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 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]**Henri Cartan and Samuel Eilenberg,*Homological algebra*, Princeton University Press, Princeton, N. J., 1956. MR**0077480****[2]**Irving Reiner,*Integral representations of cyclic groups of prime order*, Proc. Amer. Math. Soc.**8**(1957), 142–146. MR**0083493**, 10.1090/S0002-9939-1957-0083493-6**[3]**Dock Sang Rim,*Modules over finite groups*, Ann. of Math. (2)**69**(1959), 700–712. MR**0104721****[4]**Jean-Pierre Serre,*Cohomologie des groupes discrets*, Prospects in mathematics (Proc. Sympos., Princeton Univ., Princeton, N.J., 1970) Princeton Univ. Press, Princeton, N.J., 1971, pp. 77–169. Ann. of Math. Studies, No. 70 (French). MR**0385006****[5]**Richard G. Swan,*Invariant rational functions and a problem of Steenrod*, Invent. Math.**7**(1969), 148–158. MR**0244215****[6]**Richard G. Swan,*𝐾-theory of finite groups and orders*, Lecture Notes in Mathematics, Vol. 149, Springer-Verlag, Berlin-New York, 1970. MR**0308195**

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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