On the Borsuk conjecture concerning homotopy domination

In the seminal monograph "Theory of retracts", Borsuk raised the following question: suppose two compact ANR's are $h$--equal, i.e. mutually homotopy dominate each other, are they homotopy equivalent? The current paper approaches this question in two ways. On one end, we provide conditions on the fundamental group which guarantee a positive answer to the Borsuk question. On the other end, we construct various examples of compact $h$--equal, not homotopy equivalent continua, with distinct properties. The first class of these examples has trivial all known algebraic invariants (such as homology, homotopy groups etc.) The second class is given by $n$--connected continua, for any $n$, which are infinite $CW$--complexes, and hence ANR's, on a complement of a point.Comment: 18 pages, 6 figures; final version accepted for publicatio

Constructing near-embeddings of codimension one manifolds with countable dense singular sets

The purpose of this paper is to present, for all $n\ge 3$, very simple examples of continuous maps $f:M^{n-1} \to M^{n}$ from closed $(n-1)$-manifolds $M^{n-1}$ into closed $n$-manifold $M^n$ such that even though the singular set $S(f)$ of $f$ is countable and dense, the map $f$ can nevertheless be approximated by an embedding, i.e. $f$ is a {\sl near-embedding}

On manifolds with nonhomogeneous factors

We present simple examples of finite-dimensional connected homogeneous spaces (they are actually topological manifolds) with nonhomogeneous and nonrigid factors. In particular, we give an elementary solution of an old problem in general topology concerning homogeneous spaces

On continua with homotopically fixed boundary

Abstract The paper presents two subcontinua of R n , one Peano-continuum, and one cellular continuum with trivial fundamental group. Both of them have the remarkable property that neither the entire spaces nor (roughly speaking) any part of them is homotopy equivalent to a lower-dimensional space. This extends work of the last three authors and of Karimov from the planar case to the higher-dimensional case, but it also contains in the cellular case the first example with all these properties in dimension two

On uniqueness of Cartesian products of surfaces with boundary

AbstractIt is known that if one of the factors of a decomposition of a manifold into Cartesian product is an interval then the decomposition is not unique. We prove that the decomposition of a 4-manifold (possibly with boundary) into 2-dimensional factors is unique, provided that the factors are not products of 1-manifolds

CONSTRUCTING NEAR-EMBEDDINGS OF CODIMENSION ONE MANIFOLDS WITH COUNTABLE DENSE SINGULAR SETS

Abstract. We present, for all n ≥ 3, very simple examples of continuous maps f: M n−1 → M n from closed (n − 1)-manifolds M n−1 into closed n-manifolds M n such that even though the singular set S(f) of f is countable and dense, the map f can nevertheless be approximated by an embedding, i.e. f is a near-embedding. In dimension 3 one can get even a piecewise-linear approximation by an embedding. 1