Absolute Neighborhood Research Articles

Topology of Homology Manifolds

ANR homology n-manifolds are nite-dimensional absolute neighborhood retracts X with the property that for every x 2 X, Hi(X;X fxg) is 0 for i 6= n and Z for i = n. Topological manifolds are natural examples of such spaces. To obtain nonmanifold examples, we can take a manifold whose boundary consists of a union of integral homology spheres and glue on the cone on each one of the boundary components. The resulting space is not a manifold if the fundamental group of any boundary component is a nontrivial perfect group. It is a consequence of the double suspension theorem of Cannon and Edwards that, as in the examples above, the singularities of polyhedral ANR homology manifolds are isolated. There are, however, many examples of ANR homology manifolds which have no manifold points whatever. See [12] for a good exposition of the relevant theory. The purpose of this paper is to begin a surgical classi cation of ANR homology manifolds, sometimes referred to in the sequel, simply as homology manifolds. One way to approach this circle of ideas is via the problem of characterizing topological manifolds among ANR homology manifolds. In Cannon's work on the double suspension problem [6], it became clear that in dimensions greater than 4, the right transversality hypothesis is the following (weak) form of general position.

On the Space of Homeomorphisms of a Compact n‐manifold

ABSTRACTIn the study of the space of homeomorphisms H(M) (with identity on the boundary) of a compact n‐manifold M (modulo its boundary), the major unsolved problem is whether H(M) an absolute neighborhood retract. If the answer is affirmative, then it must be homeomorphic to a Hilbert manifold when combined with known results in infinite‐dimensional topology. It turns out that there is a simple reduction of the problem to the case M being an n‐ball B. That is, if H(B) is an absolute retract, then H(M) is an absolute neighborhood retract. We intend to concentrate on the study of H(B) and prove that there is dense subset of the space of homeomorphism of the Hilbert cube Q such that each of its member is a finite composition of homeomorphisms that are either Lie in H(B) or act in the direction complementary to B.

Extensions and applications of Fuller's theorem on periodic points

If $f X\arrow X$ is a continuous function, $x\in X$ is called a periodic point if some iterate $f^{k}$ of $f$ leaves $x$ fixed. In 1953 F. B. Fuller proved a basic result for the existence of periodic points. He showed that if $X$ is a compact absolute neighborhood retract with non-zero Euler characteristic and if $N(X)$ is the larger of the sums of the odd and even real Betti numbers of $X$, then for any homeomorphism $f$ on $X$, $f^{k}$ has a fixed point for some $k, $1\leq k\leq N(X)$. Actually Fuller’s result holds for any space$X$ for which the Lefschetz Fixed Point Theorem is valid.In this paper the authors first establish a number of extensions of the Fuller Theorem. They next consider G-spaces and extend results of Mann on periodic points and invariant orbits.In the last section they turn to the Riemannian category using the volume function on the orbit space to obtain additional results and examples on periodic points and invariant orbits for groups of isometries. They conclude the paper by using the Fuller Theorem, together with a result of Kobayashi relating the Lefschetz number of an isometry and the Euler characteristic of its fixed point set, to establish a new result on the existence of ${\it common}$ periodic points for ${\it commuting}$ isometries.

Counting shape and homotopy types among fundamental absolute neighborhood retracts: An elementary approach

The authors deal with the question of whether the sets of shape and homotopy types of FANRs are countable. FANRs can be considered as a natural generalization of ANRs; thus the authors' starting point consists of results on the countability of the set of homotopy types of compact metric ANRs proven by M. Mather [Topology 4 (1965), 93-94], by J. Cheeger and J. Kister [Topology 9 (1970), 149-151], and by Kister [Proc. Amer. Math. Soc. 19 (1968), 195]. Because of the fact that FANRs do not have in general the shape of finite polyhedra, these results cannot be applied to the situation of FANRs. The authors prove that the set of shape types of FANRs is countable, but that the set of homotopy types of spaces shape equivalent to a compact X is not countable. As a consequence they show that every FANR can be embedded up to shape as a retract of a movable compactum, and can be topologically embedded as a shape retract of a compactum.

On the homotopy method in the fixed point index theory of multi-valued mappings of compact absolute neighborhood retracts

On the homotopy method in the fixed point index theory of multi-valued mappings of compact absolute neighborhood retracts

The space of Lipschitz maps from a compactum to an absolute neighborhood LIP extensor

The space of Lipschitz maps from a compactum to an absolute neighborhood LIP extensor

Approximation of contractible valued correspondences by functions

Let X and Y be absolute neighborhood retracts (this is a large class of spaces) with X compact, and let F: X→ Y be an upper hemicontinuous correspondence whose values are compact and contractible. It is shown that any neighborhood of the graph of F contains the graph of a continuous function f: X→ Y. The relevance of this result to fixed point theory is indicated. It is also shown that if X is ‘locally infinite’, then F can be approximated in the stronger sense of the graph of f being close to the graph of F and every point in the graph of F being close to some point in the graph of f. A conjectured generalization of the main result is stated.

Homotopical properties of a class of nonsmooth functions

A class of extended real valued functionals, already considered for evolution problems, is studied. The set where the functional is finite is proved to be an absolute neighborhood extensor. Applications to critical point theory, involving Ljusternik-Schnirelman category and cohomological index, are shown. The stability under Γ-convergence of the homotopical type of the sublevels of the functional is also treated.

Decompositions of Manifolds.

Introduction Preliminaries The shrinkability criterion Cell-like decompositions of absolute neighborhood retracts The cell-like approximation theorem Shrinkable decompositions Nonshrinkable decompositions Applications to manifolds References Index.

A remark on the separable extension property

We present an example of a metrizable space having the separable extension property but which is not an Absolute Neighborhood Retract.

Functor exp n c , absolute retracts and Hilbert space

Recently, along with the problem of studying the properties of spaces of the form F(X), where X is compact, and F is a covariant functor from the category Comp of compacta and continuous maps to itself, more and more attention has been given to the problem of preservation by functors of certain properties of the fibers of maps. We note that the second problem is more difficult than the first as a rule. This is clear, if only from the fact that the first problem, in the case of preservation of points by the functors, can be treated as the problem of investigatin the properties of the constant map. Closely related to this them~ are questions concerning thepreservation (or change) of properties of a map after applying some covariant functor to it. At the Prague topology symposium in 1981 V. V. Fedorchuk, starting one of the first investigations in this domain, formulated some results, found in recent years, and also indicated directions of possible investigations [I]. In particular, fiberwise versions of the assertions the classical theorems of Wojdyslawski [2] and Curtis--Schori [3, 4] were cited. In [5] the author investigated properties of the functor exp~ : Comp -+ Comp. In this note we investigate conditions under which, depending on the properties of the space X and the map /: X-+ Y, the space (exp$ /)-i {y} is an absolute neighborhood retract of the class ~ of all metrizable spaces and an 12-manifold. We also study questions of the preservation by the functor exp~ of certain properties of the fibers of maps. Investigating the noncompact case we first prove theorems on spaces of the form (exp$ f)-1 {y} and afterwards, using the fiberwise versions found as a preliminary, we formulate results relating to spaces of the form exp$ X. Theorems 3 and 6 give conditions on X, which are necessary and sufficient for the space exp~ X to be an A (N) R (~) and an 12-manifold (based on Hilbert space), respectively. We note that in investigating the noncompact case, in Theorems 4 and 5 we are able to get rid of the condition of existence for the map f of so-called triviality regions. In the analogous theorems relating to the compact case it still does not seem possible to get rid of the indicated condition.

A characterization of absolute neighborhood retracts in general spaces

Some characterizationsof absolute neighborhood retracts were established in separable metric spaces by Hanner [4]. Hanner's characterizations were easily extended to the metric case. For this,see Hu [5] Chapter IV. In this paper, we shall extend one of Hanner's characterizations to more general spaces, especially, stratifiablespaces, spaces with a tr-almost locally finite base and paracomplexes. For the ANR theory of these spaces, we refer Cauty [2],Miwa [8] and Hyman [6],respectively. Throughout this paper, all spaces are assumed to be paracompact normal spaces and all maps to be continuous. / and S denote the closed unit interval [0,1] and the class of all stratifiablespaces, respectively. ANR(Q) (resp. ANE(Q)) is the abbreviation for absolute neighborhood retract (resp. extensor) for the class Q. For these definitions,see [5]. In this paper, all theorems are proved in the class SBut these theorems can be proved in some other classes. For instance, see Remark 2.3.

General position properties satisfied by finite products of dendrites

Let A be a dendrite whose endpoints are dense and let A be the complement in A of a dense er-compact collection of endpoints of A. This paper investigates various general position properties that finite products of A and A possess. In particular, it is shown that (i) if X is an LCn-space that satisfies the disjoint n-cells property, then X X A satisfies the disjoint (n + l)-cells property, (ii) An X (—1,1) is a compact (n + l)-dimensional AR that satisfies the disjoint n-cells property, (iii) An+1 is a compact (n + l)-dimensional AR that satisfies the stronger general position property that maps of n-dimensional compacta into An+1 are approximable by both Z-raaps and Zn -embeddings, and (iv) An+1 is a topologically complete (n + l)-dimensional AR that satisfies the discrete n-cells property and as such, maps from topologically complete separable n-dimensional spaces into An+1 are strongly approximable by closed Zn-embeddings. 1. Introduction. It is well known that the standard (2n + l)-cell 72n+1 is a compact absolute retract that satisfies the disjoint n-cells property and, as such, admits an embedding of every compact n-dimensional metric space. Moreover, the collection of embeddings of an n-dimensional compactum X into J2+1 forms a dense Gg in the function space C(X,I2n+1) (HW). Though there are compact n-dimensional spaces that satisfy the disjoint n-cells property and admit embed- dings of every n-dimensional compactum, there are no topologically complete n- dimensional absolute neighborhood retracts that satisfy this property, for any such space would contain uncountably many pairwise disjoint embedded n-cells and thereby violate a result of Sieklucki (Si). In this paper, we produce examples of compact (n + l)-dimensional absolute retracts that satisfy the disjoint n-cells property, and the collection of embeddings of an n-dimensional compactum X into any of these examples forms a dense Gs in the corresponding function space. The examples are particularly nice in that they arise as product spaces whose factors are compact 1-dimensional absolute retracts, equivalently, dendrites. The disjoint n-cells property is but one example of a variety of general posi- tion properties that have played an interesting and increasingly important role in Geometric Topology during the past fifteen years, particularly in the area of char- acterizations of manifolds modeled on various spaces. The disjoint disks property of

The Role of Countable Dimensionality in the Theory of Cell-Like Relations

Consider only metrizable spaces. The notion of a slice-trivial relation is introduced, and Theorem 3.2 is proved. This theorem sets forth sufficient conditions for a continuous relation with compact $U{V^\infty }$ point images to be slice-trivial. Theorem 4.5 posits a number of necessary and sufficient conditions for a map to be a hereditary shape equivalence. Several applications of these two theorems are made, including the following. Theorem 5.1. A cell-like map $f:X \to Y$ is a hereditary shape equivalence if there is a sequence $\{ {K_n}\}$ of closed subsets of $Y$ such that (1) $Y - \bigcup \nolimits _{n = 1}^\infty {{K_n}}$ is countable dimensional, and (2) $f|{f^{ - 1}}({K_n}):{f^{ - 1}}({K_n}) \to {K_n}$ is a hereditary shape equivalence for each $n \geq 1$. Theorem 5.9. If $f:X \to Y$ is a proper onto map whose point inverses are $U{V^\infty }$ sets, then $Y$ is an absolute neighborhood extensor for the class of countable dimensional spaces. Furthermore, if $Y$ is countable dimensional, then $Y$ is an absolute neighborhood retract. Theorem 5.9 is of particular interest when specialized to the identity map of a locally contractible space.

The role of countable dimensionality in the theory of cell-like relations

Consider only metrizable spaces. The notion of a slice-trivial relation is introduced, and Theorem 3.2 is proved. This theorem sets forth sufficient conditions for a continuous relation with compact U V ∞ U{V^\infty } point images to be slice-trivial. Theorem 4.5 posits a number of necessary and sufficient conditions for a map to be a hereditary shape equivalence. Several applications of these two theorems are made, including the following. Theorem 5.1. A cell-like map f : X → Y f:X \to Y is a hereditary shape equivalence if there is a sequence { K n } \{ {K_n}\} of closed subsets of Y Y such that (1) Y − ⋃ n = 1 ∞ K n Y - \bigcup \nolimits _{n = 1}^\infty {{K_n}} is countable dimensional, and (2) f | f − 1 ( K n ) : f − 1 ( K n ) → K n f|{f^{ - 1}}({K_n}):{f^{ - 1}}({K_n}) \to {K_n} is a hereditary shape equivalence for each n ≥ 1 n \geq 1 . Theorem 5.9. If f : X → Y f:X \to Y is a proper onto map whose point inverses are U V ∞ U{V^\infty } sets, then Y Y is an absolute neighborhood extensor for the class of countable dimensional spaces. Furthermore, if Y Y is countable dimensional, then Y Y is an absolute neighborhood retract. Theorem 5.9 is of particular interest when specialized to the identity map of a locally contractible space.

On Fundamental Approximative Absolute Neighborhood Retracts

AbstractIn this paper we define and study a class of compacta under the name of Fundamental Approximative Absolute Neighborhood Retracts. This class includes Borsuk’s Fundamental Absolute Neighborhood Retracts and Approximative Absolute Neighborhood Retracts in the sense of M. H. Clapp as proper subclasses. We also introduce the notion of q-strong movability and prove that Fundamental Approximative Absolute Neighborhood Retracts coincide with q-strongly movable compacta.

Mappings with 1-dimensional absolute neighborhood retract fibers

Mappings with 1-dimensional absolute neighborhood retract fibers

A survey of various modifications of the notions of absolute retracts and absolute neighborhood retracts

A survey of various modifications of the notions of absolute retracts and absolute neighborhood retracts

Refinable maps and generalized absolute neighborhood retracts

This paper studies some properties which are preserved by refinable maps when the domain is a certain kind of generalized ANR called a quasi-ANR. Among those properties are the properties of being a quasi-ANR and the disjoint n-cell property. An alternative characterization of quasi-ANR's as surjective approximative neighborhood extensors is also obtained in Theorems 3 and 4. These characterizations are analogs of similar characterizations in ANR theory.

Spaces of Aanr's

For a metric space X, let AANRC(X) denote the hyperspace of all nonempty approximative absolute neighborhood retracts in the sense of Clapp in X topologized with the metric of continuity. We show that AANRC(X) is topologically complete iff X is topologically complete. Some subsets of the first Baire category in AANRc(X) for a Q-manifold X are identified. For example, the collection AANRN(X) of all nonempty approximative absolute neighborhood retracts in the sense of Noguchi in X is such a subset. Introduction. In 1953 Noguchi [N] introduced a generalization of the notion of an absolute neighborhood retract (ANR) which is now called an approximative absolute neighborhood retract in the sense of Noguchi (AANRN). A further generalization was given in 1971 by Clapp [C] and is now known as an approximative absolute neighborhood retract in the sense of Clapp (AANRc). The importance of these generalizations is that they share many properties with ANR's (for example, certain fixed point properties). On the other side, they include compacta with local pathologies. Recently, several authors studied AANRc's using techniques of shape theory. In particular, Borsuk [B2] described them as NE-sets, Mardegic [M] characterized them as approximate polyhedra, and the author [C2] observed that AANRk's coincide with P-e-movable compacta, where P denotes the class of all finite polyhedra. The main results of this paper, described in the above abstract, are obtained in the following way. First we define the notion of a P-e-movably regular convergence for compacta in a metric space X. The limit of a P-e-movably regularly convergent sequence must be P-e-movable (i.e., an AANRC) so that a sequence {An} in AANRc(X) converges P-e-movably regularly to AO E AANRC(X) iff lim dc(An, AO) = 0, where dc is Borsuk's metric of continuity [B1]. Then we apply the method of investigating topological properties of the hyperspace of all P-movable compacta in X with the topology induced by P-movably regular convergence from [C1] to AANRc(X) using P-e-movably regular convergence. Hence, in essence, our proof of the statement about topological completeness relies on Begle's method in [Be]. Received by the editors May 16, 1980. AMS (MOS) subject classifications (1970). Primary 54B20, 54B55, 54F40.