A Classifying Family of Spaces for the Cohomology of Profinite Groups

In the study of homological algebra, one useful tool for studying the cohomology of a discrete group is that group's classifying space. In some sense, the classifying space captures both the group itself and a description of its cohomology for any action of the group on any coefficient module. While some constructions for a classifying space also apply to topological groups, the relationship of the resulting space to the group's cohomology is unclear. Profinite groups are a special case of topological groups, determined entirely as the limits of inverse systems of finite, discrete groups. The goal of this work is to construct for profinite groups as close an analog as possible to the classifying space of a discrete group. In particular, we are interested in the construction for a finite, discrete group's classifying space achieved by first producing the nerve of the group as a category and then taking its geometric realization to obtain a space with isomorphic cohomology groups. We proceed by extending each step of this process to apply to a profinite group using inverse limits, followed by correcting for a lack of continuity (in the sense of compatibility with inverse limits) in singular cohomology by applying alternative cohomology theories to the resulting sequence of spaces. The end result has a promising isomorphism to the cohomology of the group, with the possibility of a further isomorphism.

Grantor dc:publisher

University of Alabama Libraries

Year dc:date.issued

2021

Author dc:creator

• Lee, Evan Matthew

Advisor dc:contributor.advisor

• Corson, Jon

Contributors dc:contributor

• Dixon, Martyn
• Evans, Martin
• Roberts, Lawrence
• Tosun, Bulent
• Harms, Benjamin

• algebraic topology
• classifying space
• profinite groups
• sheaf theory
• topological algrebra

Statement dc:rights

• All rights reserved by the author unless otherwise indicated.

Language dc:language.iso

en_US, English

Dc Identifier Other

http://purl.lib.ua.edu/181533
u0015_0000001_0003972
Lee_alatus_0004D_14522

OAI identifier oai:identifier

oai:ir.ua.edu:123456789/8204