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.     

Refinable and monotone maps revisited

Generalizing results by J. Ford, J. W. Rogers, Jr. and H. Kato we prove that (1) a map f from a G-like continuum onto a graph G is refinable iff f is monotone; (2) a graph G is an arc or a simple closed curve iff every G-like continuum that contains no nonboundary indecomposable subcontinuum admits a monotone map onto G.We prove that if bonding maps in the inverse sequence of compact spaces are refinable then the projections of the inverse limit onto factor spaces are refinable. We use this fact to show that refinable maps do not preserve completely regular or totally regular continua.     

Pseudo-path connected homogeneous continua

The main result of this paper states that every homogeneous pseudo-path connected continuum is weakly chainable, or equivalently, every homogeneous continuum connected by continuous images of the pseudo-arc is itself a continuous image of the pseudo-arc. We notice that even though there exist homogeneous path connected continua that are not continuous images of an arc (Prajs, 2002), they all are continuous images of the pseudo-arc.     

Almost arcwise connectivity in unicoherent continua

K.R. Kellum has proved that a continuum is an almost continuous image of the interval [0, 1] if and only if it is an almost Peano continuum. Hence, a continuum is an almost continuous image of [0, 1] if it has a dense arc component.Our principal result is that any almost arcwise connected, semi-hereditarily unicoherent, metric continuum with only countably many arc components has a dense arc component. An example is given to show that this is not true for unicoherent continua in general. It is also shown that any semi-hereditarily unicoherent continuum with only countably many arc components has at most one dense arc component, and if it has a dense arc component, then every other arc component is nowhere dense. This generalizes results of Fugate and Mohler for λ-dendroids.     

Quotients of Bing spaces

A Bing space is a compact Hausdorff space whose every component is a hereditarily indecomposable continuum. We investigate spaces which are quotients of a Bing space by means of a map which is injective on components. We show that the class of such spaces does not include every compact space, but does properly include the class of compact metric spaces.     

Hereditarily indecomposable subcontinua of inverse limits of Souslin arcs are metric

We prove that if Si is a Souslin arc (a Hausdorff arc that is the compactification of a Souslin line) for each i and , then every hereditarily indecomposable subcontinuum of X is metric. Since every non-degenerate hereditarily indecomposable continuum that is an inverse limit on metric arcs is a pseudo-arc, it follows that such an X would be a pseudo-arc or a point.     

Continua for which connected partitions are compact

We investigate connected partitions of continua into compacta. In particular, we consider continua with property that every connected partition into compacta is compact. We characterize graphs which have this property as the trees and the simple closed curve. Dendrites are shown to have the property. An example of a nonlocally connected continuum with the property is also given.     

A strong characterization on the ΩEP-property

In this paper a result of A. Illanes and J.J. Charatonik obtained in [J.J. Charatonik, A. Illanes, Mappings on dendrites, Topology Appl. 144 (2004) 109-132, Corollary 5.14] is extended, by showing that a locally connected continuum X has the nonwandering-eventually-periodic property. (ΩEP-property) iff X is a dendrite that does not contain a homeomorphic copy of the null-comb. Also using “An engine breaking the ΩEP-property” constructed by P. Pyrih et al. in [P. Pyrih, J. Hladký, J. Novák, M. Sterzik, M. Tancer, An engine breaking the ΩEP-property, Topology Appl. 153 (2006) 3621-3626] the results obtained in [J.J. Charatonik, A. Illanes, Mappings on dendrites, Topology Appl. 144 (2004) 109-132; H. Méndez-Lango, On the ΩEP-property, Topology Appl. 154 (2007) 2561-2568] and [P. Pyrih, J. Hladký, J. Novák, M. Sterzik, M. Tancer, An engine breaking the ΩEP-property, Topology Appl. 153 (2006) 3621-3626] are extended, by proving that every nonlocally connected continuum X that contains a nondegenerate arc A and a point p∈A such that X is not connected in kleinen at p does not have the ΩEP-property. Answering Question 1 of [P. Pyrih, J. Hladký, J. Novák, M. Sterzik, M. Tancer, An engine breaking the ΩEP-property, Topology Appl. 153 (2006) 3621-3626]. Finally an uncountable family of non-locally connected continua containing arcs with the ΩEP-property is shown.     

Whitney preserving maps on finite graphs

For a Whitney preserving map f:X→G we show the following: (a) If X is arcwise connected and G is a graph which is not a simple closed curve, then f is a homeomorphism; (b) If X is locally connected and G is a simple closed curve, then X is homeomorphic to either the unit interval [0,1], or the unit circle S¹. As a consequence of these results, we characterize all Whitney preserving maps between finite graphs. We also show that every hereditarily weakly confluent Whitney preserving map between locally connected continua is a homeomorphism.     

Positive open book decompositions of Stein fillable 3-manifolds along prescribed links

It is known by Loi and Piergallini that a closed, oriented, smooth 3-manifold is Stein fillable if and only if it has a positive open book decomposition. In the present paper we will show that for every link L in a Stein fillable 3-manifold there exists an additional knot L^′ to L such that the link L∪L^′ is the binding of a positive open book decomposition of the Stein fillable 3-manifold. To prove the assertion, we will use the divide, which is a generalization of real morsification theory of complex plane curve singularities, and 2-handle attachings along Legendrian curves.     

Hereditarily indecomposable subcontinua of the product of two Souslin arcs are metric

We prove that if each of X and Y is a Souslin arc (a Hausdorff arc that is the compactification of a connected Souslin line), then every hereditarily indecomposable subcontinuum of X×Y is metric.     

Topological entropy of maps on regular curves

In [G.T. Seidler, The topological entropy of homeomorphisms on one-dimensional continua, Proc. Amer. Math. Soc. 108 (1990) 1025-1030], G.T. Seidler proved that the topological entropy of every homeomorphism on a regular curve is zero. Also, in [H. Kato, Topological entropy of monotone maps and confluent maps on regular curves, Topology Proc. 28 (2) (2004) 587-593] the topological entropy of confluent maps on regular curves was investigated. In particular, it was proved that the topological entropy of every monotone map on any regular curve is zero. In this paper, furthermore we investigate the topological entropy of more general maps on regular curves. We evaluate the topological entropy of maps f on regular curves X in terms of the growth of the number of components of f−n(y) (y∈X).     

Fixed point theorems for partially outward mappings

We generalize some classical theorems related to dimension. We extend Brouwer's fixed point theorem to a class of mappings whose images are not necessarily a subset of the domain. These results also generalize theorems of B.R. Halpern and G.M. Bergman. As applications, we prove some theorems for maps that pull absolute retracts outward into attached sphere collars. We note relationships to the relative Nielsen theory and show that certain of our applications can also be obtained using results of H. Schirmer.     

A study of cut points in connected topological spaces

We prove that every H(i) subset H of a connected space X such that there is no proper connected subset of X containing H, contains at least two non-cut points of X. This is used to prove that a connected space X is a COTS with endpoints if and only if X has at most two non-cut points and has an H(i) subset H such that there is no proper connected subset of X containing H. Also we obtain some other characterizations of COTS with endpoints and some characterizations of the closed unit interval.     

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.     

Brush spaces and the fixed point property

We introduce the notions of a brush space and a weak brush space. Each of these spaces has a compact connected core with attached connected fibers and may be either compact or non-compact. Many spaces, both in the Hausdorff non-metrizable setting and in the metric setting, have realizations as (weak) brush spaces. We show that these spaces have the fixed point property if and only if subspaces with core and finitely many fibers have the fixed point property. This result generalizes the fixed point result for generalized Alexandroff/Urysohn Squares in Hagopian and Marsh (2010) [4]. We also look at some familiar examples, with and without the fixed point property, from Bing (1969) [1], Connell (1959) [3], Knill (1967) [7] and note the brush space structures related to these examples.     

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 alternative proof of Lickorish–Wallace theorem

We introduce the concept of s-distance of an unstabilized Heegaard splitting. We prove if a 3-manifold admits an unstabilized genus g Heegaard splitting with s-distance m  , then surgery on some (m−1)$(m-1)$ components link may produce a 3-manifold which admits a stabilized genus g Heegaard splitting. We also give an alternative proof of the fundamental theorem of surgery theory, which states that every closed orientable 3-manifold is obtained by surgery on some link in 3-sphere.     

Closed geodesics and volume growth of open manifolds with sectional curvature bounded from below

In this paper, the relationship between the existence of closed geodesics and the volume growth of complete noncompact Riemannian manifolds is studied. First the authors prove a diffeomorphic result of such an n-m2nifold with nonnegative sectional curvature, which improves Marenich-Toponogov＇s theorem. As an application, a rigidity theorem is obtained for nonnegatively curved open manifold which contains a clesed geodesic. Next the authors prove a theorem about the nonexistence of closed geodesics for Riemannian manifolds with sectional curvature bounded from below by a negative constant.     

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).