Equivariant scanning and stable splittings of configuration spaces

Thesis

We give a definition of the scanning map for configuration spaces that is equivariant under the action of the diffeomorphism group of the underlying manifold. We use this to extend the Bödigheimer-Madsen result for the stable splittings of the Borel constructions of certain mapping spaces from compact Lie group actions to all smooth actions. Moreover, we construct a stable splitting of configuration spaces which is equivariant under smooth group actions, completing a zig-zag of equivariant stable homotopy equivalences between mapping spaces and certain wedge sums of spaces. Finally we generalise these results to configuration spaces with twisted labels (labels in a fibre bundle subject to certain conditions) and extend the Bödigheimer-Madsen result to more mapping spaces.

Authors: Manthorpe, R

• Publication date: 2012
• DOI: 10.5287/ora-vg62jav0k
• Type of award: MSc
• Level of award: Masters
• Awarding institution: University of Oxford
• Language: English
• Keywords: algebraic topology; configuration spaces; infinite loop spaces; equivariant stable homotopy; scanning map; stable splittings
• Subjects: Algebraic topology