{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/71219"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/71219","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Generalized Group Presentations and Formal Deformations of Cw Complexes","abstract":"A Peiffer-Whitehead word system W, or generalized group presentation, consists of generators for a free group and words of various orders n (GREATERTHEQ) 2 representing elements of the free group (n = 2), a free crossed module (n = 3) or a free module (n > 3). The P(,n)-equivalence relation on word systems generalizes the extended Nielsen equivalence relation on ordinary group presentations. Word systems, called homotopy readings, can be associated with any connected CW complex K by removing a maximal tree and selecting one word (or generator) per cell, via relative homotopy. Given homotopy readings W(,1) and W(,2) of finite CW complexes K(,1) and K(,2) respectively, we show that W(,1) is P(,n)-equivalent to W(,2) if and only if K(,1) formally (n + 1)-deforms to K(,2). This extends results of P. Wright (1975) and W. Metzler (1982) for the case n = 2. For n = 3, it follows that W(,1) is P(,n)-equivalent to W(,2) if and only if K(,1) and K(,2) have the same simple homotopy type.","abstract_html":"A Peiffer-Whitehead word system W, or generalized group presentation, consists of generators for a free group and words of various orders n (GREATERTHEQ) 2 representing elements of the free group (n = 2), a free crossed module (n = 3) or a free module (n > 3). The P(,n)-equivalence relation on word systems generalizes the extended Nielsen equivalence relation on ordinary group presentations. Word systems, called homotopy readings, can be associated with any connected CW complex K by removing a maximal tree and selecting one word (or generator) per cell, via relative homotopy. Given homotopy readings W(,1) and W(,2) of finite CW complexes K(,1) and K(,2) respectively, we show that W(,1) is P(,n)-equivalent to W(,2) if and only if K(,1) formally (n + 1)-deforms to K(,2). This extends results of P. Wright (1975) and W. Metzler (1982) for the case n = 2. For n = 3, it follows that W(,1) is P(,n)-equivalent to W(,2) if and only if K(,1) and K(,2) have the same simple homotopy type.","abstract_has_math":false,"creators":["Brown, Richard Arthur"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-16T06:18:10Z","date_published":"2014-12-16T06:18:10Z","updated_at":"2026-07-22T22:26:04Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8422029"],"render_values":[{"text":"(UMI)AAI8422029","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/71219","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Brown, Richard Arthur"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-16T06:18:10Z","10000-01-01","1984"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/71219","(UMI)AAI8422029"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["A Peiffer-Whitehead word system W, or generalized group presentation, consists of generators for a free group and words of various orders n (GREATERTHEQ) 2 representing elements of the free group (n = 2), a free crossed module (n = 3) or a free module (n > 3). The P(,n)-equivalence relation on word systems generalizes the extended Nielsen equivalence relation on ordinary group presentations. Word systems, called homotopy readings, can be associated with any connected CW complex K by removing a maximal tree and selecting one word (or generator) per cell, via relative homotopy. Given homotopy readings W(,1) and W(,2) of finite CW complexes K(,1) and K(,2) respectively, we show that W(,1) is P(,n)-equivalent to W(,2) if and only if K(,1) formally (n + 1)-deforms to K(,2). This extends results of P. Wright (1975) and W. Metzler (1982) for the case n = 2. For n = 3, it follows that W(,1) is P(,n)-equivalent to W(,2) if and only if K(,1) and K(,2) have the same simple homotopy type.","Made available in DSpace on 2014-12-16T06:18:10Z (GMT). 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The P(,n)-equivalence relation on word systems generalizes the extended Nielsen equivalence relation on ordinary group presentations. Word systems, called homotopy readings, can be associated with any connected CW complex K by removing a maximal tree and selecting one word (or generator) per cell, via relative homotopy. Given homotopy readings W(,1) and W(,2) of finite CW complexes K(,1) and K(,2) respectively, we show that W(,1) is P(,n)-equivalent to W(,2) if and only if K(,1) formally (n + 1)-deforms to K(,2). This extends results of P. Wright (1975) and W. Metzler (1982) for the case n = 2. For n = 3, it follows that W(,1) is P(,n)-equivalent to W(,2) if and only if K(,1) and K(,2) have the same simple homotopy type.","Made available in DSpace on 2014-12-16T06:18:10Z (GMT). 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