2026-09-19T11:51:27Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/437862022-01-13T07:54:35Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Haynes R. Miller.
Lee, Wai Kei Peter
Massachusetts Institute of Technology. Dept. of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2008-12-11T18:27:23Z
2008-12-11T18:27:23Z
2008
2008
http://hdl.handle.net/1721.1/43786
261139727
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008.
Includes bibliographical references (leaves 38-39).
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.
by Wai Kei Peter Lee.
Ph.D.
39 leaves
eng
Massachusetts Institute of Technology
M.I.T. theses are protected by
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http://dspace.mit.edu/handle/1721.1/7582
Mathematics.
Gröbner bases in rational homotopy theory
Thesis
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Publication
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<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>
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