arXiv research

Computes mapping class groups of 4-manifolds with boundary.

In this paper we study fundamental properties of geodesic mappings with respect to the smoothness class of metrics. We show that geodesic mappings preserve the smoothness class of metrics. We study geodesic mappings of Einstein spaces.

The paper studies branched surfaces and their properties.

The paper proves a smooth Birman-Hilden theorem for hyperkähler manifolds.

Residually finite groups found in manifold automorphisms.

The paper generalizes a result on smooth mapping class groups and proves a property of Dehn twists in 4-manifolds.

The study extends calibrated geometry to smooth maps and finds energy bounds.

Maps with many singularities found in complex space.

In this paper we extend Y.Eliashberg's $h$-principle to arbitrary generic smooth maps of smooth manifolds. Namely, we prove a necessary and sufficient condition for a continuous map of smooth manifolds of the same dimension to be homotopic to a generic map with a prescribed Thom-Boardman singularity $Σ^I$ at each point…

Study mapping class groups of specific 3D shapes.

In this article, we study the smooth mapping class group of a surface S relative to a given Cantor set, that is the group of isotopy classes of orientation-preserving smooth diffeomorphisms of S which preserve this Cantor set. When the Cantor set is the standard ternary Cantor set, we prove that the subgroup consisting…

Boundary Dehn twists become trivial after abelianization.

We study smooth maps between smooth manifolds with only fold points as their singularities, and clarify the obstructions to the existence of such a map in a given homotopy class for certain dimensions. The obstructions are described in terms of characteristic classes, which arise as Postnikov invariants, and can be int…

Theta graph diffeomorphism shows nontrivial mapping class of 4-sphere.

New constraints found for algebro-geometric subgroups of mapping class groups.

We obtain relations among the characteristic classes of a manifold M admitting corank one maps. Our relations yield strong restrictions on the cobordism class of M and also nonexistence results for singular maps of the projective spaces. We obtain our results through blowing up a manifold along the singular set of a sm…

Generalizes Lefschetz fibrations with rational homology disk smoothings.

The paper studies exotic diffeomorphisms of 4-manifolds with b_+ = 2.

Generators of the 4-sphere's smooth mapping class group via diffeomorphisms of Montesinos twins.

Smooth maps preserve distances on specific revolution surfaces.

Smooth functions preserve Zygmund class on curves.

Variant of previous work on smooth algebraic functions with compact and non-compact preimages.

Nontrivial boundary Dehn twist found on K3#K3 manifold.

Novikov theorem extended to rational Pontryagin classes for cyclic group $C_4$.

Let P be a connected smooth p-manifold. We describe the group of all cobordism classes of smooth maps of n-manifolds to P with singularities of a given $cal K$-invariant class in terms of certain stable homotopy groups by applying the relative homotopy principle on the existence level. We also deal with the oriented ve…

Study mapping class groups of 4-manifolds, proving non-finitely generated and splitting properties.

Study shows infinite order in mapping class groups for certain 3D shapes.

In this paper, we construct round fold maps or stable fold maps with concentric singular value sets introduced by the author on smooth bundles over spheres or bundles over more general manifolds. The class of round fold maps includes special generic maps on spheres and such maps have been constructed on smooth bundles …

Researchers solved a complex problem for a specific type of 4-manifolds.

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…

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

We will prove the relative homotopy principle for smooth maps with singularities of a given {\cal K}-invariant class with a mild condition. We next study a filtration of the group of homotopy self-equivalences of a given manifold P by considering singularities of non-negative {\cal K}-codimensions.

This paper is devoted to the study of the conformal spectrum (and more precisely the first eigenvalue) of the Laplace-Beltrami operator on a smooth connected compact Riemannian surface without boundary, endowed with a conformal class. We give a constructive proof of a critical metric which is smooth except at some coni…

Criterion for lifting smooth contact maps between Carnot groups to central extensions.

New method defines Gysin maps for stratified spaces, preserving signatures.

Let $l : Σ\to X$ be a weakly Lagrangian map of a compact orientable surface $Σ$ in a Kähler surface $X$ which is area minimizing in its homotopy class of maps in $W^{1,2}(Σ, X)$, the Sobolev space of maps of square integrable first derivative. Schoen and Wolfson showed such $l$ is Lipschitz, and it is smooth excep…

Book on infinite-dimensional Lie groups, covering basics and various classes.

For any smooth compact manifold $W$ of dimension at least two we prove that the classifying spaces of its group of diffeomorphisms which fix a set of $k$ points or $k$ embedded disks (up to permutation) satisfy homology stability. The same is true for so-called symmetric diffeomorphisms of $W$ connected sum with $k$ co…

We consider the space of smooth complex projective plane curves of degree d. Defined over this is the tautological family of plane curves, and hence there is a monodromy representation into the mapping class group of the fiber. We show two results concerning this monodromy group. First, we show that the presence of an …

Let X be a real algebraic subset of R^n and M a smooth, closed manifold. We show that all continuous maps from M to X are homotopic (in X) to C^\infty maps. We apply this result to study characteristic classes of vector bundles associated to continuous families of complex group representations, and we establish lower b…

Study shows infinite kernels in topological monodromy for curve families.

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…

Study stabilizers of smooth functions on surfaces, focusing on Morse-Bott functions.

The immersions of a smooth manifold $M$ in a symplectic manifold $(N,σ)$ inducing a given closed form $ω$ on $M$ satisfy the $C^0$-dense $h$-principle in the space of all continuous maps which pull back the deRham cohomology class of $σ$ onto that of $ω$. In this paper we prove a foliated version of this result due to …

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 …

Study on nonorientable 4-manifolds using simplified fibrations and trisections.

We prove that isomorphism classes of principal bundles over a diffeological space are in bijection to certain maps on its free loop space, both in a setup with and without connections on the bundles. The maps on the loop space are smooth and satisfy a "fusion" property with respect to triples of paths. Our bijections a…

We show the smoothness of weakly Dirac-harmonic maps from a closed spin Riemann surface into stationary Lorentzian manifolds, and obtain a regularity theorem for a class of critical elliptic systems without anti-symmetry structures.