Detecting codimension one manifold factors with topographical techniques

We prove recognition theorems for codimension one manifold factors of dimension n?4. In particular, we formalize topographical methods and introduce three ribbons properties: the crinkled ribbons property, the twisted crinkled ribbons property, and the fuzzy ribbons property. We show that X×R is a manifold in the cases when X is a resolvable generalized manifold of finite dimension n?3 with either: (1) the crinkled ribbons property; (2) the twisted crinkled ribbons property and the disjoint point disk property; or (3) the fuzzy ribbons property.     

A disjoint disks property for 3-manifolds

We show that the map separation property (MSP), a concept due to H.W. Lambert and R.B. Sher, is an appropriate analogue of J.W. Cannon’s disjoint disks property (DDP) for the class $\text{C}$ of compact generalized 3-manifolds with zero-dimensional singular set, modulo the Poincaré conjecture. Our main result is that the Poincaré conjecture (in dimension three) is equivalent to the conjecture that every X?$\text{C}$ with the MSP is a topological 3-manifold.     

Thin-type dense sets and related properties

We build on Gruenhage, Natkaniec, and Piotrowski?s study of thin, very thin, and slim dense sets in products, and the related notions of (NC) and (GC) which they introduced. We find examples of separable spaces X such that X² has a thin or slim dense set but no countable one. We characterize ordered spaces that satisfy (GC) and (NC), and we give an example of a separable space which satisfies (GC) but not witnessed by a collection of finite sets. We show that the question of when the topological sum of two countable strongly irresolvable spaces satisfies (NC) is related to the Rudin-Keisler order on βω. We also introduce and study the concepts of <κ-thin and superslim dense sets.     

Sections of Serre fibrations with 2-manifold fibers

It was proved by H. Whitney in 1933 that a Serre fibration of compact metric spaces admits a global section provided every fiber is homeomorphic to the unit interval [0,1]. Results of this paper extend Whitney's theorem to the case when all fibers are homeomorphic to a given compact two-dimensional manifold.     

Local sections of Serre fibrations with 3-manifold fibers

It was proved by H. Whitney in 1933 that a Serre fibration of compact metric spaces admits a global section provided every fiber is homeomorphic to the unit interval [0,1]. An extension of the Whitney's theorem to the case when all fibers are homeomorphic to some fixed compact two-dimensional manifold was proved by the authors (Brodsky et al. (2008) [2]). The main result of this paper proves the existence of local sections in a Serre fibration with all fibers homeomorphic to some fixed compact three-dimensional manifold.     

Blockers in hyperspaces

Given a metric continuum X, let X² and C(X) denote the hyperspaces of all nonempty closed subsets and subcontinua, respectively. For A,B∈X² we say that B does not block A if A∩B=∅ and the union of all subcontinua of X intersecting A and contained in X−B is dense in X. In this paper we study some sets of blockers for several kinds of continua. In particular, we determine their Borel classes and, for a large class of locally connected continua X, we recognize them as cap-sets.     

Decompositions into codimension two spheres and approximate fibrations

Let M denote an orientable (n+2)-manifold (n?1) and G an upper semicontinuous decomposition of M into compacta having the shape of the n-sphere. In this context it is shown that the decomposition space is a 2-manifold. Moreover, it is established that the decomposition map is an approximate fibration for n>1, while for n=1 the map is an approximate fibration over the complement of a locally finite set.     

Higher-dimensional Bruckner-Garg type theorem

In this paper, we investigate several properties of maps from a compactum X to an n-dimensional (combinatorial) manifold Mn. We introduce the notions of stable point and locally extreme point of map, and we prove a higher-dimensional Bruckner-Garg type theorem for the fiber structure of a generic map in the space C(X,Mn) of maps from a compactum X with dimX?n to an n-dimensional manifold Mn (n?1). As applications, we also study the spaces of Bing maps, Lelek maps, k-dimensional maps and Krasinkiewicz maps in C(X,Mn).     

A topological characterization of LF-spaces

We present a topological characterization of LF-spaces and detect small box-products that are (locally) homeomorphic to LF-spaces.     

Special manifolds and shape fibrator properties

Our main interest in this paper is further investigation of the concept of (PL) fibrators (introduced by Daverman [R.J. Daverman, PL maps with manifold fibers, J. London Math. Soc. (2) 45 (1992) 180-192]), in a slightly different PL setting. Namely, we are interested in manifolds that can detect approximate fibrations in the new setting. The main results state that every orientable, special (a new class of manifolds that we introduce) PL n-manifold with non-trivial first homology group is a fibrator in the new category, if it is a codimension-2 fibrator (Theorem 8.2) or has a non-cyclic fundamental group (Theorem 8.4). We show that all closed, orientable surface S with χ(S)<0 are fibrators in the new category.     

Preserving Z-sets by Dranishnikov's resolution

We prove that Dranishnikov's k-dimensional resolution is a UV^n − 1-divider of Chigogidze's k-dimensional resolution ck. This fact implies that preserves Z-sets. A further development of the concept of UV^n − 1-dividers permits us to find sufficient conditions for to be homeomorphic to the Nöbeling space νk or the universal pseudoboundary σk. We also obtain some other applications.     

Finite-to-one maps into Euclidean manifolds and spaces with disjoint disks properties

It is shown that every Euclidean manifold M has the following property for any m?1: If f:X→Y is a perfect surjection between finite-dimensional metric spaces, then the mapping space C(X,M) with the source limitation topology contains a dense Gδ-subset of maps g such that dimBm(g)?mdimf+dimY−(m−1)dimM. Here, Bm(g)={(y,z)∈Y×M||f−1(y)∩g−1(z)|?m}. The existence of residual sets of finite-to-one maps into product of manifolds and spaces having disjoint disks properties is also obtained.     

A nonaspherical cell-like 2-dimensional simply connected continuum and related constructions

We prove the existence of a 2-dimensional nonaspherical simply connected cell-like Peano continuum (the space itself was constructed in one of our earlier papers). We also indicate some relations between this space and the well-known Griffiths' space from the 1950s.     

Products of cell-like decompositions

This paper represents a survey concerning cell-like decompositions of manifolds. Primarily it summarizes the status of results and problems describing when the product of E¹ with such a decomposition space is again a manifold, and more generally it discusses conditions under which the product of two such decomposition spaces is also a manifold.     

Chainability and Hemmingsen's theorem

On the surface, the definitions of chainability and Lebesgue covering dimension ?1 are quite similar as covering properties. Using the ultracoproduct construction for compact Hausdorff spaces, we explore the assertion that the similarity is only skin deep. In the case of dimension, there is a theorem of E. Hemmingsen that gives us a first-order lattice-theoretic characterization. We show that no such characterization is possible for chainability, by proving that if κ is any infinite cardinal and A is a lattice base for a nondegenerate continuum, then A is elementarily equivalent to a lattice base for a continuum Y, of weight κ, such that Y has a 3-set open cover admitting no chain open refinement.     

Compact factors in finally compact products of topological spaces

We present instances of the following phenomenon: if a product of topological spaces satisfies some given compactness property then the factors satisfy a stronger compactness property, except possibly for a small number of factors.The first known result of this kind, a consequence of a theorem by A.H. Stone, asserts that if a product is regular and Lindelöf then all but at most countably many factors are compact. We generalize this result to various forms of final compactness, and extend it to two-cardinal compactness. In addition, our results need no separation axiom.     

Generalized Alexandroff-Urysohn squares and a characterization of the fixed point property

Given a Hausdorff continuum X, we introduce a topology on X×X that yields a Hausdorff continuum. We call the resulting space the Alexandroff-Urysohn square of X and prove that X has the fixed point property if and only if the Alexandroff-Urysohn square of X has the fixed point property.     

Remainders of arcwise connected compactifications of the plane

A. Lelek asked which continua are remainders of locally connected compactifications of the plane. In this paper we study a similar problem with local connectedness replaced by arcwise connectedness. (Each locally connected continuum is arcwise connected.) We give the following characterization: a continuum X is pointed 1-movable if and only if there is an arcwise connected compactification of the plane with X as the remainder.     

Locally G-homogeneous Busemann G-spaces

We present short proofs of all known topological properties of general Busemann G-spaces (at present no other property is known for dimensions more than four). We prove that all small metric spheres in locally G-homogeneous Busemann G-spaces are homeomorphic and strongly topologically homogeneous. This is a key result in the context of the classical Busemann conjecture concerning the characterization of topological manifolds, which asserts that every n-dimensional Busemann G-space is a topological n-manifold. We also prove that every Busemann G-space which is uniformly locally G-homogeneous on an orbal subset must be finite-dimensional.