Date of Award
Topology Manifolds (Mathematics), Conformal mapping
Doctor of Philosophy (PhD)
Louis F. McAuley
Science and Mathematics
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.
Walsh, John J., "Monotone, monotone open, and light open mappings on manifolds" (1973). Graduate Dissertations and Theses. 214.