Applications of simplicial 𝑀-sets to proper and strong shape theories

Abstract

In this paper we have tried to reduce the classical classification problems for spaces and maps of the proper category and of the strong shape category to similar problems in the homotopy category of simplicial sets or in the homotopy category of simplicial M M -sets, which M M is the monoid of proper selfmaps of the discrete space N \mathbb {N} of nonnegative integers. Given a prospace (prosimplicial set) Y Y , we have constructed a simplicial set P ¯ R Y {\overline {\mathcal {P}} ^R}Y such that the Hurewicz homotopy groups of P ¯ R Y {\overline {\mathcal {P}} ^R}Y are the Grossman homotopy groups of Y Y . For the case of the end prospace Y = ε X Y = \varepsilon X of a space X X , we obtain Brown’s proper homotopy groups; and for the Vietoris prospace Y = V X Y = VX (introduced by Porter) of a compact metrisable space X X , we have Quigley’s inward groups. The simplicial subset P ¯ R Y {\overline {\mathcal {P}} ^R}Y of a tower Y Y contains, as a simplicial subset, the homotopy limit lim R Y {\lim ^R}Y . The inclusion lim R Y → P ¯ R Y {\lim ^R}Y \to {\overline {\mathcal {P}} ^R}Y induces many relations between the homotopy and (co)homology invariants of the prospace Y Y . Using the functor P ¯ R {\overline {\mathcal {P}} ^R} we prove Whitehead theorems for proper homotopy, prohomotopy, and strong shape theories as a particular case of the standard Whitehead theorem. The algebraic condition is given in terms of Brown’s proper groups, Grossman’s homotopy groups and Quigley’s inward groups, respectively. In all these cases an equivalent cohomological condition can be given by taking twisted coefficients. The "singular" homology groups of P ¯ R Y {\overline {\mathcal {P}} ^R}Y provide homology theories for the Brown, Grossman and Quigley homotopy groups that satisfy Hurewicz theorems in the corresponding settings. However, there are other homology theories for the homotopy groups above satisfying other Hurewicz theorems. We also analyse the notion of P ¯ \overline {\mathcal {P}} -movable prospace. For a P ¯ \overline {\mathcal {P}} -movable tower we prove easily (without lim 1 {\lim ^1} functors) that the strong homotopy groups agree with the Čech homotopy groups and the Grossman homotopy groups are determined by the Čech (or strong) groups by the formula G π q = P ¯ π ˇ q ^G{\pi _q} = \overline {\mathcal {P}} \check {\pi }_q . This implies that the algebraic condition of the Whitehead theorem can be given in terms of strong (Čech) groups when the condition of P ¯ \overline {\mathcal {P}} -movability is included. We also study homology theories for the strong (Steenrod) homotopy groups which satisfy Hurewicz theorems but in general do not agree with the corresponding Steenrod-Sitnikov homology theories.