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 theory of proper shape for locally compact metric spaces
The notion of topological shape, first introduced by K. Borsuk ([1], [2]) for compact metric spaces, has been extended to larger classes of spaces in a number of ways ([3], [4], [10], [6], [8], [11]). Not all of these extensions agree, but all retain the property of the original formulation that any two spaces of the same homotopy type have the same shape. Any extension to noncompact spaces which retains this property, however, must necessarily sacrifice much of the geometric flavor of the original definition, since homotopy type is notoriously ungeometric in the noncompact case. For noncompact spaces which are locally nice (e.g., locally compact ANR's), proper homotopy type gives a much more geometric classification. Since the proper homotopy type of a compactum (in the class of compacta) is the same as its homotopy type, Borsuk's original concept of shape could, perhaps, more appropriately be considered a generalization of proper homotopy type than of homotopy type, and extended to noncompact spaces so as to retain this property. Of several natural extensions which apply to classes of spaces including noncompact ones and which generalize proper homotopy type, we have chosen to present one which, although limited to locally compact metrizable spaces, seems to emphasize geometric properties to a greater extent than do some alternatives which apply to larger classes of spaces. Moreover, this particular treatment very closely parallels Borsuk's original development, and in many instances virtually the same proofs apply. A detailed version, along with proofs of the results claimed here, will be published at a later date. In order to avoid difficulties arising from the fact that closed sets in noncompact spaces may not have countable neighborhood bases, we substitute "fundamental nets" {fx \ X e A}, A a directed set, for the fundamental sequences {fk \ k = 1, 2, . . .} of Borsuk. And in order that our extension should generalize proper homotopy type rather than homotopy type, we require that all homotopies involved in the definitions should
A LITERATURE REIEW OF TARGET COSTING IN SSCI AND SCI&SCI-EXPANDED INDEXES
Target costing as a cost management and profit planning system is used by many competitive firms since 1970s. Although the concept is simple, the implementation needs an extensive collaboration inside and outside of the organization. In contrast, many organization managers think that target costing is just related with accounting and finance staff. Our study aims to review the literature of this important concept; target costing by examining 41 English articles published in SSCI and SCI&SCI-Expanded between 1985-2008. The study aims to briefly explain the content of each article to show the reader what has accomplished so far.