Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
Proceedings of the American Mathematical Society
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, N. J., 1956. MR 17, 1040. MR 0077480 (17:1040e)
  • [2] I. Reiner, Integral representations of cyclic groups of prime order, Proc. Amer. Math. Soc. 8 (1957), 142-146. MR 18, 717. MR 0083493 (18:717a)
  • [3] D. S. Rim, Modules over finite groups, Ann. of Math. (2) 69 (1959), 700-712. MR 21 #3474. MR 0104721 (21:3474)
  • [4] J.-P. Serre, Cohomologie des groupes discrets, Ann. of Math. Studies, no. 70, Princeton Univ. Press, Princeton, N. J., 1971. MR 0385006 (52:5876)
  • [5] R. G. Swan, Invariant rational functions and a problem of Steenrod, Invent. Math. 7 (1969), 148-158. MR 39 #5532. MR 0244215 (39:5532)
  • [6] R. G. Swan and E. G. Evans, K-theory of finite groups and orders, Lecture Notes in Math., vol. 149, Springer, Berlin and New York, 1970. MR 46 #7310. MR 0308195 (46:7310)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 55C35

Retrieve articles in all journals with MSC: 55C35

Additional Information

PII: S 0002-9939(1977)0431150-9
Keywords: Cellular group actions, $ Z[{Z_p}]$ modules, Swan's invariant, Grothendieck group
Article copyright: © Copyright 1977 American Mathematical Society

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia