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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A solution of a problem of Steenrod for cyclic groups of prime order
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by James E. Arnold PDF
Proc. Amer. Math. Soc. 62 (1977), 177-182 Request permission

Abstract:

Given a $Z[G]$ module A, we will say a simply connected CW complex X is of type (A, n) if X admits a cellular G action, and ${\tilde H_i}(X) = 0,i \ne n,{H_n}(X) \simeq A$ as $Z[G]$ modules. In [5], R. Swan considers the problem posed by Steenrod of whether or not there are finite complexes of type (A, n) for all finitely generated A and finite G. Using an invariant defined in terms of ${G_0}(Z[G])$, solutions were obtained for $A = {Z_p}$ (p-prime) and $G \subseteq \operatorname {Aut}\;({Z_p})$. The question of infinite complexes of type (A, n) was left open. In this paper we obtain the following complete solution for $Z[{Z_p}]$ modules: There are complexes of type $(A,n)\;(n \geqslant 3)$, and there are finite complexes of type (A, n) if and only if the invariant which corresponds to Swan’s invariant for these modules vanishes.
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 62 (1977), 177-182
  • MSC: Primary 55C35
  • DOI: https://doi.org/10.1090/S0002-9939-1977-0431150-9
  • MathSciNet review: 0431150