#### Document Type

Dissertation

#### Date of Award

1970

#### 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 B^{n} 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, M^{n}, 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 B^{n} 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.

#### Recommended Citation

Haver, William Emery, "Cellular mappings on manifolds" (1970). *Graduate Dissertations and Theses*. 146.

https://orb.binghamton.edu/dissertation_and_theses/146