Property $SUV^{\infty }$ and proper shape theory

Abstract:

A class of spaces called the $SU{V^\infty }$ spaces has arisen in the study of a possibly noncompact variant of cellularity. These spaces play a role in this new theory analogous to that of the $U{V^\infty }$ spaces in cellularity theory. Herein it is shown that the locally compact metric space X is an $SU{V^\infty }$ space if and only if there exists a tree T such that X and T have the same proper shape. This result is then used to classify the proper shapes of the $SU{V^\infty }$ spaces, two such being shown to have the same proper shape if and only if their end-sets are homeomorphic. Also, a possibly noncompact analog of property $U{V^n}$, called $SU{V^n}$, is defined and it is shown that if X is a closed connected subset of a piecewise linear n-manifold, then X is an $SU{V^n}$ space if and only if X is an $SU{V^\infty }$ space. Finally, it is shown that a locally finite connected simplicial complex is an $SU{V^\infty }$ space if and only if all of its homotopy and proper homotopy groups vanish.