Gröbner bases in rational homotopy theory
Name
261139727-MIT.pdf
Description
Full printable version
Size
1.95 MB
Format
Adobe PDF
Checksum (MD5)
878714dba0ac04cb2b2e4aebea2eea33
Author(s)
Lee, Wai Kei Peter
Advisor(s)
Haynes R. Miller.
Date Issued
2008
Publisher
Massachusetts Institute of Technology
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.
Description
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008.
Includes bibliographical references (leaves 38-39).
Subjects
Mathematics.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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