomorphisms with natural topologies.
result Bornological isomorphisms of function spaces are established.
We study the periods mapping from the moduli space of real hyperelliptic curves with marked point on an oriented oval to the euclidean space. The mapping arises in the analysis of Chebyshev construction used in the constrained optimization of the uniform norm of polynomials and rational functions. The decomposition of …