Absolute Neighborhood Research Articles

On averaging Lefschetz numbers

Let (E, p, X) be a regular covering space where E is a connected metric ANR (absolute neighborhood retract) and let f : X → X f:X \to X be a map. This paper investigates the relationship between the Lefschetz number of f and those of its lifts, i.e. maps f ′ : E → E f’:E \to E so that p f ′ = f p pf’ = fp . In particular, it is shown that to a lift f ′ : E → E f’:E \to E one may associate a class of lifts L ( f ′ ) \mathfrak {L}(f’) with the property that the Lefschetz number of f is equal to the average of the Lefschetz numbers of maps in L ( f ′ ) \mathfrak {L}(f’) .

Concerning the unions of absolute neighborhood retracts having brick decompositions

Concerning the unions of absolute neighborhood retracts having brick decompositions

On a generalization of absolute neighborhood retracts

On a generalization of absolute neighborhood retracts

On the singularity of Mazurkiewicz in absolute neighborhood retracts

On the singularity of Mazurkiewicz in absolute neighborhood retracts

An equivalence theorem for embeddings of compact absolute neighborhood retracts

In [2] the authors showed that if A is a k-dimensional polyhedron topologically embedded in En (2k +2 _ n, n _ 5) such that En -A is 1 =ULC, then for each E>0, there is an e-push h of (En, A) such that hi A: A--En is piecewise linear. Hence, a well-known theorem of Bing and Kister [1, Theorem 5.5 ] applies to prove Theorem 1 when Ao is a polyhedron. In fact, Theorem 5.5 of [1], together with the techniques of Homma [4] and Gluck [3] and the following engulfing theorem proved in [2], make our result possible.

An Equivalence Theorem for Embeddings of Compact Absolute Neighborhood Retracts

An Equivalence Theorem for Embeddings of Compact Absolute Neighborhood Retracts

On embeddings of compacta in Euclidean space

In [5] it was shown that any two embeddings of a k-dimensional compact absolute neighborhood retract into euclidean n-space, En(n > 5 and 2k+2 <n), are equivalent by an ambient isotopy of En, provided the complement of each embedding is uniformly locally 1-connected (abbreviated 1-ULC). The purpose of this paper is to prove a generalization of the result stated above for embeddings of arbitrary compacta in En. The following theorem is the main result.

Some remarks concerning the mappings of the inverse limit into an absolute neighborhood retract and its applications to cohomotopy groups

Some remarks concerning the mappings of the inverse limit into an absolute neighborhood retract and its applications to cohomotopy groups

ANR divisors and absolute neighborhood contractibility

ANR divisors and absolute neighborhood contractibility

Retraction in 𝑚-paracompact spaces

In this paper we show that such a method is available for the class of m-paracompact normal spaces, and more generally for any classes of normal spaces which can be characterized by the normality of their product with a compact Hausdorff space. We are basically motivated by Morita's characterization of an m-paracompact normal space (Theorem 1). In Lemma 1 we establish a sort of distributive property over products which the attaching of spaces possesses, and we are then easily able to prove that the adjunction space of two m-paracompact normal spaces is m-paracompact normal. Then by some known techniques, it follows that if Q is the class of m-paracompact normal spaces and XQ and X is an AR(Q) if and only if X is a contractible ANR(Q). 2. Preliminaries. Let m be an infinite cardinal number. A space X is m-paracompact if every open cover of cardinality <m has a locally finite open refinement. We let I denote the closed unit interval [0, 1], and Im denotes the product space of m copies of I. The reader is referred to [6] for some extensive results on m-paracompact spaces, and to [3] for definitions and basic properties of an AR(Q), resp. ANR(Q), i.e., absolute retract, resp. absolute neighborhood retract for a class Q of normal spaces and an ES(Q), resp. NES(Q), i.e., extension space, resp. neighborhood extension space for a class Q of normal spaces. Morita [6, p. 229] characterizes an m-paracompact normal space in the following useful way:

Locally equiconnected spaces and absolute neighborhood retracts

Locally equiconnected spaces and absolute neighborhood retracts

Test spaces for normal spaces

Let Q be a class of topological spaces. For X in Q, let dim X denote the dimension of X defined in terms of finite open coverings. If Q is the class of all normal spaces, then the dimension of spaces X in Q is characterized by the following theorem [2, p. 212]: dimX^n if, and only if, given any closed subset C of X, any continuous mapping of C into the «-sphere Sn has a continuous extension over X with respect to S. The possibility of replacing Sn in the above theorem by a more general space Y was explored in two earlier papers [8; 9]. Such a space Y we call a test space for Q-spaces of dimension n. To be precise, F is a test space for Q-spaces of dimension n if it satisfies the statement: A Q-space X has dimension ^ra if, and only if, given any closed subset C of X, any continuous mapping of C into Y has a continuous extension over X with respect to Y. A space Y is called an absolute neighborhood retract relative to the class Q (abbreviated ANR(C)) if (a) Y is in Q, and (b) whenever Y is imbedded topologically as a closed subset of a Q-space z, then F is a retract of some neighborhood of Y in Z (see [3]). Kodama has shown in [5] that an «-dimensional metric space F is an ANR (metric) if and only if Y is LC. The results of [8 ; 9 ] may be formulated as follows : if Q is the class of all metric spaces, then an «-dimensional, w-LC, metric space Y is a test space for Q-spaces of dimension n if, and only if, Y is an ANR(metric) and is the homotopy type of a wedge of «-spheres (i.e., a union of «-spheres with a single common point). In this paper we shall prove a similar theorem for Q the class of all normal spaces. In [9] we showed that if Y and Z are both ANR(metric) and of the same homotopy type, then F is a test space for metric spaces of dimension « if and only if Z is. Needed in the proof was Borsuk's homotopy extension theorem. Dowker showed in [2, p. 205] that the homotopy extension theorem holds for mappings of a normal space into an ANR (compact metric) provided homotopy is replaced by uniform homotopy. Using this result, we can prove the following lemma:

An order relation among topological spaces

In this paper we define a partial ordering on a class of topological spaces as follows('): For two spaces X and Y in the class, we will say X Y which is an onto a-map. (Recall that f is an a-map if there is an open cover ,B of Y such that f-'(13) refines a.) It is easy to check that the relation < is reflexive and transitive. In [7], Kuratowski proved that the closed n-dimensional ball is not _ the n-sphere. More recently, Ulam [8] raised the question whether the 2-dimensional ball is < the 2-dimensional torus. Ulam's question has stimulated interest in the ordering itself, and a result of T. Ganea [5] states, roughly speaking, that if X < Y and Y is a compact n-dimensional manifold and X is an absolute neighborhood retract, then X has the same homotopy type as an n-dimensional manifold. This result then gives Ulam's question a negative answer(2), and suggests studying properties of a space Y which are inherited by a space X satisfying X _ Y. The purpose of this paper is to present certain properties which are inherited, in the above sense. Also, comparison is made between this ordering and the ordering given by dimension (Theorem 1), and (for the class of absolute neighborhood retracts) the ordering given by a-domination for every cover a (Theorem 2). Since the property of possessing a fixed point under every continuous function will be shown to be inherited, another question of Ulam's is answered here [8, Chapter IV, Problem 12].

Note on an absolute neighborhood extensor for metric spaces.

Note on an absolute neighborhood extensor for metric spaces.

A property of compact absolute neighborhood retracts

A property of compact absolute neighborhood retracts

A generalization of absolute neighborhood retracts

A generalization of absolute neighborhood retracts

On a theorem of Hanner

0LOF HANNER (See reference [4]) has shown that a separable metric space is an absolute neighborhood retract (ANR) for normal spaces if and only if it is both an ANR for separable metric spaces and an absolute G~. Using an example given in a recent paper of R. H. BING [1] we show (theorem 1) tha t if a metric space is an ANR for normal spaces it is separable, and that hence the hypothesis of separability in Itanner 's theorem can be dropped. In the same paper Bing defined a class of spaces more restricted than normal called collectionwise normal. We show (theorem 2) that Hanner's theorem extends to non-separable metric spaces if normal is replaced by collectionwise normal. Moreover (corollary 1) this form of t tanner 's theorem characterizes collectionwise normal spaces in the same way as Tietze's extension theorem characterizes normal spaces.

Some theorems concerning absolute neighborhood retracts

Some theorems concerning absolute neighborhood retracts

A note on absolute neighborhood retracts

A note on absolute neighborhood retracts

Some theorems on absolute neighborhood retracts

Some theorems on absolute neighborhood retracts