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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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On CW-complexes over groups with periodic cohomology
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by John Nicholson PDF
Trans. Amer. Math. Soc. 374 (2021), 6531-6557 Request permission

Abstract:

If $G$ has $4$-periodic cohomology, then finite D2 complexes over $G$ are determined up to polarised homotopy by their Euler characteristic if and only if $G$ has at most two one-dimensional quaternionic representations. We use this to solve Wall’s D2 problem for several infinite families of non-abelian groups and, in these cases, also show that any finite Poincaré $3$-complex $X$ with $G=\pi _1(X)$ admits a cell structure with a single $3$-cell. The proof involves cancellation theorems for $\mathbb {Z} G$ modules where $G$ has periodic cohomology.
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Additional Information
  • John Nicholson
  • Affiliation: Department of Mathematics, UCL, Gower Street, London, WC1E 6BT, United Kingdom
  • ORCID: 0000-0002-5244-5453
  • Email: j.k.nicholson@ucl.ac.uk
  • Received by editor(s): August 4, 2019
  • Received by editor(s) in revised form: January 4, 2021
  • Published electronically: May 18, 2021
  • Additional Notes: This work was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/N509577/1
  • © Copyright 2021 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 374 (2021), 6531-6557
  • MSC (2020): Primary 57K20; Secondary 20C05, 57P10, 57Q12
  • DOI: https://doi.org/10.1090/tran/8411
  • MathSciNet review: 4302168