Document Type

Dissertation

Date of Award

1975

Keywords

Conformal mapping, Homotopy theory, Manifolds (Mathematics)

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

First Advisor

Louis F. McAuley

Second Advisor

Patricia McAuley

Third Advisor

David A. Edwards

Abstract

We prove the converse of a theorem of McAuley (TOPO - 72 - General Topology and its Applications, Proc. 1972. Springer Lecture Notes, Vol. 378) and thus complete a characterization of light-open mappings between Peano continua by a sequence of special coverings of the domain. We also prove some covering homotopy theorems for a certain class of finite-to-one open maps and show that a classifying space exists for maps in this class, where point inverses consist of either n points or one point, provided a certain type of covering space exists. In addition, we have the following corollary to our work:

Theorem. A finite-to-one proper open map f:X ⇒Y between connected separable n-manifolds without boundary is the orbit map of a group action if and only if f|x - f-1(f(Bf)) is a regular covering where Bf is the set of points at which f fails to be a local homeomorphism.

This generalizes a result of Edmonds (Branched Coverings and the Geometry of n - circuits, to appear.)

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