arXiv research

New class of maps restricts manifolds strongly in algebraic topology.

This paper constructs real algebraic maps that are topologically special generic maps.

Maps with boundary definite fold points restrict manifold structure.

The study shows manifolds with special generic maps also have nice multisections.

The study restricts manifolds with certain explicit SGL maps and constructs them.

A special generic map is a smooth map regarded as a natural generalization of Morse functions with just 2 singular points on homotopy spheres. Canonical projections of unit spheres are simplest examples of such maps and manifolds admitting special generic maps into the plane are completely determined by Saeki in 1993 a…

Study on minimal torsion topological generators for mapping class groups of infinite-type surfaces.

In studies of smooth maps with good differential topological conditions such as immersions, embeddings, Morse functions and their higher dimensional versions including fold maps and application to geometry, especially algebraic and differential topology of manifolds, liftings or desingulizations of maps of appropriate …

Isothermic nets created from special maps for smooth surfaces.

A Reeb space is defined as the space of all the connected components of inverse images of a smooth map, which is a fundamental tool in studying smooth manifolds using generic smooth maps whose codimensions are not positive such as Morse functions, their higher dimensional versions including fold maps and general stable…

New method characterizes surface quadrilateral layouts as special immersions.

Energy minimizing harmonic maps between manifolds are known to be smooth outside a rectifiable set of codimension $3$, called the singular set. The possibility that this set is not a manifold, but has arbitrarily many small gaps in it, is not excluded in general. Here we prove that some part of the singular set - chara…

The paper formalizes robot environments using topological concepts.

Research explores real algebraic realization of round fold maps of codimension -1.

We present an axiomatic approach to finite- and infinite-dimensional differential calculus over arbitrary infinite fields (and, more generally, suitable rings). The corresponding basic theory of manifolds and Lie groups is developed. Special attention is paid to the case of mappings between topological vector spaces ov…

The author studies regions foliated by 1D families of functions and their applications.

In this paper we describe the cohomogeneity one special Lagrangian 3-folds in the cotangent bundle of the 3-sphere, also known in the physics literature as a deformed conifold. Our main result gives a global foliation of the deformed conifold by T^2-invariant special Lagrangian 3-folds, where the generic leaf is topolo…

Study the topological information of map germs using Euler obstruction.

Proposes a topological framework to study modular invariants and related concepts.

We consider the interplay of point counts, singular cohomology, étale cohomology, eigenvalues of the Frobenius and the Grothendieck ring of varieties for two families of varieties: spaces of rational maps and moduli spaces of marked, degree $d$ rational curves in $\mathbb{P}^n$. We deduce as special cases algebro-geome…

This paper presents and explores a theory of \emph{multiholomorphic maps}. This group of ideas generalizes the theory of pseudoholomorphic curves in a direction suggested by consideration of the kinds of compatible geometric structures that appear in the realm of special holonomy as well as some of the topological and …

We identify a deformation of the N=2 supersymmetric sigma model on a Calabi-Yau manifold X which has the same effect on B-branes as a noncommutative deformation of X. We show that for hyperkahler X such deformations allow one to interpolate continuously between the A-model and the B-model. For generic values of the non…

The paper extends properties of smooth functions to closed sets and maps.

We construct a three-dimensional topological sigma model which is induced from a generalized complex structure on a target generalized complex manifold. This model is constructed from maps from a three-dimensional manifold $X$ to an arbitrary generalized complex manifold $M$. The theory is invariant under the diffeomor…

In this paper, as a fundamental study on the theory of Morse functions and their higher dimensional versions or fold maps and applications to geometric theory of manifolds, which were started in 1950s by differential topologists such as Thom and Whitney and have been studied actively, we study algebraic and differentia…

New concept SB-generation helps classify transformation groups.

This research extends topological recursion to hyperbolic surfaces with tight boundaries and conical defects.

Let $X$ be an $(n+1)$-dimensional manifold, $Δ$ be a one-dimensional foliation on $X$, and $p: X \to X / Δ$ be a quotient map. We will say that a leaf $ω$ of $Δ$ is special whenever the space of leaves $X / Δ$ is not Hausdorff at $ω$. We present necessary and sufficient conditions for the map $p: X \to X / Δ$ to be a l…

Study of associative submanifolds in Berger space SO(5)/SO(3).

The target space geometry of abelian vector multiplets in ${\cal N}= 2$ theories in four and five space-time dimensions is called special geometry. It can be elegantly formulated in terms of Hessian geometry. In this review, we introduce Hessian geometry, focussing on aspects that are relevant for the special geometrie…

The paper examines rational homology spheres that admit special generic maps into Euclidean spaces.

Smooth maps show Gromoll filtration for spheres.

Stable fold maps are fundamental tools in a generalization of the theory of Morse functions on smooth manifolds and its application to studies of topological properties of smooth manifolds. Round fold maps were introduced as stable fold maps with singular value sets, defined as the set consisting of all the singular va…

We describe the Lorenz links generated by renormalizable Lorenz maps with reducible kneading invariant $(K_f^-,K_f^+)=(X,Y)*(S,W)$, in terms of the links corresponding to each factor. This gives one new kind of operation that permits us to generate new knots and links from old. Using this result we obtain explicit form…

Topological normal generation proved for mapping class groups of certain surfaces.

A smooth map between smooth manifolds is called a special generic map if it has only definite fold points as its singularities. In this paper, we give conditions for a special generic map into the 3-dimensional Euclidean space to be factored as the composition of an embedding and a projection for certain dimensions.

A so-called special generic map is by definition a map of smooth manifolds all of whose singularities are definite fold points. It is in general an open problem posed by Saeki in 1993 to determine the set of integers $p$ for which a given homotopy sphere admits a special generic map into $\mathbb{R}^{p}$. By means of t…

This thesis introduces big mapping class groups and their structure.

We introduce a notion of moment map adapted to actions of Lie groups that preserve a closed three-form. We show existence of our multi-moment maps in many circumstances, including mild topological assumptions on the underlying manifold. Such maps are also shown to exist for all groups whose second and third Lie algebra…

We introduce and study (strict) Schottky G-bundles over a compact Riemann surface X, where G is a connected reductive algebraic group. Strict Schottky representations are shown to be related to branes in the moduli space of G-Higgs bundles over X, and we prove that all Schottky $G$-bundles have trivial topological type…

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…

Involutions generate mapping class groups of infinite surfaces.

Study classifies 2-manifolds with special homeomorphism groups.

This paper classifies semi-equivelar maps on a special surface.

This paper finds minimal sets of generators for mapping class groups of specific surfaces.

We use Klyachko's methods to prove that the natural map G to G-hat, where G is a torsion-free group and G-hat is obtained by adding a new generator t and a new relator w, is surjective only if w is conjugate to gt or gt^{-1} for some g in G. This solves a special case of the surjectivity problem for group extensions, r…

Study classifies 3D Hessian manifolds, proving their topology.

We consider four notions of maps between smooth C^r orbifolds O, P with O compact (without boundary). We show that one of these notions is natural and necessary in order to uniquely define the notion of orbibundle pullback. For the notion of complete orbifold map, we show that the corresponding set of C^r maps between …