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.
","abstract_html":"<p>Let M<sup>m</sup> and N<sup>n</sup> 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<sub>*</sub>: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<sup>m</sup> onto N<sup>n</sup> 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<sup>m</sup> onto N<sup>n</sup> 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<sup>m</sup> to N<sup>n</sup> with f*:π1(M) -—> π1 (N) a surjection, then f is homotopic to a light open map of M onto N.</p> <p>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.</p>","abstract_has_math":false,"creators":["Walsh, John J."],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Mathematical Sciences","degree_department":null,"school":null,"contributors":["Louis F. McAuley","Patricia McAuley","Dick Wick Hall"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1973,"date_issued":"1973-01-01T08:00:00Z","date_published":"1973-01-01T08:00:00Z","updated_at":"2026-07-24T01:10:28Z","subjects":["Topology Manifolds (Mathematics)","Conformal mapping"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://orb.binghamton.edu/dissertation_and_theses/214","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Louis F. McAuley","Patricia McAuley","Dick Wick Hall"]},{"key":"dc:creator","label":"Author","values":["Walsh, John J."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematical Sciences"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Topology Manifolds (Mathematics)","Conformal mapping"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://orb.binghamton.edu/dissertation_and_theses/214"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.
"]},{"key":"dc:title","label":"Title","values":["Monotone, monotone open, and light open mappings on manifolds"]}]}],"canonical_facts":{"dc:contributor":["Louis F. McAuley","Patricia McAuley","Dick Wick Hall"],"dc:creator":["Walsh, John J."],"dc:description.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.
"],"dc:identifier":["https://orb.binghamton.edu/dissertation_and_theses/214"],"dc:subject":["Topology Manifolds (Mathematics)","Conformal mapping"],"dc:title":["Monotone, monotone open, and light open mappings on manifolds"],"thesis:degree_discipline":["Mathematical Sciences"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T01:10:28Z"}