Homology of Configuration Spaces of Surfaces as Mapping Class Group Representations

In this thesis, we study the homology of configuration spaces of surfaces viewed as representations of the mapping class group of the surface, distinguishing between various flavours: ordered and unordered configurations, of closed surfaces and surfaces with boundary, and with different homology coefficients. In Chapter 2, we prove a version of the scanning isomorphism that is “untwisted” and equivariant with the mapping class group action. We further prove that scanning remembers a product arising from superposing configurations. We apply this equivariant scanning to compute the rational cohomology of unordered configurations of surfaces with boundary. In Chapter 3, we adapt certain cellular decompositions of compactified configuration spaces to obtain the kernel of the mapping class group action on the homology of unordered configurations of both kinds of surfaces and with any coeffiecients. Finally, in Chapter 4, we geometrically construct mapping classes deep in the Johnson filtration that act non-trivially on the homology of ordered configurations, in support of a conjecture by Bianchi, Miller and Wilson.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2023

Author dc:creator

• Stavrou, Andreas

Advisor dc:contributor.advisor

• Randal-Williams, Oscar

• configuration spaces
• homology computations
• mapping class groups
• surfaces
• topology

Licence

• https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/a088c675-a536-425e-9f15-f51df02f5a2e/download
• https://www.rioxx.net/licenses/all-rights-reserved/

Language dc:language

eng

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.99769

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/353706