Quantitative Topology of Loop Space
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 clas...
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| Format: | Thesis |
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Massachusetts Institute of Technology
2022
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| Online Access: | https://hdl.handle.net/1721.1/139172 |
| _version_ | 1826190526210113536 |
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| author | Elliott, Robin |
| author2 | Guth, Larry |
| author_facet | Guth, Larry Elliott, Robin |
| author_sort | Elliott, Robin |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T08:41:37Z |
| format | Thesis |
| id | mit-1721.1/139172 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T08:41:37Z |
| publishDate | 2022 |
| publisher | Massachusetts Institute of Technology |
| record_format | dspace |
| spelling | mit-1721.1/1391722022-01-15T04:05:47Z Quantitative Topology of Loop Space Elliott, Robin Guth, Larry Massachusetts Institute of Technology. Department of Mathematics 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. 2022-01-14T14:54:35Z 2022-01-14T14:54:35Z 2021-06 2021-05-25T12:46:46.456Z Thesis https://hdl.handle.net/1721.1/139172 In Copyright - Educational Use Permitted Copyright MIT http://rightsstatements.org/page/InC-EDU/1.0/ application/pdf Massachusetts Institute of Technology |
| spellingShingle | Elliott, Robin Quantitative Topology of Loop Space |
| title | Quantitative Topology of Loop Space |
| title_full | Quantitative Topology of Loop Space |
| title_fullStr | Quantitative Topology of Loop Space |
| title_full_unstemmed | Quantitative Topology of Loop Space |
| title_short | Quantitative Topology of Loop Space |
| title_sort | quantitative topology of loop space |
| url | https://hdl.handle.net/1721.1/139172 |
| work_keys_str_mv | AT elliottrobin quantitativetopologyofloopspace |