Document Type

Dissertation

Date of Award

1970

Keywords

Homotopy theory, Manifolds (Mathematics), Cellular mappings

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

First Advisor

Louis F. McAuley

Second Advisor

Patricia McAuley

Third Advisor

Dick Wick Hall

Series

Science and Mathematics

Abstract

Cellular mappings of a manifold onto itself possess many properties of homeomorphisms. In particular; for n ≠ M, a continuous function defined from a manifold onto itself is cellular if and only if it can be uniformly approximated by homeomorphisms. This thesis is a study of cellular mappings, spaces of cellular mappings and a class of mappings, called UV ∞maps which are a natural generalization of cellular mappings for spaces which are not manifolds.

In Chapter one we prove that the space of cellular mappings from a manifold onto itself is a topological semi-group and that the space of all cellular mappings of Bn onto itself which are the identity on the boundary is locally contractible. The main theorem of Chapter two is that a mapping f of the n-sphere, n ≠ 4, onto itself is cellular if and only if f has a continuous extension which maps the interior of the n+1 ball homeomorphically onto itself. This theorem is a hither dimensional analogue of a result of Floyd and fort [11]. For higher dimensional manifolds with boundary, Mn, we show that if f mars the interior of M onto itself and the boundary of M onto itself and if f restricted to the interior is cellular, then f restricted to the boundary of M is also cellular.

Chapter three is concerned with showing that under certain conditions cellular mappings can be replaced in a canonical manner with bounded cellular mappings that agree with the original mappings on a given set. Similar techniques have proven valuable in studying homeomorphisms and spaces of homeomorphisms. In Chapter four we introduce a new type of covering property possessed by many metric spaces and show that possession of this property by the space of cellular mappings of Bn onto itself would show that the space of cellular mappings of manifold onto itself is locally contractible. In Chapter five we show that if f:X Y, Y a metric space, is a UV-map, K a locally finite complex and h:K V is any continuous function, then for any E > 0 there exists a mapping g:K X such that fg is E-homotopic to h.

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