Algebraic Topology
Lead Research Organisation:
University of Southampton
Department Name: School of Mathematics
Abstract
The project is concerned with algebraic and combinatorial invariants of topological spaces. To approach calculation of
those invariant we should first
study the existence of homotopy decompositions of topological spaces.
those invariant we should first
study the existence of homotopy decompositions of topological spaces.
People |
ORCID iD |
| George Simmons (Student) |
Publications
Grbic J
(2021)
One-relator groups and algebras related to polyhedral products
in Proceedings of the Royal Society of Edinburgh: Section A Mathematics
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/R513325/1 | 30/09/2018 | 29/09/2023 | |||
| 2127445 | Studentship | EP/R513325/1 | 30/09/2018 | 28/02/2022 | George Simmons |
| NE/W503150/1 | 31/03/2021 | 30/03/2022 | |||
| 2127445 | Studentship | NE/W503150/1 | 30/09/2018 | 28/02/2022 | George Simmons |
| Description | My main research aim is to broaden known links between distinct areas of mathematics which are underpinned by a common combinatorial object, known as a simplicial complex. This combinatorial aspect enables information about the objects of study to be enumerated and tracked clearly in a way that would not otherwise be possible. To any simplicial complex one can associate both a 'right-angled Coxeter group' and a 'moment-angle complex'. The former is an object from Geometric Group Theory, which is an abstraction of reflection groups which capture information about the symmetry of certain spaces. The latter is one of the main objects from Toric Topology, and forms an complex space whose topological properties are deeply encoded by combinatorics. Prior to commencing my research, it was known that for a certain class of simplicial complexes, called flag complexes, a certain freeness condition associated with the right-angled Coxeter group was equivalent to a freeness condition of an algebraic invariant of the moment-angle complex, known as loop homology. My first main contribution, in collaboration with my supervisor Jelena Grbic, along with Taras Panov and Marina Ilyasova, was to show another equivalence between algebraic conditions on the right-angled Coxeter group and the loop homology of the moment-angle complex. These algebraic conditions are known as one-relator conditions and are the natural thing to consider after freeness. My result, to appear in 'Proceedings of the Royal Society of Edinburgh Section A: Mathematics', adds evidence to the theory that there is a deeper link between the two main objects described, and makes further research in this direction of interest. My project will now turn to focus on the moment-angle complex in particular. It is hoped that a way to study the loop homology of the moment-angle complex directly from the combinatorics can be developed, which will allow questions, including those about freeness and one-relator conditions, to be answered for much broader classes of simplicial complexes than flag complexes. The eventual scope of the project is unclear due to the impact of COVID-19. |
| Exploitation Route | The first outcome was to establish evidence of a deeper link between geometric group theory and topology via combinatorics. This makes future research projects analysing the nature of this link viable. In particular, being able to transport tools and methods of analysis between geometric group theory and topology to supplement the study of both areas would be of very high interest. |
| Sectors | Education Other |
| Description | Thematic Program on Toric Topology and Polyhedral Products |
| Amount | $2,360 (CAD) |
| Organisation | Fields Institute for Research in Mathematical Sciences |
| Sector | Charity/Non Profit |
| Country | Canada |
| Start | 01/2020 |
| End | 06/2020 |
| Description | Fields Institute Collaboration |
| Organisation | Fields Institute for Research in Mathematical Sciences |
| Country | Canada |
| Sector | Charity/Non Profit |
| PI Contribution | We (myself and my supervisor Jelena Grbic) provided a significant contribution to the paper 'One relator groups and algebras related to polyhedral products', including the analysis of the loop homology of the moment-angle complex and the discussion of the relationship to minimally non-Golodness. I was responsible for the production of the paper. |
| Collaborator Contribution | Taras Panov and his student, Marina Ilyasova, provided the content for the above paper from the perspective of geometric group theory. |
| Impact | The paper 'One relator groups and algebras related to polyhedral products', which is listed in my Publications section. |
| Start Year | 2019 |