Nielsen equalizer theory

We extend the Nielsen theory of coincidence sets to equalizer sets, the points where a given set of (more than 2) mappings agree. On manifolds, this theory is interesting only for maps between spaces of different dimension, and our results hold for sets of k maps on compact manifolds from dimension (k−1)n to dimension n. We define the Nielsen equalizer number, which is a lower bound for the minimal number of equalizer points when the maps are changed by homotopies, and is in fact equal to this minimal number when the domain manifold is not a surface.As an application we give some results in Nielsen coincidence theory with positive codimension. This includes a complete computation of the geometric Nielsen number for maps between tori.     

On preimages of a class of generalized monotone operators

In this paper we first provide a geometric interpretation of the Minty-Browder monotonicity which allows us to extend this concept to the so called h-monotonicity, still formulated in an analytic way. A topological concept of monotonicity is also known in the literature: it requires the connectedness of all preimages of the operator involved. This fact is important since combined with the local injectivity, it ensures global injectivity. When a linear structure is present on the source space, one can ask for the preimages to even be convex. In an earlier paper, the authors have shown that Minty-Browder monotone operators defined on convex open sets do have convex preimages, obtaining as a by-product global injectivity theorems. In this paper we study the preimages of h-monotone operators, by showing that they are not divisible by closed connected hypersurfaces, and investigate them from the dimensional point of view. As a consequence we deduce that h-monotone local homeomorphisms are actually global homeomorphisms, as the proved properties of their preimages combined with local injectivity still produce global injectivity.     

Compact maps and quasi-finite complexes

The simplest condition characterizing quasi-finite CW complexes K is the implication XτhK⇒β(X)τK for all paracompact spaces X. Here are the main results of the paper:

Theorem 0.1. If{Ks}_s∈Sis a family of pointed quasi-finite complexes, then their wedge_?s∈SKsis quasi-finite.     

Algorithms for finding connected separators between antipodal points

A set (or a collection of sets) contained in the Euclidean space Rm is symmetric if it is invariant under the antipodal map. Given a symmetric unicoherent polyhedron X (like an n-dimensional cube or a sphere) and an odd real function f defined on vertices of a certain symmetric triangulation of X, we algorithmically construct a connected symmetric separator of X by choosing a subcollection of the triangulation. Each element of the subcollection contains the vertices v and u such that f(v)f(u)?0.     

On two problems in extension theory

In this note we introduce the concept of a quasi-finite complex. Next, we show that for a given countable simplicial complex L the following conditions are equivalent:

•

L is quasi-finite.

•

There exists a [L]-invertible mapping of a metrizable compactum X with e-dimX?[L] onto the Hilbert cube.Finally, we construct an example of a quasi-finite complex L such that its extension type [L] does not contain a finitely dominated complex.

     

A combinatorial approach to coarse geometry

Using ideas from shape theory we embed the coarse category of metric spaces into the category of direct sequences of simplicial complexes with bonding maps being simplicial. Two direct sequences of simplicial complexes are equivalent if one of them can be transformed to the other by contiguous factorizations of bonding maps and by taking infinite subsequences. This embedding can be realized by either Rips complexes or analogs of Roe?s anti-?ech approximations of spaces.In this model coarse n-connectedness of K={K₁→K₂→?} means that for each k there is m>k such that the bonding map from Kk to Km induces trivial homomorphisms of all homotopy groups up to and including n.The asymptotic dimension being at most n means that for each k there is m>k such that the bonding map from Kk to Km factors (up to contiguity) through an n-dimensional complex.Property A of G. Yu is equivalent to the condition that for each k and for each ?>0 there is m>k such that the bonding map from |Kk| to |Km| has a contiguous approximation g:|Kk|→|Km| which sends simplices of |Kk| to sets of diameter at most ?.     

Extension dimension and quasi-finite CW-complexes

We extend the definition of quasi-finite complexes from countable complexes to arbitrary ones and provide a characterization of quasi-finite complexes in terms of L-invertible maps and dimensional properties of compactifications. Several results related to the class of quasi-finite complexes are established, such as completion of metrizable spaces, existence of universal spaces and a version of the factorization theorem. Furthermore, we define UV(L)-spaces in the realm of metrizable spaces and show that some properties of UV(n)-spaces and UV(n)-maps remain valid for UV(L)-spaces and UV(L)-maps, respectively.     

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.     

Subsequential filters

Following N. Noble, we say that a space is subsequential if it is a subspace of a sequential space. A free filter F on ω is called subsequential if the space ω∪{F} is subsequential. In this paper, we state several properties of these filters.     

Extraordinary dimension of maps

We consider the extraordinary dimension dimL introduced recently by Shchepin [E.V. Shchepin, Arithmetic of dimension theory, Russian Math. Surveys 53 (5) (1998) 975-1069]. If L is a CW-complex and X a metrizable space, then dimLX is the smallest number n such that ΣnL is an absolute extensor for X, where ΣnL is the nth suspension of L. We also write dimLf?n, where is a given map, provided dimLf⁻¹(y)?n for every y∈Y. The following result is established: Supposeis a perfect surjection between metrizable spaces, Y a C-space and L a countable CW-complex. Then conditions (1)-(3) below are equivalent:

(1)

dimLf?n;

(2)

There exists a dense andGδsubsetGofC(X,In)with the source limitation topology such thatdimL(f×g)=0for everyg∈G;

(3)

There exists a mapis such thatdimL(f×g)=0;If, in addition, X is compact, then each of the above three conditions is equivalent to the following one;

(4)

There exists anFσsetA⊂Xsuch thatdimLA?n−1and the restriction mapf|(X?A)is of dimensiondimf|(X?A)?0.

     

Selective screenability in topological groups

We examine the selective screenability property in topological groups. In the metrizable case we also give characterizations of Sc(Onbd,O) and Smirnov-Sc(Onbd,O) in terms of the Haver property and finitary Haver property respectively relative to left-invariant metrics. We prove theorems stating conditions under which Sc(Onbd,O) is preserved by products. Among metrizable groups we characterize the countable dimensional ones by a natural game.     

Making holes in hyperspaces

Let X be a metric continuum and C(X) the hyperspace of all nonempty subcontinua of X. Let A∈C(X), A is said to make a hole in C(X), if C(X)−{A} is not unicoherent. In this paper we study the following problem.Problem: For which A∈C(X), A makes a hole in C(X).In this paper we present some partial solutions to this problem in the following cases: (1) A is a free arc; (2) A is a one-point set; (3) A is a free simple closed curve; (4) A=X.     

An n-dimensional version of Steinhaus' chessboard theorem

The main result of this paper is an n-dimensional version of the Steinhaus' chessboard theorem. Our theorem implies the Poincaré theorem as well as its parametric extension. But it is known that the Poincaré theorem is equivalent to the Brouwer Fixed-Point theorem.     

Almost arcwise connected continua without dense arc components

A continuum M is almost arcwise connected if each pair of nonempty open subsets of M can be joined by an arc in M. An almost arcwise connected plane continuum without a dense arc component can be defined by identifying pairs of endpoints of three copies of the Knaster indecomposable continuum that has two endpoints. In [7] K.R. Kellum gave this example and asked if every almost arcwise connected continuum without a dense arc component has uncountably many arc components. We answer Kellum's question by defining an almost arcwise connected plane continuum with only three arc components none of which are dense. A continuum M is almost Peano if for each finite collection $\text{C}$ of nonempty open subsets of M there is a Peano continuum in M that intersects each element of $\text{C}$. We define a hereditarily unicoherent almost Peano plane continuum that does not have a dense arc component. We prove that every almost arcwise connected planar λ-dendroid has exactly one dense arc component. It follows that every hereditarily unicoherent almost arcwise connected plane continuum without a dense arc component has uncountably many arc components. Using an example of J. Krasinkiewicz and P Minc [8], we define an almost Peano λ-dendroid that do not have a dense arc component. Using a theorem of J.B. Fugate and L. Mohler [3], we prove that every almost arcwise connected λ-dendroid without a dense arc component has uncountably many arc components. In Euclidean 3-space we define an almost Peano continuum with only countably many arc components no one of which is dense. It is not known if the plane contains a continuum with these properties.     

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.     

On continuous choice of retractions onto nonconvex subsets

For a Banach space B and for a class A of its bounded closed retracts, endowed with the Hausdorff metric, we prove that retractions on elements A∈A can be chosen to depend continuously on A, whenever nonconvexity of each A∈A is less than . The key geometric argument is that the set of all uniform retractions onto an α-paraconvex set (in the spirit of E. Michael) is -paraconvex subset in the space of continuous mappings of B into itself. For a Hilbert space H the estimate can be improved to and the constant can be replaced by the root of the equation α+α²+α³=1.