Email: ar750@cam.ac.uk

Nilpotent Groups, Spaces and G-Spaces - PhD Thesis.

Miscellaneous

Model structures on topological spaces (First year PhD project) - overview of model categories, ordinals and the small object argument, introduction to simplicial sets, geometric realization takes Kan fibrations to Hurewicz fibrations, derivation of the Quillen model structure on simplicial sets and the q, h and m-model structures on spaces.

Finitely generated nilpotent spaces - a nilpotent space is equivalent to a CW complex with finite skeleta iff its homotopy groups are finitely generated iff its homology groups are finitely generated.

Simplicial spaces and fibrations - we prove that the realisation of a 'locally trivial' map of simplicial spaces is a Hurewicz fibration. As a consequence, we deduce that the orbit map EG to BG is a Hurewicz fibration whenever G is a topological group with a nondegenerate basepoint.

Papers and Preprints

Completion preserves homotopy fibre squares of connected nilpotent spaces - New York J. Math. 30(2024) 502-512.

Localisations and completions of nilpotent G-spaces - Journal of Homotopy and Related Structures (2025).

The equivariant covering homotopy property - an alternative proof of the ECHP of generalised equivariant bundles.

A double coset formula for the genus of a nilpotent group