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# A Solution of a Problem of Steenrod for Cyclic Groups of Prime Order

James E. Arnold, Jr.
Proceedings of the American Mathematical Society
Vol. 62, No. 1 (Jan., 1977), pp. 177-182
DOI: 10.2307/2041971
Stable URL: http://www.jstor.org/stable/2041971
Page Count: 6
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## Abstract

Given a $Z\lbrack G \rbrack$ module $A$, we will say a simply connected CW complex $X$ is of type $(A, n)$ if $X$ admits a cellular $G$ action, and $\tilde{H}_i(X) = 0, i \neq n, H_n(X) \simeq A$ as $Z\lbrack G \rbrack$ 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 $G_0(Z\lbrack G \rbrack)$, solutions were obtained for $A = Z_p(p-\text{prime})$ 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 $Z\lbrack Z_p \rbrack$ 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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