{"id":{"repo_id":"durham","oai_identifier":"oai:etheses.durham.ac.uk:83"},"canonical_url":"https://search.dev.ndltd.org/etd/durham/oai:etheses.durham.ac.uk:83","repository":{"repo_id":"durham","name":"Durham University","base_url":"http://etheses.dur.ac.uk/cgi/oai2"},"display":{"title":"Topology of Closed 1-Forms on Manifolds with Boundary","abstract":"The topological structure of a manifold can be eectively revealed by studying the critical points of a nice function assigned on it. This is the essential motivation of Morse theory and many of its generalisations from a modern viewpoint. One fruitful direction of the generalisation of the theory is to look at the zeros of a closed 1-form which can be viewed locally as a real function up to an additive constant, initiated by S.P. Novikov, see [32] and [33]. Extensive literatures have been devoted to the study of so-called Novikov theory on closed manifolds, which consists of interesting objects such as Novikov complex, Morse-Novikov inequalities and Novikov ring. On the other hand, the topology of a space, e.g. a manifold, provides vital information on the number of the critical points of a function. Along this line, a whole dierent approach was suggested in the 1930s by Lusternik and Schnirelman [25] and [26]. M. Farber in [9], [10], [11] and [12] generalised this concept with respect to a closed 1-form, and used it to study the critical points and existence of homoclinic cycles on a closed manifold in much more degenerate settings. This thesis combines the two aspects in the context of closed 1-forms and attempts a systematic treatment on smooth compact manifolds with boundary in the sense that the transversality assumptions on the boundary is consistent thoroughly. Overall, the thesis employs a geometric approach to the generalisation of the existing results.","abstract_html":"The topological structure of a manifold can be eectively revealed by studying the critical points of a nice function assigned on it. This is the essential motivation of Morse theory and many of its generalisations from a modern viewpoint. One fruitful direction of the generalisation of the theory is to look at the zeros of a closed 1-form which can be viewed locally as a real function up to an additive constant, initiated by S.P. Novikov, see [32] and [33]. Extensive literatures have been devoted to the study of so-called Novikov theory on closed manifolds, which consists of interesting objects such as Novikov complex, Morse-Novikov inequalities and Novikov ring. On the other hand, the topology of a space, e.g. a manifold, provides vital information on the number of the critical points of a function. Along this line, a whole dierent approach was suggested in the 1930s by Lusternik and Schnirelman [25] and [26]. M. Farber in [9], [10], [11] and [12] generalised this concept with respect to a closed 1-form, and used it to study the critical points and existence of homoclinic cycles on a closed manifold in much more degenerate settings. This thesis combines the two aspects in the context of closed 1-forms and attempts a systematic treatment on smooth compact manifolds with boundary in the sense that the transversality assumptions on the boundary is consistent thoroughly. Overall, the thesis employs a geometric approach to the generalisation of the existing results.","abstract_has_math":false,"creators":["Li, Tieqiang Tan"],"institution":"Durham University","degree_name":"PhD","degree_level":"doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2009,"date_issued":"2009","date_published":"2009","updated_at":"2026-07-24T02:11:01Z","subjects":[],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":null,"outbound_label":null,"outbound_source":null},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Li, Tieqiang Tan"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2009"]},{"key":"dc:date.issued","label":"Date","values":["2009"]},{"key":"dc:publisher.department","label":"Dc Publisher Department","values":["Mathematical Sciences, Department of"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["Durham University"]},{"key":"dc:relation.isreferencedby","label":"Dc Relation Isreferencedby","values":["https://etheses.durham.ac.uk/id/eprint/83/"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["PhD"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://etheses.durham.ac.uk/id/eprint/83/1/thesis.pdf"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The topological structure of a manifold can be eectively revealed by studying the critical points of a nice function assigned on it. This is the essential motivation of Morse theory and many of its generalisations from a modern viewpoint. One fruitful direction of the generalisation of the theory is to look at the zeros of a closed 1-form which can be viewed locally as a real function up to an additive constant, initiated by S.P. Novikov, see [32] and [33]. Extensive literatures have been devoted to the study of so-called Novikov theory on closed manifolds, which consists of interesting objects such as Novikov complex, Morse-Novikov inequalities and Novikov ring. On the other hand, the topology of a space, e.g. a manifold, provides vital information on the number of the critical points of a function. Along this line, a whole dierent approach was suggested in the 1930s by Lusternik and Schnirelman [25] and [26]. M. Farber in [9], [10], [11] and [12] generalised this concept with respect to a closed 1-form, and used it to study the critical points and existence of homoclinic cycles on a closed manifold in much more degenerate settings. This thesis combines the two aspects in the context of closed 1-forms and attempts a systematic treatment on smooth compact manifolds with boundary in the sense that the transversality assumptions on the boundary is consistent thoroughly. Overall, the thesis employs a geometric approach to the generalisation of the existing results."]},{"key":"dc:format","label":"Dc Format","values":["text"]},{"key":"dc:title","label":"Title","values":["Topology of Closed 1-Forms on Manifolds with Boundary"]}]}],"canonical_facts":{"dc:creator":["Li, Tieqiang Tan"],"dc:date":["2009"],"dc:date.issued":["2009"],"dc:description.abstract":["The topological structure of a manifold can be eectively revealed by studying the critical points of a nice function assigned on it. This is the essential motivation of Morse theory and many of its generalisations from a modern viewpoint. One fruitful direction of the generalisation of the theory is to look at the zeros of a closed 1-form which can be viewed locally as a real function up to an additive constant, initiated by S.P. Novikov, see [32] and [33]. Extensive literatures have been devoted to the study of so-called Novikov theory on closed manifolds, which consists of interesting objects such as Novikov complex, Morse-Novikov inequalities and Novikov ring. On the other hand, the topology of a space, e.g. a manifold, provides vital information on the number of the critical points of a function. Along this line, a whole dierent approach was suggested in the 1930s by Lusternik and Schnirelman [25] and [26]. M. Farber in [9], [10], [11] and [12] generalised this concept with respect to a closed 1-form, and used it to study the critical points and existence of homoclinic cycles on a closed manifold in much more degenerate settings. This thesis combines the two aspects in the context of closed 1-forms and attempts a systematic treatment on smooth compact manifolds with boundary in the sense that the transversality assumptions on the boundary is consistent thoroughly. Overall, the thesis employs a geometric approach to the generalisation of the existing results."],"dc:format":["text"],"dc:identifier.uri":["https://etheses.durham.ac.uk/id/eprint/83/1/thesis.pdf"],"dc:publisher.department":["Mathematical Sciences, Department of"],"dc:publisher.institution":["Durham University"],"dc:relation.isreferencedby":["https://etheses.durham.ac.uk/id/eprint/83/"],"dc:title":["Topology of Closed 1-Forms on Manifolds with Boundary"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["doctoral"],"dc:type.qualificationname":["PhD"]},"updated_at":"2026-07-24T02:11:01Z"}