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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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Cohomology of metacyclic groups
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by Johannes Huebschmann PDF
Trans. Amer. Math. Soc. 328 (1991), 1-72 Request permission

Abstract:

Let ${\mathbf {e}}:1 \to N \to G \to K \to 1$ be an extension of a finite cyclic group $N$ by a finite cyclic group $K$. Using homological perturbation theory, we introduce the beginning of a free resolution of the integers ${\mathbf {Z}}$ over the group ring ${\mathbf {Z}}G$ of $G$ in such a way that the resolution reflects the structure of $G$ as an extension of $N$ by $K$, and we use this resolution to compute the additive structure of the integral cohomology of $G$ in many cases. We proceed by first establishing a number of special cases, thereafter constructing suitable cohomology classes thereby obtaining a lower bound, then computing characteristic classes introduced in an earlier paper, and, finally, exploiting these classes, obtaining upper bounds for the cohomology via the integral cohomology spectral sequence of the extension ${\mathbf {e}}$. The calculation is then completed by comparing the two bounds.
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Additional Information
  • © Copyright 1991 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 328 (1991), 1-72
  • MSC: Primary 20J06
  • DOI: https://doi.org/10.1090/S0002-9947-1991-1031239-1
  • MathSciNet review: 1031239