Absolute Neighborhood Research Articles

Twisted Products: Enveloping Actions and Equivariant Absolute Neighborhood Extensors

Twisted Products: Enveloping Actions and Equivariant Absolute Neighborhood Extensors

Free Banach lattices generated by a lattice and projectivity

In this article we deal with the free Banach lattice F B L ⟨ L ⟩ FBL\langle \mathbb {L} \rangle generated by a lattice L \mathbb {L} . We prove that if F B L ⟨ L ⟩ FBL\langle \mathbb {L} \rangle is projective then L \mathbb {L} has a maximum and a minimum. On the other hand, we show that if L \mathbb {L} has maximum and minimum then F B L ⟨ L ⟩ FBL\langle \mathbb {L} \rangle is 2 2 -lattice isomorphic to a C ( K ) C(K) -space. As a consequence, F B L ⟨ L ⟩ FBL\langle \mathbb {L} \rangle is projective if and only if it is lattice isomorphic to a C ( K ) C(K) -space with K K being an absolute neighborhood retract. As an application, we characterize those linearly ordered sets and Boolean algebras for which the corresponding free Banach lattice is projective.

Recent advances in the Lefschetz fixed point theory for multivalued mappings

In 1923 S. Lefschetz proved the famous fixed point theorem which is now known as the Lefschetz fixed point theorem (comp. [5], [9], [20], [21]. The multivalued case was considered for the first time in 1946 by S. Eilenberg and D. Montgomery ([10]). They proved the Lefschetz fixed point theorem for acyclic mappings of compact ANR-spaces (absolute neighborhood retracts (see [4] or [13]) using Vietoris mapping theorem (see [4], [13], [16]) as the main tool. In 1970 Eilenberg, Montgomery's result was generalized for acyclic mappings of complete ANR-s (see [17]). Next, a class of admissible multivalued mappings was introduced ([13] or [16]).
 Note that the class of admissible mappings is quite large and contains as a special case not only acyclic mappings but also infinite compositions of acyclic mappings. For this class of multivalued mappings several versions of the Lefschetz fixed point theorem were proved (comp. [11], [13]-15], [18], [19], [27]). In 1982 G. Skordev and W. Siegberg ([26]) introduced the class of multivalued mappings so-called now (1 − n)-acyclic mappings. Note that the class (1−n)-acyclic mappings contain as a special case n-valued mappings considered in [6], [12], [28]. We recommend [8] for the most important results connected with (1 − n)-acyclic mappings. Finally, the Lefschetz fixed point theorem was considered for spheric mappings (comp. [3], [2], [7], [23]) and for random multivalued mappings (comp. [1], [2], [13]). Let us remark that the main classes of spaces for which the Lefschetz fixed point theorem was formulated are the class of ANR-spaces ([4]) and MANR-spaces (multi absolute neighborhood retracts (see [27]). The aim of this paper is to recall the most important results concerning the Lefschetz fixed point theorem for multivalued mappings and to prove new versions of this theorem mainly for AANR-spaces (approximative absolute neighborhood retracts (see [4] or [13]) and for MANR-s. We believe that this article will be useful for analysts applying topological fixed point theory for multivalued mappings in nonlinear analysis, especially in differential inclusions.

Terrain Hazard Risk Analysis for Flood Disaster Management in Chaohu Basin, China, Based on Two-Dimensional Cloud

Terrain analysis is essential to flood disaster risk evaluation. It is a complicated evaluation process, involving both quantitative and qualitative data analysis. However, quantitative and qualitative data cannot be put into operation directly. Based on stochastic and fuzzy mathematics, cloud models allow interchange between qualitative and quantitative data, dealing with randomness and ambiguity. Two- or multi-dimensional cloud models can solve the problem of multivariable analysis. This study used absolute elevation and neighborhood elevation standard deviation as main factors. Using the model, it demonstrated the construction of qualitative conditions and risk evaluation clouds and established a set of two-dimensional cloud reasoning rules to calculate the joint certainties with all the grids in reasoning rules. By selecting the highest certainty of cloud reasoning, preliminary evaluation results were obtained. For more accurate results, the model algorithm was improved, and further iterations were performed. The results of two-dimensional cloud reasoning showed better dispersion and precision than traditional methods did. The terrain risk distribution of Chaohu Basin, China, agreed with reality with great detail. A new method regarding the risk assessment of flood disaster was also proposed.

Abstract P398: Relative versus Absolute Neighborhood Economic Conditions and Risk of Stroke in a United States National Cohort

Introduction: Neighborhood economic conditions are important predictors of incident stroke. Neighborhood economic conditions can be characterized differently but studies have not identified which characterizations are most predictive of incident stroke. Previous work has suggested that the income distribution within the population may be more relevant for chronic disease than the absolute value of median income of the population. This project aims to estimate the effects of neighborhood income and neighborhood income inequality on incidence stroke. Hypothesis: Neighborhood income inequality and absolute neighborhood income will predict incident stroke above and beyond demographic characteristics and individual socioeconomic status. Methods: We used data from the REasons for Geographic and Racial Differences in Stroke (REGARDS) study, a national longitudinal study of black and white older adults (N=29,443). Baseline median household income and the Gini coefficient of income inequality at the Census Tract level were obtained from the 2000 decennial census. Both variables were standardized to have a mean of 0 and a standard deviation of 1; z-score of median household income was multiplied by -1 (representing neighborhood income deprivation) to allow for comparability. Using Cox Proportional Hazards analysis we examined the risk of stroke associated with these exposures in separate models. Model 1 was unadjusted, Model 2 was adjusted for demographic characteristics (age, race, sex, marital status), and Model 3 was adjusted for individual socioeconomic characteristics (household income and education). Results: Within Model 1, neighborhood income deprivation (HR: 1.18; 95% CI 1.11-1.24) and neighborhood income inequality (HR: 1.10; 95% CI: 1.05-1.16) were both significantly associated with risk of stroke. The association persisted in Model 2: neighborhood income deprivation (HR: 1.13; 95% CI 1.07-1.20) and neighborhood income inequality (HR: 1.06; 95% CI: 1.00-1.11). Within Model 3, neighborhood income deprivation maintained its positive association (HR: 1.06; 95% CI 1.00-1.13), while neighborhood income inequality was no longer predictive of stroke risk (HR:1.04; 95% CI 0.99-1.10). Conclusions: Neighborhood income deprivation and neighborhood income inequality were similar predictors of incident stroke. The effect of absolute economic resources in a Census Tract was slightly larger than the relative economic standards within a Census Tract. There is a need to investigate other characterizations of neighborhood economic conditions, such as residential income segregation. The findings from this project suggest that there are multiple policy levels that can be used to decrease incident stroke, through decreasing income inequality and improving the absolute economic conditions of deprived neighborhoods.

Alternative to Morse–Novikov theory for a closed 1-form. I

This paper extends the Alternative to Morse–Novikov theory we have proposed in Burghelea (New topological invariants for real- and angle-valued maps, Word Scientific, Hackensack, 2018) from real- and angle-valued maps to closed 1-forms. For a topological closed 1-form $$\omega $$ on a compact ANR ( $$=$$ absolute neighborhood retract) X, a concept generalizing closed differential 1-form on a compact manifold, under the mild hypothesis of tameness, a field $$\kappa $$ and a non-negative integer r, we propose two configurations $$\varvec{\delta }^\omega _r{:}\,\mathbb {R}\rightarrow \mathbb {Z}_{\geqslant 0}$$ and which recover Novikov–Betti numbers and the Novikov complex associated with a Morse closed 1-form with non-degenerated zeros. Precisely, the sum of the multiplicities of the points in the support of $$\varvec{\delta }^\omega _r$$ equals the rth Novikov–Betti number and that of the points in the support of $$\varvec{\gamma }^\omega _r$$ equals the rank of the boundary map in the Novikov complex. We formulate the basic properties of these configurations, the stability and the Poincare duality when X is a $$\kappa $$ -orientable closed topological manifold, which in full generality will be proven in the second part of this work.

Homotopy Properties of the Space If(X) of Idempotent Probability Measures

A subspace If(X) of the space of idempotent probability measures on a given compact space X is constructed. It is proved that if the initial compact space X is contractible, then If(X) is a contractible compact space as well. It is shown that the shapes of the compact spaces X and If(X) are equal. It is also proved that, given a compact space X, the compact space If(X) is an absolute neighborhood retract if and only if so is X.

Isovariant fibrant spaces

A G-map f:X→Y is called isovariant if it preserves isotropy groups (stabilizers), that is, Gx=Gf(x) for all x∈X. For a given compact group G we consider the category Isov-M of metrizable G-spaces and isovariant maps. In a natural way the notion of an isovariant absolute (neighborhood) extensor (Isov-A(N)E) can be introduced; this was done in [1]. Using this notion we define the concept of an isovariant fibrant space and study its properties. In particular, we show that every compact metric G-space admits an isovariant fibrant extension.

Proper homotopy types and [formula omitted]-boundaries of spaces admitting geometric group actions

We extend several techniques and theorems from geometric group theory so they apply to geometric actions on arbitrary proper metric ARs (absolute retracts). Previous versions often required actions on CW complexes, manifolds, or proper CAT(0) spaces, or else included a finite-dimensionality hypothesis. We remove those requirements, providing proofs that simultaneously cover all of the usual variety of spaces. A second way that we generalize earlier results is by eliminating a freeness requirement often placed on the group actions. In doing so, we allow for groups with torsion.The main theorems are new in that they generalize results found in the literature, but a significant aim is expository. Toward that end, brief but reasonably comprehensive introductions to the theories of ANRs (absolute neighborhood retracts) and Z-sets are included, as well as a much shorter introduction to shape theory. Here is a sampling of the theorems proved here.Theorem.If quasi-isometric groupsG andH act geometrically on proper metric ARsX andY, resp., thenX is proper homotopy equivalent toY.Theorem.If quasi-isometric groupsG andH act geometrically on proper metric ARsX andY, resp., andY can be compactified to aZ-structureY¯,Z forH, then the same boundary can be added toX to obtain aZ-structure forG.Theorem.If quasi-isometric groupsGandH admitZ-structuresX¯,Z1 andY¯,Z2, resp., thenZ1 andZ2 are shape equivalent.

Topological properties in Whitney blocks

Let C(X) be the hyperspace of subcontinua of a continuum X. A Whitney block is a set of the form μ−1([s,t]), where μ:C(X)→[0,1] is a Whitney map and 0≤s<t≤1. In this paper, we study the following implication: if X has property P, then each Whitney block in C(X) has property P. We consider the following properties: connectedness im kleinen, being absolute neighborhood retract, local contractibility, and m-mutual aposyndesis.

Different types of relative contractibility and their applications

In this article the new properties of relative retracts in the context of relative homotopy are studied. The results of the studies are particularly applied to the characterization of connected and locally connected spaces and absolute neighborhood retracts.

On Essential Fixed Points of Compact Mappings on Arbitrary Absolute Neighborhood Retracts and Their Application to Multivalued Fractals

The existence of essential fixed points is proved for compact self-maps of arbitrary absolute neighborhood retracts, provided the generalized Lefschetz number is nontrivial and the topological dimension of a fixed point set is equal to zero. Furthermore, continuous self-maps of some special compact absolute neighborhood retracts, whose Lefschetz number is nontrivial, are shown to possess pseudo-essential fixed points even without the zero dimensionality assumption. Both results are applied to the existence of essential and pseudo-essential multivalued fractals. An illustrative example of this application is supplied.

Fixed points of parameterized perturbations

Let X be a convex subset of a locally convex topological vector space, let U⊂X be open with U¯ compact, let F:U¯→X be an upper semicontinuous convex valued correspondence with no fixed points in U¯∖U, let P be a compact absolute neighborhood retract, and let ρ:U¯→P be a continuous function. We show that if the fixed point index of F is not zero, then there is a neighborhood V of F in the (suitably topologized) space of upper semicontinuous convex valued correspondences from U¯ to X such that for any continuous function g:P→V there is a p∈P and a fixed point x of g(p) such that ρ(x)=p. This implies that no normal form game satisfies the conditions specified in Section 4.6 of Levy (2013).

Internally [formula omitted]-like spaces and internal inverse limits

We establish equivalences between compacta that admit mappings that limit to the identity, and compacta that are inverse limits of the images under these maps. Our results have relationships to Mardešić and Segalʼs equivalence between polyhedra-like compacta and inverse limits of polyhedra, to the Anderson–Choquet Embedding Theorem, to approximative absolute neighborhood retracts, and to continua that are approximated from within as defined by C.A. Eberhart and J.B. Fugate.

Fixed point homomorphisms for parameterized maps

Let X be an ANR (absolute neighborhood retract), \({\Lambda}\) a k-dimensional topological manifold with topological orientation \({\eta}\) , and \({f : D \rightarrow X}\) a locally compact map, where D is an open subset of \({X \times \Lambda}\) . We define Fix(f) as the set of points\({{(x, \lambda) \in D}}\) such that \({x = f(x, \lambda)}\) . For an open pair (U, V) in \({X \times \Lambda}\) such that \({{\rm Fix}(f) \cap U \backslash V}\) is compact we construct a homomorphism \({\Sigma_{(f,U,V)} : H^{k}(U, V ) \rightarrow R}\) in the singular cohomologies H* over a ring-with-unit R, in such a way that the properties of Solvability, Excision and Naturality, Homotopy Invariance, Additivity, Multiplicativity, Normalization, Orientation Invariance, Commutativity, Contraction, Topological Invariance, and Ring Naturality hold. In the case of a \({C^{\infty}}\) -manifold \({\Lambda}\) , these properties uniquely determine \({\Sigma}\) . By passing to the direct limit of \({\Sigma_{(f,U,V)}}\) with respect to the pairs (U, V) such that \({K = {\rm Fix}(f) \cap U \backslash V}\) , we define a homomorphism \({\sigma_{(f,K)} : {H}_{k}({\rm Fix}(f), Fix(f) \backslash K) \rightarrow R}\) in the Cech cohomologies. Properties of \({\Sigma}\) and \({\sigma}\) are equivalent each to the other. We indicate how the homomorphisms generalize the fixed point index.

The properties of the multi-valued domination of metric spaces

In this paper the properties of the multi-valued domination of metric spaces are presented, which allows to observe interesting relationships and similarities between absolute neighborhood multi-retracts (absolute multi-retracts) and absolute neighborhood retracts (absolute retracts).

Retracts and extensors of compact transformation groups

In this paper, we study G-spaces, i.e., topological spaces with actions of a compact group G, from the point of view of the theory of retracts. We recall definitions and properties of absolute retracts, absolute neighborhood retracts, absolute extensors, and absolute neighborhood extensors with respect to the equivariant weakly hereditary class \( \mathfrak{G} \) of G-spaces. Also, we give basic notions of the theory of topological transformation groups.

A characterization of semiprojectivity for commutative C *-algebras

Given a compact, metric space X, we show that the commutative C*-algebra C(X) is semiprojective if and only if X is an absolute neighborhood retract of dimension at most one. This confirms a conjecture of Blackadar. Generalizing to the non-unital setting, we derive a characterization of semiprojectivity for separable, commutative C*-algebras. As further application of our findings we verify two conjectures of Loring and Blackadar in the commutative case, and we give a partial answer to the question, when a commutative C*-algebra is weakly (semi-)projective.

Periodic point free continuous self-maps on graphs and surfaces

We prove the following three results. We denote by Per(f) the set of all periods of a self-map f.Let G be a connected compact graph such that dimQH1(G,Q)=r, and let f:G→G be a continuous map. If Per(f)=∅, then the eigenvalues of f⁎1 are 1 and 0, this last with multiplicity r−1, where f⁎1 is the induced action of f on the first homological space.Let Mg,b be an orientable connected compact surface of genus g⩾0 with b⩾0 boundary components, and let f:Mg,b→Mg,b be a continuous map. The degree of f is d if b=0. If Per(f)=∅, then the eigenvalues of f⁎1 are 1, d and 0, this last with multiplicity 2g−2 if b=0; and 1 and 0, this last with multiplicity 2g+b−2 if b>0.Let Ng,b be a non-orientable connected compact surface of genus g⩾1 with b⩾0 boundary components, and let f:Ng,b→Ng,b be a continuous map. If Per(f)=∅, then the eigenvalues of f⁎1 are 1 and 0, this last with multiplicity g+b−2.The tools used for proving these results can be applied for studying the periodic point free continuous self-maps of many other compact absolute neighborhood retract spaces.

The point in weak semiprojectivity and AANR compacta

We initiate the study of pointed approximative absolute neighborhood retracts. Our motivation is to generate examples of C ⁎ -algebras that behave in unexpected ways with respect to weak semiprojectivity. We consider both weak semiprojectivity (WSP) and weak semiprojectivity with respect to the class of unital C ⁎ -algebras (WSP1). For a non-unital C ⁎ -algebra, these are different properties. One example shows a C ⁎ -algebra A can fail to be WSP while its unitization A ˜ is WSP. Another example shows WSP1 is not closed under direct sums.