John Nicholson

I am currently a Rankin-Sneddon 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 organise the Geometry and Topology Seminar at the University of Glasgow.

Email:

Research Interests

My research is on classification problems in low-dimensional 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 closely-related topics:

• The homotopy/homeomorphism classification of 4-dimensional manifolds, as well as in higher dimensions [Papers: 3,6,8,10]
• Various problems on the topology of 2-dimensional cell complexes (see Wall's 1979 Problems List) [Papers: 2,4,5,7,9]
• Projective modules over integral group rings, particularly the existence of non-free stably free modules [Papers: 1,2,5,7,11]

Publications and Preprints

11. The cancellation property for projective modules over integral group rings
    Preprint, 39 pages (arXiv:2406.08692)


12. Stable equivalence relations on 4-manifolds, with Daniel Kasprowski and Simona Veselá
    Preprint, 25 pages (arXiv:2405.06637)


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


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


15. Stably free modules and the unstable classification of 2-complexes
    Preprint, 29 pages (arXiv:2108.02220)

1. Homotopy classification of 4-manifolds whose fundamental group is dihedral, with Daniel Kasprowski and Benjamin Ruppik
   Algebraic & Geometric Topology, 22 (2022), 2915-2949 (journal version, arXiv:2011.03520)


2. Cancellation for (G,n)-complexes and the Swan finiteness obstruction
   International Mathematical Research Notices, to appear, 30 pages (arXiv:2005.01664)


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


4. Connected sum decompositions of high-dimensional manifolds, with Imre Bokor, Diarmuid Crowley, Stefan Friedl, Fabian Hebestreit, Daniel Kasprowski, Markus Land
   2019-20 MATRIX Annals (2021), 5-30 (journal version, arXiv:1909.02628)


5. On CW-complexes over groups with periodic cohomology
   Transactions of the American Mathematical Society, 374 (2021), 6531-6557 (journal version, arXiv:1905.12018)


6. A cancellation theorem for modules over integral group rings
   Mathematical Proceedings of the Cambridge Philosophical Society, 171 (2021), 317-327 (journal version, arXiv:1807.00307)

• 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 Randal-Williams, University of Cambridge (2017) (link)

Talks

• University of Cambridge, Differential Geometry and Topology seminar (28th February 2024)
  Title: Simple homotopy types of even dimensional manifolds (link)
• University of Glasgow, Algebra and Number Theory Seminar (10th January 2024)
  Title: The cancellation property for projective modules over integral group rings (link)
• University of Glasgow, Geometry and Topology Seminar (27th November 2023)
  Title: Homotopy classification of 4-manifolds (link)
• University of Augsburg, Differential Geometry Seminar (20th July 2023)
  Title: Simple homotopy types of 4-manifolds (link)
• University of Southampton, Topology Seminar (15th May 2023)
  Title: Simple homotopy types of 4-manifolds (link)
• McMaster University, Geometry and Topology Seminar (6th April 2023)
  Title: Simple homotopy types of 4-manifolds (link)
• University of Toronto, Geometry and Topology Seminar (3rd April 2023)
  Title: Simple homotopy types of 4-manifolds (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 2-complexes (link)
• University of Oxford, Topology Seminar (24th October 2022)
  Title: Simple homotopy types of 4-manifolds (link)
• University of Münster, Geometric Group Theory Seminar (30th June 2022)
  Title: Stably free modules and the homotopy type of a finite 2-complex (link)
• Indiana University Bloomington, Topology Seminar (28th April 2022)
  Title: 4-manifolds and the homotopy type of a finite 2-complex (link)
• The London Geometry and Topology Seminar (21st January 2022)
  Title: 4-manifolds and the homotopy type of a finite 2-complex (link)
• The University of Alabama, Algebra/Topology Seminar (17th November 2021)
  Title: The homotopy type of a finite 2-complex with non-minimal Euler characteristic (link)
• Universidad de Buenos Aires, Topology Seminar (20th October 2021)
  Title: The homotopy type of a finite 2-complex with non-minimal 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: 4-manifolds 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 CW-complexes
• Ohio State University, Topology and Geometric Group Theory Seminar (14th July 2020)
  Title: Projective modules and the homotopy classification of CW-complexes (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 stably-free modules over integral group rings (link)
• University of Sydney, Geometry and Topology Seminar (22nd August 2018)
  Title: Poincaré 3-complexes and stably-free modules over integral group rings (link)

• Topology, Representation Theory and Higher Structures (Contributed Talk) (29th May 2024)
  Title: Simple homotopy types of 4-manifolds (link)
• Transapline Topology Tetrahedron, University of Warwick (Invited Talk) (25th April 2024)
  Title: Simple homotopy types of even dimensional manifolds (link)
• A Festival Remembering Victor Snaith: Topology, Number Theory and Interactions (Flash Talk) (13th July 2023)
  Title: Simple homotopy types of 4-manifolds (link)
• Algebraic Methods in 4-manifold Topology, University of Glasgow (Talk as Co-organiser) (20th June 2023)
  Title: Simple homotopy types of 4-manifolds
• 3rd International Meeting on Geometric Group Theory and Low Dimensional Topology, Karlsruhe University of Education (Invited) (3rd November 2022)
  Title: Wall's problems on 2-complexes, some progress and new directions (link)
• 4-manifolds: From Above and Below (Flash Talk) (12th September 2022)
  Title: The simple homotopy type of a 4-manifold (link)
• Early Career Researchers in Mathematics 2022, University College London (Plenary) (21st June 2022)
  Title: The topology of 2-complexes (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 2-complex (link)
• AMS Special Session on Geometric Topology in the Middle Dimensions, Purdue University (Invited) (26th March 2022)
  Title: 4-manifolds and the homotopy type of a finite 2-complex (link)
• Four Manifolds: Confluence of High and Low Dimensions, Fields Institute (Contributed) (9th July 2019)
  Title: On CW-complexes over groups with periodic cohomology (link)

Conferences and Workshops

• The Interplay of Geometric Group Theory and K-theory
  University of Southampton, UK (June 2024)
• Focused Research Group on 4-manifolds
  University of Glasgow, UK (June 2024)
• Topology, Representation Theory and Higher Structures (Week 1)
  Isle of Skye, UK (May 2024)
• A Festival Remembering Victor Snaith: Topology, Number Theory and Interactions
  University of Bristol, UK (July 2023)
• Algebraic Methods in 4-Manifold Topology (Organised jointly with Mark Powell) (Website)
  University of Glasgow, UK (June 2023)
• EPSRC Pure Mathematics Community Workshop
  University of Bristol, UK (March 2023)
• 4-manifolds: 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)

• 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 K-theory: 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)

Teaching

• School of Mathematics and Statistics, University of Glasgow
• Lecturer: Galois Theory (4H, MATHS4105) (link)
  
• Lecturer: Linear Algebra (2B, MATHS2004) (link)
  

• Department of Mathematics, Imperial College London
• Lecturer: Algebra III (Rings and Modules)
  Lecture notes: pdf
  Problem sheets: 1, 2, 3, 4
  Solutions to problem sheets: 1, 2, 3, 4

• Ross Mathematics Program (Online)
• Lecturer: Introduction to Algebraic Topology (short course)
  Slides: 1, 2, 3, 4

• Department of Mathematics, University College London
  • Problems class instructor: Algebra 4: Groups and Rings (MATH0053)
  • Tutor: Analysis 1/2 (MATH0003/4), Algebra 1/2 (MATH0005/6)
• Advanced Mathematics Support Programme
  • Teaching assistant: STEP II and III course
• Ross Mathematics Program USA
• Lecturer: Introduction to Algebraic Topology (short course)
  Lecture notes: pdf

• Department of Mathematics, University College London
  • Problems class assistant: Algebra 2 (MATH0006)
• Advanced Mathematics Support Programme
  • Teaching assistant: STEP II and III course
• Ross Mathematics Program Asia
• Counselor, Seminar instructor

Miscellaneous

From April 2019 to 2020, I was a part of the editorial team of Chalkdust.

I am a coauthor of the textbook Edexcel A-level 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.