Representing homology classes by locally flat 2-spheres

We prove an existence result for topologically locally flat embeddings of 2-spheres in simply connected 4-manifolds. This topological result is deduced from a splitting theorem for pointed Hermitian modules over a cyclic group ring. A stability result for such modules is also proved. This applies to the isotopy classification of locally flat embeddings.Partially supported by the NSF.     

On polynomial isotopy of knot-types

We have proved that every knot-type ???³ can be uniquely represented by polynomials up to polynomial isotopy i.e. if two polynomial embeddings of ? in ?³ represent the same knot-type, then we can join them by polynomial embeddings.     

完全二部图$K_{m,n}$的框架表示的唯一性

Kobayashi讨论了空间图在三维流形中的一类标准嵌入,给出了书册表示的概念,并证明了空间图$K_n$的书册表示的最少页是页变换与固边合痕意义下的不变量.本文给出了完全二部图$K_{m,n}$的框架表示的概念,并且证明了完全二部图的框架表示的最小层是层变换与固边合痕意义下的不变量.     

Isotopy invariants for smooth tori in 4-manifolds

In this paper we define invariants under smooth isotopy for certain two-dimensional knots using some refined Cerf theory. One of the invariants is the knot type of some classical knot generalizing the string number of closed braids. The other invariant is a generalization of the unique invariant of degree 1 for classical knots in 3-manifolds. Possibly, these invariants can be used to distinguish smooth embeddings of tori in some 4-manifolds but which are equivalent as topological embeddings.     

Isotopies vis-à-vis level-preserving embeddings

Various standard texts on differential topology maintain that the level-preserving map defined by the track of an isotopy of embeddings is itself an embedding. This note describes a simple counterexample to this assertion.     

Totally real embeddings of the torus intoC 2

In this short note we combine a construction of Viro and a result of Eliashberg and Harlamov to prove that there exist smooth totally real embeddings of the torus intoC ² which are isotopic but not so within the class of totally real surfaces. We also show how Viro's construction can be used to define an isotopy invariant for a certain class of complex curves inC P ².     

Linking Class of Lagrangian or Totally Real Embeddings

Given a totally real embedding j of the 2-torus into ², one defines a 1-class ₁ – its linking class – which is a tool to detect arcwise connected components of the space of totally real embeddings EmbTr( , ²). We generalize the construction of the linking class to any totally real embedding j of a connected, oriented, compact manifold without boundary M ⁿ into ⁿ. We obtain an (n – 1)-class _{n– 1} which is still an invariant for isotopy classes of totally real embeddings. We show that this class is nontrivial by computing it for some families of totally real embeddings. We then study the relationship between isotopy classes of ordinary embeddings and the linking class. With additional assumptions on M ⁿ (n 4 and M ⁿ parallelizable) we obtain the following: two totally real embeddings of M ⁿ into ⁿ which belong to the same isotopy class of totally real immersion, belong to the same isotopy class of ordinary embedding if and only if (1) their linking classes are the same (if n odd); (2) the images of their linking classes by the coefficient homomorphism : H ^{n– 1} (M ⁿ , ) H ^{n– 1} (M ⁿ , ₂) are the same (if n even).     

Horizontal loops in Engel space

A simple proof is given of the following result first observed by Adachi: embedded circles tangent to the standard Engel structure on are classified, up to isotopy via such embeddings, by their rotation number.     

Symmetrization inequalities and Sobolev embeddings

We prove new extended forms of the Pólya-Szegö symmetrization principle. As a consequence new sharp embedding theorems for generalized Besov spaces are proved, including a sharpening of the limiting cases of the classical Sobolev embedding theorem. In particular, a surprising self-improving property of certain Sobolev embeddings is uncovered.

     

Proper holomorphic embeddings into stein manifolds with the density property

We prove that a Stein manifold of dimension d admits a proper holomorphic embedding into any Stein manifold of dimension at least 2d + 1 satisfying the holomorphic density property. This generalizes classical theorems of Remmert, Bishop and Narasimhan, pertaining to embeddings into complex euclidean spaces, as well as several other recent results.     

Extrapolation results of Lions-Peetre type

We establish compactness results for extrapolation constructions which correspond to the well-known Lions-Peetre compactness theorems of interpolation theory. Applications are given to compactness of certain limiting Sobolev embeddings.     

Un invariant d'isotopie lisse pour des surfaces dans une variété de dimension 4An invariant of smooth isotopy for surfaces in a 4-manifold

We construct an invariant of smooth isotopy for surfaces smoothly embeded in 4-manifolds. This invariant is used to distinguish smooth embeddings of tori or Klein bottles that are regular homotopic in $\text{C}{}^1\text{×}\text{C}{}^1$, and that have the same complement and the same fundamental rack. To cite this article: C. Darolles, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 811–815.     

Testing Graph Isotopy on Surfaces

We investigate the following problem: Given two embeddings G ₁ and G ₂ of the same abstract graph G on an orientable surface S, decide whether G ₁ and G ₂ are isotopic; in other words, whether there exists a continuous family of embeddings between G ₁ and G ₂. We provide efficient algorithms to solve this problem in two models. In the first model, the input consists of the arrangement of G ₁ (resp., G ₂) with a fixed graph cellularly embedded on S; our algorithm is linear in the input complexity, and thus, optimal. In the second model, G ₁ and G ₂ are piecewise-linear embeddings in the plane, minus a finite set of points; our algorithm runs in O(n ^3/2logn) time, where n is the complexity of the input. The graph isotopy problem is a natural variation of the homotopy problem for closed curves on surfaces and on the punctured plane, for which algorithms have been given by various authors; we use some of these algorithms as a subroutine. As a by-product, we reprove the following mathematical characterization, first observed by Ladegaillerie (Topology 23:303–311, 1984): Two graph embeddings are isotopic if and only if they are homotopic and congruent by an oriented homeomorphism.     

On a double boundary layer in a nonlinear boundary value problem

The question of the isotopy of a quasiconformal mapping and its special aspects in dimension greater than 2 are considered. It is shown that an arbitrary quasiconformal mapping of a ball has an isotopy to the identity map such that the coefficient of quasiconformality (dilatation) of the mapping varies continuously and monotonically. In contrast to the planar case, in dimension higher than 2 such an isotopy is not possible in an arbitrary domain. Examples showing specific features of the multidimensional case are given. In particular, they show that even when such an isotopy exists, it is not always possible to perform an isotopy so that the coefficient of quasiconformality approaches 1 monotonically at each point in the source domain.     

Embedding theorems of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type

In this paper, the author establishes the embedding theorems for different metrics of inhomoge-neous Besov and Triebel-Lizorkin spaces on spaces of homogeneous type. As an application, the author obtains some estimates for the entropy numbers of the embeddings in the limiting cases between some Besov spaces and some logarithmic Lebesgue spaces.     

Multipliers for Besov spaces on graded Lie groups

In this note, we give embeddings and other properties of Besov spaces, as well as spectral and Fourier multiplier theorems, in the setting of graded Lie groups. We also present a Nikolskii-type inequality and the Littlewood–Paley theorem that play a role in this analysis and are also of interest on their own.     

A controlled-topology proof of the product structure theorem

The controlled end and h-cobordism theorems (Ends of maps I, 1979) are used to give quick proofs of the Top/PL and PL/DIFF product structure theorems.     

Leavitt path algebras are Bézout

We study mapping class group orbits of homotopy and isotopy classes of curves with self-intersections. We exhibit the asymptotics of the number of such orbits of curves with a bounded number of self-intersections, as the complexity of the surface tends to infinity.We also consider the minimal genus of a subsurface that contains the curve. We determine the asymptotic number of orbits of curves with a fixed minimal genus and a bounded self-intersection number, as the complexity of the surface tends to infinity.As a corollary of our methods, we obtain that most curves that are homotopic are also isotopic. Furthermore, using a theorem by Basmajian, we get a bound on the number of mapping class group orbits on a given hyperbolic surface that can contain short curves. For a fixed length, this bound is polynomial in the signature of the surface.The arguments we use are based on counting embeddings of ribbon graphs.