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.
Duality and Eilenberg-Mac Lane spectra
Stable cohomotopy groups of Eilenberg-Mac Lane spectra of finite groups are shown to be trivial. This implies that the stable homotopy category, which is large enough to represent ordinary cohomology theory, cannot be self-dual. It can also be interpreted as an evidence to support Freyd's generating hypothesis and a proof of a stable version of a conjecture of D. Sullivan.