Topologizing homeomorphism groups of rim-compact spaces

Let X be a Tychonoff space, H(X) the group of all self-homeomorphisms of X with the usual composition and the evaluation function. Topologies on H(X) providing continuity of the evaluation function are called admissible. Topologies on H(X) compatible with the group operations are called group topologies. Whenever X is locally compact T₂, there is the minimum among all admissible group topologies on H(X). That can be described simply as a set-open topology, further agreeing with the compact-open topology if X is also locally connected. We show the same result in two essentially different cases of rim-compactness. The former one, where X is rim-compact T₂ and locally connected. The latter one, where X agrees with the rational number space Q equipped with the euclidean topology. In the first case the minimal admissible group topology on H(X) is the closed-open topology determined by all closed sets with compact boundaries contained in some component of X. Moreover, whenever X is also separable metric, it is Polish. In the rational case the minimal admissible group topology on H(Q) is just the closed-open topology. In both cases the minimal admissible group topology on H(X) is closely linked to the Freudenthal compactification of X. The Freudenthal compactification in rim-compactness plays a key role as the one-point compactification does in local compactness. In the rational case we investigate whether the fine or Whitney topology on H(Q) induces an admissible group topology on H(Q) stronger than the closed-open topology.     

Classifying homeomorphism groups of infinite graphs

In this paper, we classify topologically the homeomorphism groups H(Γ) of infinite graphs Γ with respect to the compact-open and the Whitney topologies.     

Groups of measure-preserving homeomorphisms of noncompact 2-manifolds

Suppose M is a noncompact connected 2-manifold and μ is a good Radon measure of M with μ(∂M)=0. Let H(M) denote the group of homeomorphisms of M equipped with the compact-open topology and H₀(M) denote the identity component of H(M). Let H(M;μ) denote the subgroup of H(M) consisting of μ-preserving homeomorphisms of M and H₀(M;μ) denote the identity component of H(M;μ). We use results of A. Fathi and R. Berlanga to show that H₀(M;μ) is a strong deformation retract of H₀(M) and classify the topological type of H₀(M;μ).     

The mapping class group of powers of the long ray and other non-metrisable spaces

We identify the mapping class group, i.e. the space of homeomorphisms modulo isotopy, of powers of the long ray and long line as well as generalisations of the long plane obtained by taking copies of the first octant of the long plane and identifying them along their boundaries. We show that every countable group is the mapping class group of such a space. We also consider homotopy classes of continuous functions between these spaces.     

Examples of cell-like decompositions of the infinite-dimensional manifolds σ and Σ

Denote by σ the subspace of Hilbert space {(x_i)?l₂:x_i=0 for all but finitely many i}. Examples of cell-like decompositions of σ are constructed that have decomposition spaces that are not homeomorphic to σ. At one extreme is a cell-like decomposition G of σ produced using ghastly finite dimensional examples such that the decomposition space σ?G contains no embedded 2-cell but (σ?G)×$\text{R}$ is homeomorphic to σ. At the other extreme is a cell-like decomposition G of σ satisfying: (a) the nondegeneracy set N_G={g?G:g≠point} consists of countably many arcs (necessarily tame); (b) the nondegeneracy set N_G is a closed subset of the decomposition space σ?G; (c) each map f:B²→σ?G of a 2-cell into σ?G can be approximated arbitrarily closely by an embedding; (d) σ?G is not homeomorphic to σ but (σ?G)×$\text{R}$ is homeomorphic to σ. The fact that both conditions (a) and (b) can be satisfied (and have (d) hold) is directly attributable to σ’s incompleteness as a topological space.     

Folding a surface to a polygon

A concept of folding for compact connected surfaces, involving the partition of the surface into combinatorially identical n-sided topological polygons, is defined. The existence of such foldings for given n and given surfaces is explored, with definitive results for the sphere and the torus. We obtain necessary conditions for the existence of such foldings in all other cases.Supported by Kuwait University Grant SM 043.     

Topology of manifolds modeled on countable direct limits of Menger compacta

We construct n-dimensional counterparts of manifolds modeled on the space ?² equipped by the bounded weak topology (-manifolds). For -manifolds we prove the characterization, triangulation and classification theorems. In addition, a universal map of onto Q^∞ (the countable direct limit of Hilbert cubes and Z-embeddings) is constructed and characterized.     

On the topology of graph picture spaces

We study the space of pictures of a graph G in complex projective d-space. The main result is that the homology groups (with integer coefficients) of are completely determined by the Tutte polynomial of G. One application is a criterion in terms of the Tutte polynomial for independence in the d-parallel matroids studied in combinatorial rigidity theory. For certain special graphs called orchards, the picture space is smooth and has the structure of an iterated projective bundle. We give a Borel presentation of the cohomology ring of the picture space of an orchard, and use this presentation to develop an analogue of the classical Schubert calculus.     

Separable complete ANR's admitting a group structure are Hilbert manifolds

Let X be a complete-metrizable, separable ANR. The following two facts are shown: (a) if X admits a topological group structure, then either this is a Lie group structure or X is an l₂-manifold; (b) If X is a closed convex set in a complete metric linear space, then X is either locally compact or homeomorphic to l₂.     

Existence of three solutions for a non-homogeneous Neumann problem through Orlicz-Sobolev spaces

The aim of this paper is to establish a multiplicity result for an eigenvalue non-homogeneous Neumann problem which involves a nonlinearity fulfilling a nonstandard growth condition. Precisely, a recent critical points result for differentiable functionals is exploited in order to prove the existence of a determined open interval of positive eigenvalues for which the problem admits at least three weak solutions in an appropriate Orlicz-Sobolev space.     

The disconnection number of a graph

The disconnection number d(X) is the least number of points in a connected topological graph X such that removal of d(X) points will disconnect X (Nadler, 1993 [6]). Let Dn denote the set of all homeomorphism classes of topological graphs with disconnection number n. The main result characterizes the members of Dn+1 in terms of four possible operations on members of Dn. In addition, if X and Y are topological graphs and X is a subspace of Y with no endpoints, then d(X)?d(Y) and Y obtains from X with exactly d(Y)−d(X) operations. Some upper and lower bounds on the size of Dn are discussed.The algorithm of the main result has been implemented to construct the classes Dn for n?8, to estimate the size of D₉, and to obtain information on certain subclasses such as non-planar graphs (n?9) and regular graphs (n?10).     

The D-property of some Lindelöf spaces and related conclusions

It is shown that the space Cp(τω) is a D-space for any ordinal number τ, where . This conclusion gives a positive answer to R.Z. Buzyakova's question. We also prove that another special example of Lindelöf space is a D-space. We discuss the D-property of spaces with point-countable weak bases. We prove that if a space X has a point-countable weak base, then X is a D-space. By this conclusion and one of T. Hoshina's conclusion, we have that if X is a countably compact space with a point-countable weak base, then X is a compact metrizable space. In the last part, we show that if a space X is a finite union of θ-refinable spaces, then X is a αD-space.     

Boundaries and complements of infinite- dimensional manifolds in the model space

Let M be a manifold modeled on a locally convex linear metric space E≌E^ω (or ≌E^ω_f and N a Z-submanifold of M. Then N is collared in M. In this paper, we study the following problem [1, 3]: Under what conditions can M be embedded in E so that N is the topological boundary of M in E? We gain a more mild sufficient condition than the previous papers [7, 8] and a necessary and sufficient condition in the case M has the homotopy type of Sⁿ (and each component of N is simply connected if n?2) and in the case N has the homotopy type of Sⁿ (n?2). Also we obtain a necessary and sufficient condition under which M can be embedded in E so that bd M = N and cl(E\M) has the homotopy type of Sⁿ (we assume that M and N are simply connected if n ? 2).     

A pair of spaces of upper semi-continuous maps and continuous maps

For a Tychonoff space X, we use ↓USC(X) and ↓C(X) to denote the families of the regions below all upper semi-continuous maps and of the regions below all continuous maps from X to I=[0,1], respectively. In this paper, we consider the spaces ↓USC(X) and ↓C(X) topologized as subspaces of the hyperspace Cld(X×I) consisting of all non-empty closed sets in X×I endowed with the Vietoris topology. We shall prove that ↓USC(X) is homeomorphic (≈) to the Hilbert cube Q=ω[−1,1] if and only if X is an infinite compact metric space. And we shall prove that (↓USC(X),↓C(X))≈(Q,c₀), where , if and only if ↓C(X)≈c₀ if and only if X is a compact metric space and the set of isolated points is not dense in X.     

The random-cluster model on the complete graph

Summary The random-cluster model of Fortuin and Kasteleyn contains as special cases the percolation, Ising, and Potts models of statistical physics. When the underlying graph is the complete graph onn vertices, then the associated processes are called mean-field. In this study of the mean-field random-cluster model with parametersp=/n andq, we show that its properties for any value ofq(0, ) may be derived from those of an Erds-Rényi random graph. In this way we calculate the critical point _c (q) of the model, and show that the associated phase transition is continuous if and only ifq2. Exact formulae are given for _C (q), the density of the largest component, the density of edges of the model, and the free energy. This work generalizes earlier results valid for the Potts model, whereq is an integer satisfyingq2. Equivalent results are obtained for a fixed edge-number random-cluster model. As a consequence of the results of this paper, one obtains large-deviation theorems for the number of components in the classical random-graph models (whereq=1).     

Quasilinear elliptic non-homogeneous Dirichlet problems through Orlicz–Sobolev spaces

In this paper, we are interested in the existence of infinitely many weak solutions for a non-homogeneous eigenvalue Dirichlet problem. By using variational methods, in an appropriate Orlicz–Sobolev setting, we determine intervals of parameters such that our problem admits either a sequence of non-negative weak solutions strongly converging to zero provided that the non-linearity has a suitable behaviour at zero or an unbounded sequence of non-negative weak solutions if a similar behaviour occurs at infinity.     

Compactifications determined by subsets of C1(X)

It is well known that every compactification of a completely regular space X can be generated, via a Tychonoff-type embedding, by some suitably chosen subset of C¹(X). Different subsets may give rise to equivalent compactifications, and we are concerned with the problem of finding all subsets of C¹(X) which yield a given compactification αX. The problem is easier if generalized: we say that a subset F of C¹(X) “determines” the compactification αX if αX is the smallest compactification to which every element of F extends, and give a simple necessary and sufficient condition for F to determine a given compactification αX. A number of sufficient conditions for two sets to determine the same compactification are given, and the relation between sets which determine αX and those which generate αX (via an embedding) is considered. Generally, a much smaller set of functions is required to determine αX than to generate it; the number needed to determine αX is never more than the weight of αX?X, while the number required to generate it is, if infinite, equal to the weight of αX.