Duality between stable strong shape morphisms and stable homotopy classes
Let SStrShn be the full subcategory of the stable strong shape category SStrSh of pointed compacta [H-N] whose objects are all pointed subcompacta of S and let SOn be the full subcategory of the stable homotopy category S ([S-W] or [S]) whose objects are all open subsets of S. In this paper it is shown that there exists a contravariant additive functor Dn : SStrShn → SOn such that Dn(X) = S \ X for every subcompactum X of S and Dn : SStrShn(X, Y ) → SOn(S \Y,S \X) is an isomorphism of abelian groups for all compacta X,Y ⊂ S. Moreover, if X ⊂ Y ⊂ S, j : S \ Y → S \ X is an inclusion and α ∈ SStrShn(X, Y ) is induced by the inclusion of X into Y then Dn(α) = {j}. Introduction. Basic definitions. In [N] it has been proved that the stable shape category of subcompacta of S is isomorphic to the stable weak homotopy category of their complements. This theorem generalizes the Spanier-Whitehead Duality Theorem and corresponds to the Chapman Complement Theorem [S1]. In [H-N] the authors have constructed a stable strong shape category of pointed metric compacta (see also [M]). The purpose of the present note is to prove that there exists a contravariant functor from the stable strong shape category of pointed subcompacta of 2000 Mathematics Subject Classification. 55P55, 55P25.
A strong shape theory admitting an S-dual
Abstract If one replaces continuous mappings by strong shape morphisms with compact support (also called compact-open strong shape mappings) (Definition 2.1) and compact polyhedra by arbitrary subspaces of some Sn, then every object X in the corresponding S-category B has an S-dual DX extending classical S-duality to arbitrary separable metrizable, finite-dimensional spaces.
Some theorems on absolute neighborhood retracts
1. In this paper we shall s tudy A1NR's (absolute neighborhood retracts). The general problem will be as follows. Suppose we have proved that all ANR's have a certain property. Then we may ask, if this proper~ty is characteristic for ANR%, or in other words if it is true tha t a separable metric space having this property necessarily is an ANR. Thus we shall s tudy necessary conditions for a space to be an ANR, and we shall find tha t some of these conditions are also sufficient. Using KURATOWSKI'S modification ([7] p. 270) of BORSUK'S original definition ([1] p. 222), we mean by an ANR a separable metric space X such that, whenever X is imbedded as a closed subset of another separable metric space Z, it is a retract of some neighborhood in Z. First, we take up the study of local properties of ANR's. I t is known tha t an ANR is locally contractible (cf. [4] p. 273) and BORSUK proved tha t local contractability is sufficient for a finite dimensional compact space to be an ANR ([1] p. 240). In a recent paper, however, he has given an example of a locally contractible infinite dimensional space, which is not an ANR [3]. So the question then arises, if the property of a space to be an ANR is a local property. That the answer is affirmative is shown by theorem 3.3. In the case of a compact space this has already been proved by YAJIIMA [10]. Thereafter we prove some theorems on homotopy of mappings into an ANR. Briefly the result can be stated by saying that two mappings of the same space into an ANR which are "near" enough to each other, are homotopic, and tha t if the homotopy is already given on a closed subset and is "small" enough, then this homotopy is extendable. For a compact ANR we can give an exact meaning to the words near and small in terms of some metric. But the uniformity structure implied by a metric does not seem to be a suitable tool for handling non-compact ANR's. Instead of a metric we therefore use open coverings of the space. BORSUK proved [2] tha t any compact ANR X is dominated by a finite polyhedron P. This means tha t there exists two mappings q : X -+ P and ~v : P -+ X such tha t ~o ~0 : X -+ X is homotopic to the identity mapping i : X ~ X. We now prove tha t the polyhedron P and the mappings ~ and yJ can be chosen so tha t this homotopy between ~ ~ and i is arbitrarily small, and we show tha t in this way we get a sufficient condition. This result is generalized in a natural way to n?n-compact spaces by using infinite locally finite polyhedra. Since these polyhedra are ANR's (see corollary 3.5) we thus see tha t any ANR is dominated b y a locally compact A~R.