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.
Stable cohomotopy groups of compact spaces
We show that one can reduce the study of global (in particular cohomological) properties of a compact Hausdorff space X to the study of its stable cohomotopy groups π s (X). Any cohomology functor on the homotopy category of compact spaces factorizes via the stable shape category ShStab. This is the main reason why the language and technique of stable shape theory can be used to describe and analyze the global structure of compact spaces. For a given Hausdorff compact space X, there exists a metric compact space with the same stable shape iff the stable cohomotopy groups of X are countable. If π s (X) = 0 for almost all n > 0 and the integral cohomology groups of X are countable (respectively finitely generated) for all n, then the k-fold suspension of X has the same stable shape as a finite-dimensional compact metric space (respectively a finite CW complex) for sufficiently large k. There is a duality between compact Hausdorff spaces and CW spectra under which stable cohomotopy groups of X correspond to homotopy groups of the CW spectrum WX assigned to X and the class of all X with C(X) = max{k : π s (X) 6= 0} <∞ corresponds to the class of spectra bounded below. The notion of the cohomological dimension H-dimX with respect to a generalized cohomology theory H is studied. In particular we show that π-dimX ≥ H-dimX for every H and π-dimX =∞ if π-dimX > dimZX, where π is the stable cohomotopy theory and dimZX is the integral cohomological dimension. The following question remains open: does π-dimX coincide with dimX? Consider a reduced generalized cohomology theory H∗ = {hn} defined on the homotopy categoryHCWf of finite pointed CW complexes. It consists of a family of contravariant functors hn : HCWf → AB together with a family of natural equivalences εn : hn+1 → hn · S, where AB denotes the category of Abelian groups and S the reduced suspension functor ([Sw, p. 124]). The Čech cohomology groups hn(X) of a compact Hausdorff space X are equal to the direct limit of the system {hn(|N(α)|)}, where α varies over the 2000 Mathematics Subject Classification: Primary 55P25, 55P55, 55N20; Secondary 55N07.
Duality in manifolds
SummaryFor the purpose of presenting general Alexander-duality-theorems (§7), strong-shape-homology Ē* and cohomology Ē* with coefficients in an arbitrary spectrum E turns out to be the appropriate implement. Therefore the main properties of Ē* and Ē* are (analogously to those of ordinary (co-) homology with coefficients in E) developed (§§3-5). In order to be able to perform the necessary constructions, strong-shape-theory and in particular two different kinds of smash-products in this shape-category are treated (§§1,2, appendix). All previously known Alexander-duality theorems appear as special cases of the main theorems of this paper (§8).