广义Cantor流形与齐性

Simply connected open 3-manifolds with rigid genus one ends

We construct uncountably many simply connected open 3-manifolds with genus one ends homeomorphic to the Cantor set. Each constructed manifold has the property that any self homeomorphism of the manifold (which necessarily extends to a homeomorphism of the ends) fixes the ends pointwise. These manifolds are complements of rigid generalized Bing-Whitehead (BW) Cantor sets. Previous examples of rigid Cantor sets with simply connected complement in $R^{3}$ had infinite genus and it was an open question as to whether finite genus examples existed. The examples here exhibit the minimum possible genu…

流形：Hausdorff性与齐性

Manifolds: Hausdorffness versus homogeneity

我们在流形的框架下分析了Hausdorff性与齐性的关系，而不局限于Hausdorff性。我们展示了齐次非豪斯多夫流形的例子，并证明了Lindelöf齐次流形是豪斯多夫流形。

Effective forcing with Cantor manifolds

A set $A$ of integers is called total if there is an algorithm which, given an enumeration of $A$, enumerates the complement of $A$, and called cototal if there is an algorithm which, given an enumeration of the complement of $A$, enumerates $A$. Many variants of totality and cototality have been studied in computability theory. In this note, by an effective forcing construction with strongly infinite dimensional Cantor manifolds, which can be viewed as an effectivization of Zapletal's "half-Cohen" forcing (i.e., the forcing with Henderson compacta), we construct a set of integers whose enumer…

Foliations on non-metrisable manifolds: absorption by a Cantor black hole

We investigate contrasting behaviours emerging when studying foliations on non-metrisable manifolds. It is shown that Kneser's pathology of a manifold foliated by a single leaf cannot occur with foliations of dimension-one. On the other hand, there are open surfaces admitting no foliations. This is derived from a qualitative study of foliations defined on the long tube $\mathbb S^1\times {\mathbb L}_+$ (product of the circle with the long ray), which is reminiscent of a `black hole', in as much as the leaves of such a foliation are strongly inclined to fall into the hole in a purely vertical w…

The non-existence of common models for some classes of higher-dimensional hereditarily indecomposable continua

A continuum $K$ is a common model for the family ${\mathcal K}$ of continua if every member of ${\mathcal K}$ is a continuous image of $K$. We show that none of the following classes of spaces has a common model: 1) the class of strongly chaotic hereditarily indecomposable $n$-dimensional Cantor manifolds, for any given natural number $n$, 2) the class of strongly chaotic hereditarily indecomposable hereditarily strongly infinite-dimensional Cantor manifolds, 3) the class of strongly chaotic hereditarily indecomposable continua with transfinite dimension (small or large) equal to $α$, for any …

关于纽结与平面简单闭曲线的超空间

On hyperspaces of knots and planar simple closed curves

本文研究了 $\mathbb R^n$ 中简单闭曲线与纽结的 Vietoris 超空间，证明了这些超空间具有强局部可缩、弧连通及无穷维 Cantor 流形等拓扑性质，并阐明了特定子空间的绝对邻域收缩核属性。

Decomposition space theory

In these notes we give a brief introduction to decomposition theory and we summarize some classical and well-known results. The main question is that if a partitioning of a topological space (in other words a decomposition) is given, then what is the topology of the quotient space. The main result is that an upper semi-continuous decomposition yields a homeomorphic decomposition space if the decomposition is shrinkable (i.e. there exist self-homeomorphisms of the space which shrink the partitions into arbitrarily small sets in a controllable way). This is called Bing shrinkability criterion. I…