Property 𝑆𝑈𝑉^{∞} and proper shape theory

ABSTRACT. Various conditions of contractibility and extensibility at °°f or locally compact metric spaces are studied. These are shown to be equivalent if the space under consideration is an absolute neighborhood retract (ANR) and an ANR satisfying them is called docile at °°. Docility at °° is invariant under proper homotopy domination. The ANR X is docile at °° if and only if FX (the Freudenthal compactification of X) is an ANR and FX -X is unstable in FX; the inclusion of X into FX is a homotopy equivalence.1. Introduction. In recent years there has been considerable interest in the geometry of noncompact spaces, particularly "at °°". In this paper we consider various conditions of contractibility and extensibility at °° (alternatively, "at the ends") for locally compact metric spaces. These conditions are shown, in §3, to be equivalent provided the space in question is an absolute neighborhood retract, and an absolute neighborhood retract satisfying them will be called docile at °°. Docility at «> js shown to be invariant under proper homotopy domination.In §4 it is shown that if X is an absolute neighborhood retract which is docile at °°, then FX, the Freudenthal compactification of X, is an absolute neighborhood retract, and that EX, the end-set of X, is unstable in FX. The converse is also shown to hold, and it is shown that the inclusion of X into FX is a homotopy equivalence.

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Property 𝑆𝑈𝑉^{∞} and proper shape theory

Cited by 3 publications

References 12 publications

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