Homotopy properties of locally compact spaces at infinity-calmness and smoothness

Abstract

We define two properties of noncompact locally compact spaces called ^-calmness at oo and (^, ^-smoothness at oo for arbitrary classes of topological spaces ^ and &. A number of theorems and examples concerning these properties are given. By considering complements of ϋΓ-sets in the Hubert cube from them we get three new shape invariant conditions for compact metric spaces named calmness, %-calmness, and %-smoothness. Calmness is a condition while ^-smoothness implies that (and under some additional assumptions is also implied by) the kth shape pro-group of a compactum in question is trivial, for all k>n. 1* Introduction* This paper continues the study of homotopy properties of noncompact locally compact spaces at oo from [6], [7], and [8]. In [8] we introduced concepts of calm at oo, w-calm at oo, and ^-smooth at oo locally compact spaces. In the present paper these notions are investigated in much the same way as at oo and tameness at oo were investigated in [6] and [7], respectively. We prove analogous theorems and give a number of examples illustrating those concepts. By a standard .ZΓ-set complement device [10] (see also [6]) we get three new shape invariant properties of compact metric spaces called calmness, w-calmness, and ^-smoothness. The usefulness of these properties in the future development of shape theory remains to be seen. Our results show that they are rather natural and that one can prove theorems about them resembling some statements about and fundamental dimension of compact metric spaces. We assume the reader is familiar with shape theory of compact metric spaces [2] and with the most elementary concepts and results of infinite dimensional topology [11]. The paper is organized as follows. In § 2 we collect definitions (mostly from [6]) to be used in later sections. The § 3 investigates ^-calm at oo noncompact locally compact spaces, for an arbitrary class of topological spaces ^. ^-calmness at oo is a movability at oo type condition for homotopies weaker than the condition SANR(oo) (or strong at oo) introduced in the author's thesis [9]. The short § 4 lists properties of calm compact metric