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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Author: R. B. Sher
Journal: Trans. Amer. Math. Soc. 190 (1974), 345-356
MSC: Primary 54C56
MathSciNet review: 0341389
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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.

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Keywords: Property $ SU{V^\infty }$, property $ U{V^\infty }$, cellularity, quasi-cellularity, proper shape, proper homotopy type, tree, ends, proper homotopical domination, property $ SU{V^n}$, property $ U{V^n}$, proper homotopy group
Article copyright: © Copyright 1974 American Mathematical Society