John Nicholson
I am currently a RankinSneddon Research Fellow at the University of Glasgow. Previously, I was a Heilbronn Research Fellow at Imperial College London. In October 2021, I completed my PhD at University College London under the supervision of F. E. A. Johnson. Before that, I was a student at the University of Cambridge.
I coorganise the Geometry and Topology Seminar at the University of Glasgow.
Email:
Research Interests
My research is on classification problems in lowdimensional topology and their interactions with areas of algebra such as homological group theory and integral representation theory. My time is currently split between the following closelyrelated topics:

The homotopy/homeomorphism classification of 4dimensional manifolds, as well as in higher dimensions [Papers: 3,4,8]

Various problems on the topology of 2dimensional cell complexes (see Wall's 1979 Problems List) [Papers: 2,5,6,7]

Projective modules over integral group rings, particularly the existence of nonfree stably free modules [Papers: 1,2,6,7]
Publications and Preprints
Preprints

On the universal pairing for 2complexes, with Mikhail Khovanov and Slava Krushkal
Submitted, 17 pages (arXiv:2312.07429)

Simple homotopy types of even dimensional manifolds, with Csaba Nagy and Mark Powell
Submitted, 76 pages (arXiv:2312.00322)

Stably free modules and the unstable classification of 2complexes
Submitted, 29 pages (arXiv:2108.02220)

Cancellation for (G,n)complexes and the Swan finiteness obstruction
Submitted, 29 pages (arXiv:2005.01664)
Publications

Projective modules and the homotopy classification of (G,n)complexes
Algebraic & Geometric Topology, to appear, 28 pages (arXiv:2004.04252)

Homotopy classification of 4manifolds whose fundamental group is dihedral, with Daniel Kasprowski and Benjamin Ruppik
Algebraic & Geometric Topology, 22 (2022), 29152949 (journal version, arXiv:2011.03520)

Connected sum decompositions of highdimensional manifolds, with Imre Bokor, Diarmuid Crowley, Stefan Friedl, Fabian Hebestreit, Daniel Kasprowski, Markus Land
201920 MATRIX Annals (2021), 530 (journal version, arXiv:1909.02628)

On CWcomplexes over groups with periodic cohomology
Transactions of the American Mathematical Society, 374 (2021), 65316557 (journal version, arXiv:1905.12018)

A cancellation theorem for modules over integral group rings
Mathematical Proceedings of the Cambridge Philosophical Society, 171 (2021), 317327 (journal version, arXiv:1807.00307)
Theses

Projective modules over integral group rings and Wall's D2 problem, PhD Thesis supervised by F. E. A. Johnson, University College London (2021) (link)

Exotic spheres, Part III Essay supervised by Oscar RandalWilliams, University of Cambridge (2017) (link)
Talks
Invited seminar talks

University of Cambridge, Differential Geometry and Topology seminar (28th February 2024)
Title: Simple homotopy types of even dimensional manifolds

University of Glasgow, Algebra and Number Theory Seminar (10th January 2024)
Title: The cancellation property for projective modules over integral group rings

University of Glasgow, Geometry and Topology Seminar (27th November 2023)
Title: Homotopy classification of 4manifolds (link)

University of Augsburg, Differential Geometry Seminar (20th July 2023)
Title: Simple homotopy types of 4manifolds (link)

University of Southampton, Topology Seminar (15th May 2023)
Title: Simple homotopy types of 4manifolds (link)

McMaster University, Geometry and Topology Seminar (6th April 2023)
Title: Simple homotopy types of 4manifolds (link)

University of Toronto, Geometry and Topology Seminar (3rd April 2023)
Title: Simple homotopy types of 4manifolds (link)

The London Algebra Colloquium (2nd February 2023)
Title: Stably free modules and the topology of group presentations (link)

University of Exeter, Number Theory and Algebra Seminar (15th December 2022)
Title: Integral representation theory and the topology of 2complexes (link)

University of Oxford, Topology Seminar (24th October 2022)
Title: Simple homotopy types of 4manifolds (link)

University of Münster, Geometric Group Theory Seminar (30th June 2022)
Title: Stably free modules and the homotopy type of a finite 2complex (link)

Indiana University Bloomington, Topology Seminar (28th April 2022)
Title: 4manifolds and the homotopy type of a finite 2complex (link)

The London Geometry and Topology Seminar (21st January 2022)
Title: 4manifolds and the homotopy type of a finite 2complex (link)

The University of Alabama, Algebra/Topology Seminar (17th November 2021)
Title: The homotopy type of a finite 2complex with nonminimal Euler characteristic (link)

Universidad de Buenos Aires, Topology Seminar (20th October 2021)
Title: The homotopy type of a finite 2complex with nonminimal Euler characteristic (link)

Ludwig Maximilian University of Munich, Arithmetic and Algebraic Geometry Seminar (1st June 2021)
Title: Cancellation for projective modules over integral group rings (link)

Université du Québec à Montréal (CIRGET), Geometry and Topology Seminar (23rd April 2021)
Title: Projective modules and exotic group presentations (link)

Max Planck Institute for Mathematics, Topology Seminar (1st March 2021)
Title: 4manifolds and exotic group presentations (link)

5th Meeting of "New Directions in Group Theory and Triangulated Categories" (8th December 2020)
Title: Projective modules and exotic group presentations

University of Southampton, Pure Mathematics Colloquium (13th November 2020)
Title: Projective modules and exotic group presentations (link)

University of Regensburg, Low Dimensional Topology Seminar (8th October 2020)
Title: Projective modules and the homotopy classification of CWcomplexes

Ohio State University, Topology and Geometric Group Theory Seminar (14th July 2020)
Title: Projective modules and the homotopy classification of CWcomplexes (link)

Durham University, Geometry and Topology Seminar (31st October 2019)
Title: Cancellation theorems in algebra with applications to topology (link)

University of Melbourne, Topology Seminar (27th August 2018)
Title: Wall's D2 problem and stablyfree modules over integral group rings (link)

University of Sydney, Geometry and Topology Seminar (22nd August 2018)
Title: Poincaré 3complexes and stablyfree modules over integral group rings (link)
Conference talks

A Festival Remembering Victor Snaith: Topology, Number Theory and Interactions (Flash Talk) (13th July 2023)
Title: Simple homotopy types of 4manifolds (link)

3rd International Meeting on Geometric Group Theory and Low Dimensional Topology, Karlsruhe University of Education (Invited) (3rd November 2022)
Title: Wall's problems on 2complexes, some progress and new directions (link)

4manifolds: From Above and Below (Flash Talk) (12th September 2022)
Title: The simple homotopy type of a 4manifold (link)

Early Career Researchers in Mathematics 2022, University College London (Plenary) (21st June 2022)
Title: The topology of 2complexes (link)

Topological and Homological Methods in Group Theory 2022, Bielefeld University (Contributed) (29th March 2022)
Title: Stably free modules and the homotopy type of a finite 2complex (link)

AMS Special Session on Geometric Topology in the Middle Dimensions, Purdue University (Invited) (26th March 2022)
Title: 4manifolds and the homotopy type of a finite 2complex (link)

Four Manifolds: Confluence of High and Low Dimensions, Fields Institute (Contributed) (9th July 2019)
Title: On CWcomplexes over groups with periodic cohomology (link)
Conferences and Workshops
In person

A Festival Remembering Victor Snaith: Topology, Number Theory and Interactions
University of Bristol, UK (July 2023)

Algebraic Methods in 4Manifold Topology (Organised jointly with Mark Powell) (Website)
University of Glasgow, UK (June 2023)

EPSRC Pure Mathematics Community Workshop
University of Bristol, UK (March 2023)

4manifolds: From Above and Below
Centre International de Rencontres Mathématiques, France (September 2022)

Heilbronn Annual Conference 2022
University of Bristol, UK (September 2022)

Early Career Researchers in Mathematics 2022
University College London, UK (June 2022)

Topological and Homological Methods in Group Theory 2022
Bielefeld University, Germany (March 2022)

Workshop on Finiteness Properties
Karlsruhe Institute of Technology, Germany (February 2022)

Four Manifolds: Confluence of High and Low Dimensions
Fields Institute, Toronto, Canada (July 2019)

Topology of Manifolds: Interactions Between High and Low Dimensions
MATRIX Institute, Melbourne, Australia (January 2019)

Harnessing Higher Structures in Homotopy: Workshop on Manifolds
Isaac Newton Institute, Cambridge, UK (December 2018)

Manifolds, Groups and Homotopy
Isle of Skye, UK (June 2018)
Online

3rd International Meeting on Geometric Group Theory and Low Dimensional Topology
Karlsruhe University of Education, Germany (November 2022)

AMS Spring Central Sectional Meeting
Purdue University, USA (March 2022)

Manifolds and Ktheory: the legacy of Andrew Ranicki
International Centre for Mathematical Sciences, Edinburgh, UK (June 2021)

British Mathematical Colloquium (BMC)
University of Glasgow, UK (April 2021)

2nd International Meeting on Geometric Group Theory and Low Dimensional Topology
Karlsruhe University of Education, Germany (December 2020)
202324
202223
202021
201819
201718
Miscellaneous
From April 2019 to 2020, I was a part of the editorial team of Chalkdust.
I am a coauthor of the textbook Edexcel Alevel Further Mathematics Core Pure Mathematics Book 2 which was published by Pearson in March 2018.
I am a contributor to the material on the website Underground Mathematics, which I worked on during the summer of 2015.
Here are some instructions on How to Set Up a UCL webpage.