arXiv research

Smooth isotopy on cube saves energy with extra dimensions.

New smoothing techniques for topological surfaces in 4-manifolds.

Classifies smooth isotopy of a specific Lagrangian embedding.

Improved proofs for topological and smooth pseudo-isotopies of simply connected 4-manifolds.

New obstructions show some 4-manifold homeomorphisms are pseudo-isotopic but not isotopic.

We present a grid diagram analogue of Carter, Rieger and Saito's smooth movie theorem. Specifically, we give definitions for grid movies, grid movie isotopies and present a definition of grid planar isotopy as a particular subset of the grid diagram moves: stabilization, destabilization and commutation. We show that gr…

Legendrian Lavrentiev links are shown to be equivalent to smooth links.

Proves a Thom Isotopy Theorem for nonproper semialgebraic maps.

Approaches 4D Schoenflies via pseudo-isotopy.

The symplectic isotopy conjecture states that every smooth symplectic surface in $CP^2$ is symplectically isotopic to a complex algebraic curve. Progress began with Gromov's pseudoholomorphic curves [Gro85], and progressed further culminating in Siebert and Tian's proof of the conjecture up to degree 17 [ST05], but fur…

Example shows smooth vs topological isotopy in a 4-manifold.

Smoothly isotopic surfaces in a 4-manifold are stable isotopic if trivial in the 2-homology.

Example of non-smooth isotopy using K3 surfaces.

We prove that a topological contact isotopy uniquely defines a topological contact Hamiltonian. Combined with previous results from [MS11], this generalizes the classical one-to-one correspondence between smooth contact isotopies and their generating smooth contact Hamiltonians and conformal factors to the group of top…

New gauge theory invariant detects non-smooth isotopy of $P^2$-knots.

New example of non-smooth isotopy after stabilization in 4-manifolds.

Embeddings of surfaces in 5-manifolds are classified by an invariant.

If a continuous map f: X->Q is approximable arbitrary closely by embeddings X->Q, can some embedding be taken onto f by a pseudo-isotopy? This question, called Isotopic Realization Problem, was raised by Shchepin and Akhmet'ev. We consider the case where X is a compact n-polyhedron, Q a PL m-manifold and show that the …

Study on topological pseudo-isotopies of 4-manifolds, proving some cases and finding counterexamples.

In this paper, we establish compactness for various geometric curvature energies including integral Menger curvature, and tangent-point repulsive potentials, defined a priori on the class of compact, embedded $m$-dimensional Lipschitz submanifolds in ${\mathbb{R}}^n$. It turns out that due to a smoothing effect any seq…

David Gabai recently proved a smooth 4-dimensional "Light Bulb Theorem" in the absence of 2-torsion in the fundamental group. We extend his result to 4-manifolds with arbitrary fundamental group by showing that an invariant of Mike Freedman and Frank Quinn gives the complete obstruction to "homotopy implies isotopy" fo…

We develop a new approach to the classical problem on isotopy classification of embeddings of manifolds into Euclidean spaces. This approach involves studying of a new embedding invariant, of almost-embeddings and of smoothing, as well as explicit constructions of embeddings. Using this approach we obtain complete conc…

Generalizing Milnor's result that an FTC (finite total curvature) knot has an isotopic inscribed polygon, we show that any two nearby knotted FTC graphs are isotopic by a small isotopy. We also show how to obtain sharper constants when the starting curve is smooth. We apply our main theorem to prove a limiting result f…

Solves a problem about deforming symplectic forms on a Klein bottle.

Study equivariant isotopy in higher dimensions, finding exceptions.

Smooth approximations lead to homotopy equivalences in manifold spaces.

New method constructs Stein surfaces using topological isotopy.

The paper constructs infinitely many $G$-smoothings of a $G$-manifold.

The paper introduces an inequality to detect when surfaces in 4-manifolds are not smoothly isotopic.

Smoothly isotopic 3-discs in 4-sphere become identical after 5-dimensional push.

Let $M_1$ and $M_2$ be closed connected orientable $3$-manifolds. We classify the sets of smooth and piecewise linear isotopy classes of embeddings $M_1\sqcup M_2\rightarrow S^6$.

New method for classifying disk embeddings in 4-manifolds.

The study finds non-simple isotopy classes of links in 3-manifolds, including Legendrian and pseudo-Legendrian examples.

For a fixed regular cone in Euclidean space with small entropy we show that all smooth self-expanding solutions of the mean curvature flow that are asymptotic to the cone are in the same isotopy class.

The paper studies elliptic surfaces and proves unique fibered structures.

The paper gives topological as well as rigid isotopy classification of smooth irreducible algebraic curves in the real projective 3-space for the case when the degree of the curve is at most six and its genus is at most one.

Symplectic Khovanov homology is an invariant of oriented links defined by Seidel and Smith and conjectured to be isomorphic to Khovanov homology. I define morphisms (up to a global sign ambiguity) between symplectic Khovanov homology groups, corresponding to isotopy classes of smooth link cobordisms in 4D between a fix…

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

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…

Regular neighborhoods of singular submanifolds are isotopic to bundle morphisms.

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

We work in the smooth category. Let $N$ be a closed connected orientable 4-manifold with torsion free $H_1$, where $H_q := H_q(N; \mathbb Z)$. Our main result is a readily calculable classification of embeddings $N\to\mathbb R^7$ up to isotopy, with an indeterminancy. Such a classification was only known before for $H_…

Lips and swallow-tails are generic local moves of singularities of a smooth map to a 2-manifold. We prove that these moves of singularities of the product map of two functions on a 3-manifold can be realized by isotopies of the functions.

We construct an enhanced version of knot contact homology, and show that we can deduce from it the group ring of the knot group together with the peripheral subgroup. In particular, it completely determines a knot up to smooth isotopy. The enhancement consists of the (fully noncommutative) Legendrian contact homology a…

Smooth knots can be embedded into a specific Menger continuum.

A real 3-manifold is a smooth 3-manifold together with an orientation preserving smooth involution, called a real structure. In this article we study open book decompositions on smooth real 3-manifolds that are compatible with the real structure. We call them real open book decompositions. We show that each real open b…

New algebraic theory classifies symplectic curves in complex projective space.

The paper solves isotopy problems on 4-manifolds and classifies symplectic structures.