{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/88015"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/88015","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"A classification of toric, folded-symplectic manifolds","abstract":"Given a $G$-toric, folded-symplectic manifold with co-orientable folding hypersurface, we show that its orbit space is naturally a manifold with corners $W$ equipped with a smooth map $\\psi: W \\to \\frak{g}^*$, where $\\frak{g}^*$ is the dual of the Lie algebra of the torus, $G$. The map $\\psi$ has fold singularities at points in the image of the folding hypersurface under the quotient map and it is a unimodular local embedding away from these points. Thus, to every $G$-toric, folded-symplectic manifold we can associate its orbit space data $\\psi:W \\to \\fg^*$, a unimodular map with folds. We fix a unimodular map with folds $\\psi:W \\to \\fg^*$ and show that isomorphism classes of $G$-toric, folded-symplectic manifolds whose orbit space data is $\\psi:W \\to \\fg^*$ are in bijection with $H^2(W; \\mathbb{Z}_G\\times \\R)$, where $\\mathbb{Z}_G= \\ker(\\exp:\\frak{g} \\to G)$ is the integral lattice of $G$. Thus, there is a pair of characteristic classes associated to every $G$-toric, folded-symplectic manifold. This result generalizes a classical theorem of Delzant as well as the classification of toric, origami manifolds, due to Cannas da Silva, Guillemin, and Pires, in the case where the folding hypersurface is co-orientable. We spend a significant amount of time discussing the fundamentals of equivariant and non-equivariant folded-symplectic geometry. In particular, we characterize folded-symplectic forms in terms of their induced map from the sheaf of vector fields into a distinguished sheaf of one-forms, we relate the existence of an orientation on the folding hypersurface of a fold-form to the intrinsic derivative of the contraction mapping from the tangent bundle to the cotangent bundle, and we show that $G$-toric, folded-symplectic manifolds are stratified by $K$-toric, folded-symplectic submanifolds, where $K$ varies over the subtori of $G$ and the action is principal on each stratum. We show how these structures give rise to the rigid orbit space structure of a toric, folded-symplectic manifold used in the classification. We also give a robust description of folded-symplectic reduction, which we use to construct local models of toric, folded-symplectic manifolds.","abstract_html":"Given a $G$-toric, folded-symplectic manifold with co-orientable folding hypersurface, we show that its orbit space is naturally a manifold with corners $W$ equipped with a smooth map <span class=\"etd-inline-math\">\\psi: W \\to \\frak{g}<sup>*</sup></span>, where <span class=\"etd-inline-math\">\\frak{g}<sup>*</sup></span> is the dual of the Lie algebra of the torus, $G$. The map $\\psi$ has fold singularities at points in the image of the folding hypersurface under the quotient map and it is a unimodular local embedding away from these points. Thus, to every $G$-toric, folded-symplectic manifold we can associate its orbit space data <span class=\"etd-inline-math\">\\psi:W \\to \\fg<sup>*</sup></span>, a unimodular map with folds. We fix a unimodular map with folds <span class=\"etd-inline-math\">\\psi:W \\to \\fg<sup>*</sup></span> and show that isomorphism classes of $G$-toric, folded-symplectic manifolds whose orbit space data is <span class=\"etd-inline-math\">\\psi:W \\to \\fg<sup>*</sup></span> are in bijection with <span class=\"etd-inline-math\">H<sup>2</sup>(W; \\mathbb{Z}<sub>G</sub>\\times \\R)</span>, where <span class=\"etd-inline-math\">\\mathbb{Z}<sub>G</sub>= \\ker(\\exp:\\frak{g} \\to G)</span> is the integral lattice of $G$. Thus, there is a pair of characteristic classes associated to every $G$-toric, folded-symplectic manifold. This result generalizes a classical theorem of Delzant as well as the classification of toric, origami manifolds, due to Cannas da Silva, Guillemin, and Pires, in the case where the folding hypersurface is co-orientable. We spend a significant amount of time discussing the fundamentals of equivariant and non-equivariant folded-symplectic geometry. In particular, we characterize folded-symplectic forms in terms of their induced map from the sheaf of vector fields into a distinguished sheaf of one-forms, we relate the existence of an orientation on the folding hypersurface of a fold-form to the intrinsic derivative of the contraction mapping from the tangent bundle to the cotangent bundle, and we show that $G$-toric, folded-symplectic manifolds are stratified by $K$-toric, folded-symplectic submanifolds, where $K$ varies over the subtori of $G$ and the action is principal on each stratum. We show how these structures give rise to the rigid orbit space structure of a toric, folded-symplectic manifold used in the classification. We also give a robust description of folded-symplectic reduction, which we use to construct local models of toric, folded-symplectic manifolds.","abstract_has_math":true,"creators":["Hockensmith, Daniel Lawrence"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Lerman, Eugene","Kerman, Ely","Tolman, Susan","Watts, Jordan"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-09-29T20:38:14Z","date_published":"2015-09-29T20:38:14Z","updated_at":"2026-07-22T22:26:31Z","subjects":["folded-symplectic","toric","Delzant","origami manifolds","classification","completely integrable system"],"languages":["en"],"rights":["Copyright 2015 Daniel Hockensmith"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/88015","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Lerman, Eugene","Kerman, Ely","Tolman, Susan","Watts, Jordan"]},{"key":"dc:creator","label":"Author","values":["Hockensmith, Daniel Lawrence"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-09-29T20:38:14Z","2015-08","2015-07-15","2015-8"]},{"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":["folded-symplectic","toric","Delzant","origami manifolds","classification","completely integrable system"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2015 Daniel Hockensmith"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/88015"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Given a $G$-toric, folded-symplectic manifold with co-orientable folding hypersurface, we show that its orbit space is naturally a manifold with corners $W$ equipped with a smooth map $\\psi: W \\to \\frak{g}^*$, where $\\frak{g}^*$ is the dual of the Lie algebra of the torus, $G$. The map $\\psi$ has fold singularities at points in the image of the folding hypersurface under the quotient map and it is a unimodular local embedding away from these points. Thus, to every $G$-toric, folded-symplectic manifold we can associate its orbit space data $\\psi:W \\to \\fg^*$, a unimodular map with folds. We fix a unimodular map with folds $\\psi:W \\to \\fg^*$ and show that isomorphism classes of $G$-toric, folded-symplectic manifolds whose orbit space data is $\\psi:W \\to \\fg^*$ are in bijection with $H^2(W; \\mathbb{Z}_G\\times \\R)$, where $\\mathbb{Z}_G= \\ker(\\exp:\\frak{g} \\to G)$ is the integral lattice of $G$. Thus, there is a pair of characteristic classes associated to every $G$-toric, folded-symplectic manifold. This result generalizes a classical theorem of Delzant as well as the classification of toric, origami manifolds, due to Cannas da Silva, Guillemin, and Pires, in the case where the folding hypersurface is co-orientable. We spend a significant amount of time discussing the fundamentals of equivariant and non-equivariant folded-symplectic geometry. In particular, we characterize folded-symplectic forms in terms of their induced map from the sheaf of vector fields into a distinguished sheaf of one-forms, we relate the existence of an orientation on the folding hypersurface of a fold-form to the intrinsic derivative of the contraction mapping from the tangent bundle to the cotangent bundle, and we show that $G$-toric, folded-symplectic manifolds are stratified by $K$-toric, folded-symplectic submanifolds, where $K$ varies over the subtori of $G$ and the action is principal on each stratum. We show how these structures give rise to the rigid orbit space structure of a toric, folded-symplectic manifold used in the classification. We also give a robust description of folded-symplectic reduction, which we use to construct local models of toric, folded-symplectic manifolds.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2015-09-29 without embargo terms","The student, Daniel Hockensmith, accepted the attached license on 2015-07-09 at 15:47.","The student, Daniel Hockensmith, submitted this Dissertation for approval on 2015-07-09 at 15:57.","This Dissertation was approved for publication on 2015-07-15 at 09:08.","DSpace SAF Submission Ingestion Package generated from Vireo submission #8395 on 2015-09-29 at 13:22:25","Made available in DSpace on 2015-09-29T20:38:14Z (GMT). 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The map $\\psi$ has fold singularities at points in the image of the folding hypersurface under the quotient map and it is a unimodular local embedding away from these points. Thus, to every $G$-toric, folded-symplectic manifold we can associate its orbit space data $\\psi:W \\to \\fg^*$, a unimodular map with folds. We fix a unimodular map with folds $\\psi:W \\to \\fg^*$ and show that isomorphism classes of $G$-toric, folded-symplectic manifolds whose orbit space data is $\\psi:W \\to \\fg^*$ are in bijection with $H^2(W; \\mathbb{Z}_G\\times \\R)$, where $\\mathbb{Z}_G= \\ker(\\exp:\\frak{g} \\to G)$ is the integral lattice of $G$. Thus, there is a pair of characteristic classes associated to every $G$-toric, folded-symplectic manifold. This result generalizes a classical theorem of Delzant as well as the classification of toric, origami manifolds, due to Cannas da Silva, Guillemin, and Pires, in the case where the folding hypersurface is co-orientable. We spend a significant amount of time discussing the fundamentals of equivariant and non-equivariant folded-symplectic geometry. In particular, we characterize folded-symplectic forms in terms of their induced map from the sheaf of vector fields into a distinguished sheaf of one-forms, we relate the existence of an orientation on the folding hypersurface of a fold-form to the intrinsic derivative of the contraction mapping from the tangent bundle to the cotangent bundle, and we show that $G$-toric, folded-symplectic manifolds are stratified by $K$-toric, folded-symplectic submanifolds, where $K$ varies over the subtori of $G$ and the action is principal on each stratum. We show how these structures give rise to the rigid orbit space structure of a toric, folded-symplectic manifold used in the classification. We also give a robust description of folded-symplectic reduction, which we use to construct local models of toric, folded-symplectic manifolds.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2015-09-29 without embargo terms","The student, Daniel Hockensmith, accepted the attached license on 2015-07-09 at 15:47.","The student, Daniel Hockensmith, submitted this Dissertation for approval on 2015-07-09 at 15:57.","This Dissertation was approved for publication on 2015-07-15 at 09:08.","DSpace SAF Submission Ingestion Package generated from Vireo submission #8395 on 2015-09-29 at 13:22:25","Made available in DSpace on 2015-09-29T20:38:14Z (GMT). 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