Alternate Author Name(s)

Dr. John Walsh, PhD '74

Document Type

Dissertation

Date of Award

1973

Keywords

Topology Manifolds (Mathematics), Conformal mapping

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

Let Mm and Nn be compact connected p.1. (piecewise linear) manifolds with m ≥ 3. The first result is that a mapping f from M to N is homotopic to a monotone map of M onto N if and only if f*:π1(M) -—> π1 (N) is a surjection (mapping = map = continuous function). There are two results of a technical nature which contain sufficient conditions for the existence of monotone open and light open maps between p.1. manifolds. These provide the following two results. If f is a monotone map of Mm onto Nn and m ≥ 3, then f can be uniformly approximated by monotone open maps of M onto N. If f is an open map of Mm onto Nn and n≥ m ≥ 3, then f can be uniformly approximated by light open maps of M onto N. An immediate corollary is that if n≥ m ≥ 3 and f is a map from Mm to Nn with f*:π1(M) -—> π1 (N) a surjection, then f is homotopic to a light open map of M onto N.

This work is motivated by the recent work of David C. Wilson where he constructs such mappings from manifolds onto cells; the methods of proof are similar.

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