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 with S-duality
If in the classical S-category P, 1) continuous mappings are replaced by compact-open strong shape (= coss) morphisms (cf. §1 or [1], §2), and 2) ∧-products are properly reinterpreted, then an S-duality theorem for arbitrary subsets X ⊂ S (rather than for compact polyhedra) holds (Theorem 2.1). 0. Introduction. In a previous paper [1] we introduced the concept of coss-shape (compact-open strong shape or strong shape with compact support) and established for each X = (X,m), with m ∈ Z and finitedimensional X ∈Met (= category of separable metric spaces), i.e. for any object in an S-category P (cf. [1, §4]), an S-dual DX ∈ P which is, up to isomorphism in P, uniquely determined and natural, and satisfies D2X ≈ X and P(X,Y) ≈ P(DY, DX). In the present paper we introduce in §1 and §3 new, additional cossmorphisms X → Y ∧ Z and X ∧ Y → Z (Y ∧ Z and X ∧ Y are pairs of objects in P) which for special choices of X, Y, Z can be interpreted as coss-morphisms between X and Y ∧ Z ∈ P (cf. §3) resp. X ∧Y and Z, or sometimes even as strong shape morphisms (1.2 and §5). The main objective of this paper is the verification of Theorem 2.1 asserting the existence of a natural isomorphism (1) {X ∧Y,Z}c ≈ {X, DY ∧ Z}c ({. . . , . . .}c denoting the respective sets of coss-morphisms in the S-category, cf. §3). Among the corollaries in §2 we recover the main theorem of [1] and two other S-duality theorems for special choices of X, Y, Z. In particular, classical S-duality (cf. [9], [10]) turns out to be a corollary of (1) (cf. §2(8)). 1991 Mathematics Subject Classification: Primary 55P25, 55P55; Secondary 55N20, 55M05.
Best approximation and moduli of smoothness: Computation and equivalence theorems
In this paper we investigate the trigonometric series with the @b-general monotone coefficients. First, we study the uniform convergence criterion. The estimates of best approximations and moduli of smoothness of the series in uniform metrics are obtained in terms of coefficients. These results imply several important relations between moduli of smoothness of different orders (in particular, Marchaud-type inequality) and best approximations.