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 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.

Name thesis:degree_name

Doctoral

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Mathematics

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2021

Author dc:creator

• Elliott, Robin

Advisor dc:contributor.advisor

• Guth, Larry

Statement dc:rights

• In Copyright - Educational Use Permitted
• Copyright MIT

Licence dc:rights.uri

• http://rightsstatements.org/page/InC-EDU/1.0/

Handle dc:identifier.uri

https://hdl.handle.net/1721.1/139172

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/139172