{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/43786"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/43786","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Gröbner bases in rational homotopy theory","abstract":"The Mayer-Vietoris sequence in cohomology has an obvious Eckmann-Hilton dual that characterizes the homotopy of a pullback, but the Eilenberg-Moore spectral sequence has no dual that characterizes the homotopy of a pushout. The main obstacle is the lack of an Eckmann-Hilton dual to the Kiinneth theorem with which to understand the homotopy of a coproduct. This difficulty disappears when working rationally, and we dualize Rector's construction of the Eilenberg-Moore spectral sequence to produce a spectral sequence converging to the homotopy of a pushout. We use Gröbner-Shirshov bases, an analogue of Gröbner bases for free Lie algebras, to compute directly the E2 term for pushouts of wedges of spheres. In particular, for a cofiber sequence A --> X --> C where A and X are wedges of spheres, we use this calculations to generalize a result of Anick by giving necessary and sufficient conditions for the map X --> C to be surjective in rational homotopy. More importantly, we are able to avoid the use of differential graded algebra and minimal models, and instead approach simple but open problems in rational homotopy theory using a simplicial perspective and the combinatorial properties of Gröbner-Shirshov bases.","abstract_html":"The Mayer-Vietoris sequence in cohomology has an obvious Eckmann-Hilton dual that characterizes the homotopy of a pullback, but the Eilenberg-Moore spectral sequence has no dual that characterizes the homotopy of a pushout. The main obstacle is the lack of an Eckmann-Hilton dual to the Kiinneth theorem with which to understand the homotopy of a coproduct. This difficulty disappears when working rationally, and we dualize Rector's construction of the Eilenberg-Moore spectral sequence to produce a spectral sequence converging to the homotopy of a pushout. We use Gröbner-Shirshov bases, an analogue of Gröbner bases for free Lie algebras, to compute directly the E2 term for pushouts of wedges of spheres. In particular, for a cofiber sequence A --> X --> C where A and X are wedges of spheres, we use this calculations to generalize a result of Anick by giving necessary and sufficient conditions for the map X --> C to be surjective in rational homotopy. More importantly, we are able to avoid the use of differential graded algebra and minimal models, and instead approach simple but open problems in rational homotopy theory using a simplicial perspective and the combinatorial properties of Gröbner-Shirshov bases.","abstract_has_math":false,"creators":["Lee, Wai Kei Peter"],"institution":"Massachusetts Institute of Technology","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Dept. of Mathematics.","school":null,"contributors":[],"advisors":["Haynes R. Miller."],"committee_chairs":[],"committee_members":[],"year":2008,"date_issued":"2008","date_published":"2008","updated_at":"2026-07-22T22:21:21Z","subjects":["Mathematics."],"languages":["eng"],"rights":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."],"rights_urls":["http://dspace.mit.edu/handle/1721.1/7582"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1721.1/43786","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Haynes R. Miller."]},{"key":"dc:contributor.department","label":"Department","values":["Massachusetts Institute of Technology. Dept. of Mathematics."]},{"key":"dc:contributor.other","label":"Dc Contributor Other","values":["Massachusetts Institute of Technology. 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They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://dspace.mit.edu/handle/1721.1/7582"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1721.1/43786"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008.","Includes bibliographical references (leaves 38-39)."]},{"key":"dc:description.abstract","label":"Abstract","values":["The Mayer-Vietoris sequence in cohomology has an obvious Eckmann-Hilton dual that characterizes the homotopy of a pullback, but the Eilenberg-Moore spectral sequence has no dual that characterizes the homotopy of a pushout. The main obstacle is the lack of an Eckmann-Hilton dual to the Kiinneth theorem with which to understand the homotopy of a coproduct. This difficulty disappears when working rationally, and we dualize Rector's construction of the Eilenberg-Moore spectral sequence to produce a spectral sequence converging to the homotopy of a pushout. We use Gröbner-Shirshov bases, an analogue of Gröbner bases for free Lie algebras, to compute directly the E2 term for pushouts of wedges of spheres. In particular, for a cofiber sequence A --> X --> C where A and X are wedges of spheres, we use this calculations to generalize a result of Anick by giving necessary and sufficient conditions for the map X --> C to be surjective in rational homotopy. More importantly, we are able to avoid the use of differential graded algebra and minimal models, and instead approach simple but open problems in rational homotopy theory using a simplicial perspective and the combinatorial properties of Gröbner-Shirshov bases."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:title","label":"Title","values":["Gröbner bases in rational homotopy theory"]}]}],"canonical_facts":{"dc:contributor.advisor":["Haynes R. Miller."],"dc:contributor.department":["Massachusetts Institute of Technology. Dept. of Mathematics."],"dc:contributor.other":["Massachusetts Institute of Technology. Dept. of Mathematics."],"dc:creator":["Lee, Wai Kei Peter"],"dc:date.accessioned":["2008-12-11T18:27:23Z"],"dc:date.available":["2008-12-11T18:27:23Z"],"dc:date.issued":["2008"],"dc:description":["Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008.","Includes bibliographical references (leaves 38-39)."],"dc:description.abstract":["The Mayer-Vietoris sequence in cohomology has an obvious Eckmann-Hilton dual that characterizes the homotopy of a pullback, but the Eilenberg-Moore spectral sequence has no dual that characterizes the homotopy of a pushout. The main obstacle is the lack of an Eckmann-Hilton dual to the Kiinneth theorem with which to understand the homotopy of a coproduct. This difficulty disappears when working rationally, and we dualize Rector's construction of the Eilenberg-Moore spectral sequence to produce a spectral sequence converging to the homotopy of a pushout. We use Gröbner-Shirshov bases, an analogue of Gröbner bases for free Lie algebras, to compute directly the E2 term for pushouts of wedges of spheres. In particular, for a cofiber sequence A --> X --> C where A and X are wedges of spheres, we use this calculations to generalize a result of Anick by giving necessary and sufficient conditions for the map X --> C to be surjective in rational homotopy. More importantly, we are able to avoid the use of differential graded algebra and minimal models, and instead approach simple but open problems in rational homotopy theory using a simplicial perspective and the combinatorial properties of Gröbner-Shirshov bases."],"dc:description.degree":["Ph.D."],"dc:identifier.uri":["http://hdl.handle.net/1721.1/43786"],"dc:language.iso":["eng"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."],"dc:rights.uri":["http://dspace.mit.edu/handle/1721.1/7582"],"dc:subject":["Mathematics."],"dc:title":["Gröbner bases in rational homotopy theory"],"dc:type":["Thesis"]},"updated_at":"2026-07-22T22:21:21Z"}