Algebraic models for non-simply connected manifolds
Lead Research Organisation:
University College London
Department Name: Mathematics
Abstract
Research Area: Algebra
We consider closed manifolds M of dimension n > 4 having fundamental group G and whose universal coverings highly connected in the sense of Wall. When the fundamental group is trivial such manifolds were classified by Wall and, in the case n = 5, by Barden.
We consider the extent to which a corresponding classification can be achieved when G is a finite metacyclic group.
We consider closed manifolds M of dimension n > 4 having fundamental group G and whose universal coverings highly connected in the sense of Wall. When the fundamental group is trivial such manifolds were classified by Wall and, in the case n = 5, by Barden.
We consider the extent to which a corresponding classification can be achieved when G is a finite metacyclic group.
Organisations
People |
ORCID iD |
| Francis Johnson (Primary Supervisor) | |
| John Nicholson (Student) |
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509577/1 | 01/10/2016 | 24/03/2022 | |||
| 1916669 | Studentship | EP/N509577/1 | 01/10/2017 | 30/09/2021 | John Nicholson |
| Description | I have developed a near-complete classification of when a certain object from algebra (projective modules over integral group rings) has the property that their classification reduces to the much-simpler 'stable' classification (known as the cancellation property). I have built these results up enough that they have interesting consequences for several long-standing problems in algebraic topology. Firstly, I can resolve a special case of the D2 problem of C. T. C. Wall, which was first posed in 1965. Secondly, I extend the work of Michael Dyer on the classification of (G,n)-complexes: a certain simple family of n-dimensional spaces which are only about as complicated as 2-dimensional spaces. Finally, and partly through their application to these results, they also apply to the classification of manifolds. In particular, I obtain many families of examples of manifolds of dimension >4 which are not equivalent but are equivalent 'stably'. |
| Exploitation Route | The algebraic results should have further applications in Number Theory through the theory of Galois modules. I believe the topological results will, with some work, be of interest to those working on the classification of manifolds. This will be especially true if I can extend my results to 4 dimensions which is something I will be working on during the end of my PhD. More generally, I should remark that understanding problems like this one which are very special in 4 dimensions is considered of central importance in theoretical physics, especially when dealing with questions such as ``why is the universe we live in 4-dimensional?" |
| Sectors | Other |
| Description | Mini-course on summer program (Ross Mathematics Program) |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Schools |
| Results and Impact | Gave a course consisting of four 1-1.5 hour online lectures for an audience predominantly from the US and China, most of whom had previously attended the Ross Mathematics Program (a summer program in Columbus, Ohio). The course was an introduction to algebraic topology and was tailer-made to the audience and made use of the perspective on the subject I have from doing my PhD. |
| Year(s) Of Engagement Activity | 2021 |
| Description | Mini-course on summer program (Ross Mathematics Program) |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Study participants or study members |
| Results and Impact | Gave a course consisting of five 1.5 hour lectures on a summer program in Columbus in Ohio, USA. The audience was primarily aged 15-19 though also included undergraduate and postgraduate students aged 19-26 as well, on a few occasions, a professor and a senior lecturer. The course was an introduction to algebraic topology and was tailer-made to the audience and made use of the perspective on the subject I have from doing my PhD. |
| Year(s) Of Engagement Activity | 2019 |
| URL | https://rossprogram.org/alumni/previous-summers/ |