2026-09-19T01:40:03Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/1391722022-01-15T04:05:47Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Guth, Larry
Elliott, Robin
Massachusetts Institute of Technology. Department of Mathematics
2022-01-14T14:54:35Z
2022-01-14T14:54:35Z
2021-06
2021-05-25T12:46:46.456Z
https://hdl.handle.net/1721.1/139172
In this thesis we investigate how the size of a cycle in the based loop space of a simply connected Riemannian manifold controls its topology. Analogous to Gromov’s notion of distortion of higher homotopy groups of underlying Riemannian manifold, we define notions of distortion for (co)homology classes in the loop space with real coefficients, and study the asymptotics of these distortions. Upper bounds for cohomological distortion are obtained using K.-T. Chen’s theory of iterated integrals to set up differential forms on the loop space. Lower bounds, matching the upper bounds up to a log factor, are given by exhibiting an efficient family of cycles built out of the cells of a cell decomposition the underlying manifold.
Ph.D.
Massachusetts Institute of Technology
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http://rightsstatements.org/page/InC-EDU/1.0/
Quantitative Topology of Loop Space
Thesis
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Doctoral
Doctor of Philosophy
Publication
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<Abstract>In this thesis we investigate how the size of a cycle in the based loop space of a simply connected Riemannian manifold controls its topology. Analogous to Gromov’s notion of distortion of higher homotopy groups of underlying Riemannian manifold, we define notions of distortion for (co)homology classes in the loop space with real coefficients, and study the asymptotics of these distortions. Upper bounds for cohomological distortion are obtained using K.-T. Chen’s theory of iterated integrals to set up differential forms on the loop space. Lower bounds, matching the upper bounds up to a log factor, are given by exhibiting an efficient family of cycles built out of the cells of a cell decomposition the underlying manifold.</Abstract>
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