## 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

## Series

Science and Mathematics

## Abstract

Let M^{m} and N^{n} 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 M^{m} onto N^{n} and m ≥ 3, then f can be uniformly approximated by monotone open maps of M onto N. If f is an open map of M^{m} onto N^{n} 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 M^{m} to N^{n} 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.

## Recommended Citation

Walsh, John J., "Monotone, monotone open, and light open mappings on manifolds" (1973). *Graduate Dissertations and Theses*. 214.

https://orb.binghamton.edu/dissertation_and_theses/214