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Proceedings of the American Mathematical Society

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

 

 

A solution of a problem of Steenrod for cyclic groups of prime order


Author: James E. Arnold
Journal: Proc. Amer. Math. Soc. 62 (1977), 177-182
MSC: Primary 55C35
MathSciNet review: 0431150
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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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DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0431150-9
Keywords: Cellular group actions, $ Z[{Z_p}]$ modules, Swan's invariant, Grothendieck group
Article copyright: © Copyright 1977 American Mathematical Society