On stable cohomotopy groups of compact spaces. II. (Q617722)

A shape category comes together with a category of spaces (eventually with stable homotopy classes of mappings as morphisms) and a subcategory of \textit{good} spaces (e.g. a category of CW spaces or of ANRs). The present author works with compact spaces and \(n\)-connected CW complexes. So the author defines \({\infty\choose n}\)-homotopy and \({\infty \choose n}\)-homotopy classes of mappings and then a stable shape category \(Sh\pmb S ^\infty_n\). He compares \(Sh\pmb S^\infty_n\) with the stable shape category \(ShStab\): There is a functor which transforms \({\infty \choose n}\)-equivalences into isomorphisms in \(Sh\mathbf S^\infty_n\). The author investigates the concept of movability (called \({\infty\choose\ast}\)-movability) in his new category. \({\infty \choose\ast}\) -spaces have many pleasant properties. There are much more new movable objects than the classical movable ones. We have \(c(X) = \max\{n:\pi^n_s(X))\}= \max\{n: H^n (X;\mathbb Z)\}\) for \({\infty\choose \ast}\)-movable spaces \(X\). As can be expected, morphisms in \(Sh\pmb S^\infty_n\) induce morphisms for the stable cohomotopy group functor. [For part I of this paper see Zbl 1083.55006.]