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

DOI:
https://doi.org/10.1090/S0002-9939-1977-0431150-9

MathSciNet review:
0431150

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Given a module *A*, we will say a simply connected CW complex *X* is of type (*A, n*) if *X* admits a cellular *G* action, and as 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 , solutions were obtained for (*p*-prime) and . The question of infinite complexes of type (*A, n*) was left open. In this paper we obtain the following complete solution for modules:

There are complexes of type , 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.

**[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)**

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

Retrieve articles in all journals with MSC: 55C35

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1977-0431150-9

Keywords:
Cellular group actions,
modules,
Swan's invariant,
Grothendieck group

Article copyright:
© Copyright 1977
American Mathematical Society