A separable manifold failing to have the homotopy type of a CW-complex

Carrier and Nerve Theorems in the Extension Theory

We show that a regular cover of a general topological space provides structure similar to a triangulation. In this general setting we define analogues of simplicial maps and prove their existence and uniqueness up to homotopy. As an application we give simple proofs of sharpened versions of nerve theorems of K. Borsuk and A. Weil, which state that the nerve of a regular cover is homotopy equivalent to the underlying space. Next we prove a nerve theorem for a class of spaces with uniformly bounded extension dimension. In particular we prove that the canonical map from a separable metric n-dimen…

用普通映射的同伦类表示局部紧空间的细形状

Representing fine shape of local compacta by homotopy classes of ordinary maps

本文为每个局部紧空间构造一个伦型唯一的空间，使细形状类与普通映射同伦类一一对应，从而用普通映射表示细形状。

同调不变量的可定义内容 II：Čech 上同调与同伦分类

The definable content of homological invariants II: Čech cohomology and homotopy classification

本文将描述集合论应用于同调代数与代数拓扑，提出可定义的Čech上同调函子并证明其能精化经典函子，解决了映射望远镜同伦分类与Borsuk-Eilenberg问题的非光滑性。

A construction of Nöbeling manifolds of arbitrary weight

For each cardinal $κ$, each natural number $n$ and each simplicial complex $K$ we construct a space $ν^n_κ(K)$ and a map $π\colon ν^n_κ(K) \to K$ such that the following conditions are satisfied. 1. $ν^n_κ(K)$ is a complete metric $n$-dimensional space of weight $κ$. 2. $ν^n_κ(K)$ is an absolute neighborhood extensor in dimension $n$. 3. $ν^n_κ(K)$ is strongly universal in the class of $n$-dimensional complete metric spaces of weight~$κ$. 4. $π$ is an $n$-homotopy equivalence. For $κ= ω$ the constructed spaces are $n$-dimensional separable Nöbeling manifolds. The constructed spaces have very i…

Fine shape I

We introduce and develop fine shape, which has a very simple definition and aims to supersede all previously known shape theories for metrizable spaces. The problem with known shape theories of metrizable spaces is illustrated by the following bizarre situation. Čech cohomology is an invariant of shape, and a fortiori of strong shape. But Steenrod-Sitnikov homology is not shape invariant (already for compacta) and has not been proved to be strong shape invariant (in more than 40 years). Worse yet, there is an ordinary homology theory that is strong shape invariant by design, but it cannot be c…