Date of Award

6-1986

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

John J. Walsh

Committee Members

Robert J. Daverman, Kwangh Jeon

Abstract

Recently, R.J. Daverman and J.J. Walsh modified an example due to J. Taylor to obtain an example of a cell-like map from a compactum with non-trivial shape onto the Hilbert cube Q such that the non-degeneracy set is contained in the countable union of finite dimensional closed subsets of Q. Previously, G. Kozlowski proved that a cell-like map f: X' → X from a compact ANR X' onto a metric space X is a hereditary shape equivalence if there exists a sequence {Bn}∞n=1 of finite dimensional closed subsets of X such that the non-degeneracy set is contained in ∞Un=1 Bn and {f-1 (Bn)}∞n=1 forms a pairwise disjoint null-sequence. Here we raise two questions, which we show are equivalent. First: Is a cell-like map f: X' → X a hereditary shape equivalence if there exists a sequence {Bn}∞n=1 of finite dimensional closed subsets of X such that Un≠m (Bn ∩ Bm) has a strong transfinite dimension and the non-degeneracy set is contained in ∞Un=1 Bn ? Even though we are not able to answer these questions, we give affirmative answers to the questions for special cases, and , furthermore, we are able to extend the aforementioned result of Kozlowski's. Also, we attempt to extend certain analyses of cell-like maps, which are proper by definition, to (non-proper) UV∞-maps. We prove that for a UV∞-map f: X' → X from an ANR X' to a metric space X the following are equivalent: (1) X is an ANR; (2) f is a hereditary homotopy equivalence; (3) f is a hereditary shape equivalence; (4) f is a fine homotopy equivalence. Since UV∞-maps are generally not onto, the notion of a "hereditary shape equivalence" is a variation of that for cell-like maps, though it agrees with, say, Kozlowski's for cell-like maps.

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