On tautological classes of fibre bundles and self-embedding calculus

In this thesis we study the ring of characteristic classes of smooth fibre bundles with a particular focus on tautological classes and the ring that they generate. In the first part we use tools from rational homotopy theory to compute analogues of tautological rings over fibrations, which provides upper bounds on the tautological rings of fibre bundles. In some cases we find an upper bound on the Krull dimension that is sharp. In the second part, we study the classifying space using the calculus of embeddings which provides a homotopy theoretic approximation. We construct cohomology classes on the self-embedding tower which extend certain characteristic classes that were introduced by Kontsevich. This construction is based on introducing configuration space integrals over the tower itself, which also has some consequences for tautological classes that we explore.

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2020

Author dc:creator

• Prigge, Nils

Advisor dc:contributor.advisor

• Randal-Williams, Oscar

• Algebraic Topology
• Characteristic Classes
• Embedding Calculus
• Fibre Bundles

Licence

• https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/942a89d7-901d-47ba-a385-411a08859691/download
• https://www.rioxx.net/licenses/all-rights-reserved/

Language dc:language

eng

Author Identifier

0000-0002-5233-7866
0000-0002-7479-2878

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/317892