The homotopy theory of cone attachments
Lead Research Organisation:
University of Southampton
Department Name: School of Mathematics
Abstract
Cone attachments are fundamental in homotopy theory;
the construction of CW-complexes being one example. A basic
question is to describe the effect on the homotopy groups of a space X
after attaching a cone. In the 1990s rational homotopy theorists
made good progress on this problem by considering what they call
inert maps. Recent work by Beben and Theriault has described an
integral version of a inert map. The project is to build on these recent
advances in order to (i) broaden the theory and (ii) analyse
interesting examples.
the construction of CW-complexes being one example. A basic
question is to describe the effect on the homotopy groups of a space X
after attaching a cone. In the 1990s rational homotopy theorists
made good progress on this problem by considering what they call
inert maps. Recent work by Beben and Theriault has described an
integral version of a inert map. The project is to build on these recent
advances in order to (i) broaden the theory and (ii) analyse
interesting examples.
Organisations
People |
ORCID iD |
| Guy Boyde (Student) |
Publications
Boyde G
Z/pr-hyperbolicity via homology
Boyde G
(2022)
p-Hyperbolicity of homotopy groups via K-theory
in Mathematische Zeitschrift
Boyde G
(2020)
Bounding size of homotopy groups of Spheres
in Proceedings of the Edinburgh Mathematical Society
Guy Boyde
(2021)
Growth of homotopy groups
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509747/1 | 01/10/2016 | 30/09/2021 | |||
| 2126011 | Studentship | EP/N509747/1 | 01/10/2018 | 30/09/2021 | Guy Boyde |
| Description | Algebraic topology studies geometric objects using algebraic invariants. One particular invariant, the so-called homotopy groups, has proven over the last 100 years to be very difficult to compute and describe. The work in this thesis adresses certain "asymptotic" structural aspects of homotopy groups, especially growth behaviour. A research program initiated by Huang and Wu in 2017 has been expanded upon, and we have begun to uncover general principles. We give the first tractable general criteria for a space to have "exponentially growing" homotopy groups. The thesis comprises three papers, all of which have been accepted into peer-reviewed journals. |
| Exploitation Route | The three papers which comprise the thesis have already been cited 5 times in total. Interest in these questions has been generated - other authors have since resolved a 50-year open problem. New connections between different parts of mathematics have been made, which should lead to new ideas. |
| Sectors | Other |