Codimension one spheres in Rn with double tangent balls

In contrast to the situation in $\text{R}$³, where a 2-sphere with double tangent balls at each point must be tamely embedded in $\text{R}$³, there exist wild (n?1)-spheres in $\text{R}$ⁿ for n>3 with this same geometric property. However, if the sphere Σ is tame moduio a subset X that lies in a polyhedron P that is tame in Σ, the dimension of P is less than n?2, n>4, and Σ has double tangent balls over X, then Σ must be tame in $\text{R}$ⁿ. Also if the tangent balls extend over P and are pairwise congruent, the dimensional restriction on P can be dropped. Examples are given to support the necessity of the hypotheses of the included theorems.     

Embedding infinite cyclic covers of knot spaces into 3-space

We say a knot k in the 3-sphere S³ has PropertyIE if the infinite cyclic cover of the knot exterior embeds into S³. Clearly all fibred knots have Property IE.There are infinitely many non-fibred knots with Property IE and infinitely many non-fibred knots without property IE. Both kinds of examples are established here for the first time. Indeed we show that if a genus 1 non-fibred knot has Property IE, then its Alexander polynomial Δ_k(t) must be either 1 or 2t²−5t+2, and we give two infinite families of non-fibred genus 1 knots with Property IE and having Δ_k(t)=1 and 2t²−5t+2 respectively.Hence among genus 1 non-fibred knots, no alternating knot has Property IE, and there is only one knot with Property IE up to ten crossings.We also give an obstruction to embedding infinite cyclic covers of a compact 3-manifold into any compact 3-manifold.     

How strange can an attractor for a dynamical system in a 3-manifold look?

The aim of this paper is to characterise those compact subsets K of 3-manifolds M that are (stable and not necessarily global) attractors for some flow on M. We will show that it is the topology of M−K, rather than that of K, the one that plays a relevant role in this problem.A necessary and sufficient condition for a set K to be an attractor is that it must be an “almost tame” subset of M in a sense made precise under the equivalent notions of “weakly tame” and “tamely embedded up to shape”, defined in the paper. These are complemented by a further equivalent condition, “algebraic tameness”, which has the advantage of being checkable by explicit computation.A final section of the paper is devoted to a partial analysis of the same question when one replaces flows by discrete dynamical systems.     

Classifying foliations of 3-manifolds via branched surfaces

We use branched surfaces to define an equivalence relation on C¹ codimension one foliations of any closed orientable 3-manifold that are transverse to some fixed nonsingular flow. There is a discrete metric on the set of equivalence classes with the property that foliations that are sufficiently close (up to equivalence) share important topological properties.     

Modifying a branched surface to carry a foliation

We consider C¹ nonsingular flows on a closed 3-manifold under which there is no transverse disk that flows continuously back into its own interior. We provide an algorithm for modifying any branched surface transverse to such a flow ? that terminates in a branched surface carrying a foliation F precisely when F is transverse to ?. As a corollary, we find branched surfaces that do not carry foliations but that lift to ones that do.     

On the Lusternik–Schnirelmann category of Peano continua

We define the LS-category _catg${\mathrm{cat}}_g$ by means of covers of a space by general subsets, and show that this definition coincides with the classical Lusternik–Schnirelmann category for compact metric ANR spaces. We apply this result to give short dimension theoretic proofs of the Grossman–Whitehead theorem and Dranishnikov?s theorem. We compute _catg${\mathrm{cat}}_g$ for some fractal Peano continua such as Menger spaces and Pontryagin surfaces.     

Equivariant embeddings of principal Z-bundles into complex vector bundles

Let π: E→X be a principal $\text{Z}$_n-bundle and p:V→X an m-dimensional complex vector bundle over, say, a connected CW-complex X. An equivariant embedding of π into p is an embedding h:E → V commuting with projections such that h(e · z)=zh(e) for all eεE and $\text{zε}\text{Z}{}_{\mathrm{n}}\text{?S}{}^1\text{?}\text{Z}$. We compute the primary obstruction $\text{cεH}{}^{\mathrm{2m}}\text{(X;}\text{Z}\text{)}$ to embedding π equivariantly into p. If dim X?2m, then c=0 if and only if π admits an equivariant embedding into p. If dim X>2m and π embeds equivariantly into p, then c=0. Other embedding criteria exist in case p is the trivial m-plane bundle ε^m. We use these criteria for a discussion of the classification of the equivalence classes of principal $\text{Z}$-bundles that admit equivariant embeddings into ε^m. Finally, we offer an example of a principal $\text{Z}$-bundle that admit an ordinary but not an equivariant embedding into ε¹.     

The proper L-S category of Whitehead manifolds

It is shown that the proper L-S category of an eventually end-irreducible, R²-irreducible Whitehead 3-manifold is 4. For this we prove, in the category of germs at infinity of proper maps, a partial analogue of the characterization by Eilenberg and Ganea of the L-S category of an aspherical space.     

Compressing thin spheres in the complement of a link

Let L be a link in S³ that is in thin position but not in bridge position and let P be a thin level sphere with compressing disk D. We introduce the idea of alternating level spheres for D and show that all such spheres are thin and their widths are monotone decreasing. This allows us to generalize a result of Wu by giving a bound on the number of disjoint irreducible compressing disks P can have in terms of the width of P, including identifying thin spheres with unique compressing disks. We also give conditions under which P must be incompressible on some side or be weakly incompressible. In particular we show that the thin level sphere of second lowest width is weakly incompressible. If P is strongly compressible we describe how a pair of compressing disks must lie relative to the link.     

Rational homology spheres and the four-ball genus of knots

Using the Heegaard Floer homology of Ozsváth and Szabó we investigate obstructions to a rational homology sphere bounding a four-manifold with a definite intersection pairing. As an application we obtain new lower bounds for the four-ball genus of Montesinos links.     

Partial rigidity of degenerate CR embeddings into spheres

In this paper, we study degenerate CR embeddings f$f$ of a strictly pseudoconvex hypersurface M⊂Cⁿ⁺¹$M\subset {\mathbb{C}}^{n+1}$ into a sphere S$\mathbb{S}$ in a higher dimensional complex space C^N+1${\mathbb{C}}^{N+1}$. The degeneracy of the mapping f$f$ will be characterized in terms of the ranks of the CR second fundamental form and its covariant derivatives. In 2004, the author, together with X. Huang and D. Zaitsev, established a rigidity result for CR embeddings f$f$ into spheres in low codimensions. A key step in the proof of this result was to show that degenerate mappings are necessarily contained in a complex plane section of the target sphere (partial rigidity). In the 2004 paper, it was shown that if the total rank d$d$ of the second fundamental form and all of its covariant derivatives is <n$<n$ (here, n$n$ is the CR dimension of M$M$), then f(M)$f\left(M\right)$ is contained in a complex plane of dimension n+d+1$n+d+1$. The converse of this statement is also true, as is easy to see. When the total rank d$d$ exceeds n$n$, it is no longer true, in general, that f(M)$f\left(M\right)$ is contained in a complex plane of dimension n+d+1$n+d+1$, as can be seen by examples. In this paper, we carry out a systematic study of degenerate CR mappings into spheres. We show that when the ranks of the second fundamental form and its covariant derivatives exceed the CR dimension n$n$, then partial rigidity may still persist, but there is a “defect” k$k$ that arises from the ranks exceeding n$n$ such that f(M)$f\left(M\right)$ is only contained in a complex plane of dimension n+d+k+1$n+d+k+1$. Moreover, this defect occurs in general, as is illustrated by examples.     

A new a posteriori error estimate for convection–reaction–diffusion problems

A new a posteriori error estimate is derived for the stationary convection–reaction–diffusion equation. In order to estimate the approximation error in the usual energy norm, the underlying bilinear form is decomposed into a computable integral and two other terms which can be estimated from above using elementary tools of functional analysis. Two auxiliary parameter-functions are introduced to construct such a splitting and tune the resulting bound. If these functions are chosen in an optimal way, the exact energy norm of the error is recovered, which proves that the estimate is sharp. The presented methodology is completely independent of the numerical technique used to compute the approximate solution. In particular, it is applicable to approximations which fail to satisfy the Galerkin orthogonality, e.g. due to an inconsistent stabilization, flux limiting, low-order quadrature rules, round-off and iteration errors, etc. Moreover, the only constant that appears in the proposed error estimate is global and stems from the Friedrichs–Poincaré inequality. Numerical experiments illustrate the potential of the proposed error estimation technique.     

Milnor's μ-invariants,Massey products and Whitney's trick in 4 dimensions

We present the failure of Whitney's lemma in dimension 4 from the homotopical and topological viewpoints. Those are detected by Massey products. The invariants for the examples represented by framed links are computed in terms of Milnor's $\text{μ}$-invariants.     

Lifted infinitesimal holomorphic representation for the n-dimensional complex hyperbolic ball and for Cartan domains of type I

The Laplacian and Ornstein–Uhlenbeck operators on the finite dimensional complex ball are obtained from the infinitesimal holomorphic representation of the group U(n,1)$U(n,1)$. We compare the invariant measures for these operators with the unitarizing measures of the discrete series representation. Then with Hua differential calculus, we show how to extend the results to domains with matrix elements.     

Hyperbolic sections in surface bundles

We give conditions assuring that the given section in a surface bundle over the circle is hyperbolic in terms of the ‘‘projection" in the fiber surface according to the Nielsen-Thurston types of the monodromies.     

The loop expansion of the Kontsevich integral, the null-move and S-equivalence

The Kontsevich integral of a knot is a graph-valued invariant which (when graded by the Vassiliev degree of graphs) is characterized by a universal property; namely it is a universal Vassiliev invariant of knots. We introduce a second grading of the Kontsevich integral, the Euler degree, and a geometric null-move on the set of knots. We explain the relation of the null-move to S-equivalence, and the relation to the Euler grading of the Kontsevich integral. The null-move leads in a natural way to the introduction of trivalent graphs with beads, and to a conjecture on a rational version of the Kontsevich integral, formulated by the second author and proven in Geom. Top 8 (2004) 115 (see also Kricker, preprint 2000, math/GT.0005284).     

On the Lusternik-Schnirelmann category of Stiefel manifolds

We determine the Lusternik-Schnirelmann category of real Stiefel manifolds Vn,k and quaternionic Stiefel manifolds Xn,k for n?2k which is equal to the cup-length of the mod 2 cohomology of Vn,k and the integer cohomology of Xn,k, respectively.