Vol. 106, No. 1, 1983

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Topological spherical space form problem. III. Dimensional bounds and smoothing

Ib Henning Madsen, Charles B. Thomas and C. Terence C. (Charles) (Clegg) Wall

Vol. 106 (1983), No. 1, 135–143

In the two earlier papers in this series we showed, that if the finite group π has period 2d in cohomology, and if for all primes p a subgroup of order 2p is cyclic, then there exists a free topological action by π on a sphere of dimension S2nd1, for some positive integer n. Two questions remained open, namely whether there also existed smooth actions, and whether n could be taken equal to one. In this short paper we prove that there exists a free smooth action of π on S2e(π)1, the sphere with the standard differentiable structure. Here e(π) is the Artin-Lam induction exponent, that is, the least positive integer such that e(π)1 belongs to the ideal of the rational representation ring, generated by representations induced from cyclic subgroups. It turns out that e(π) = d(π) or 2d(π), and that our result is geometrically the best possible, except for one class of groups.

Mathematical Subject Classification 2000
Primary: 57S25
Received: 31 December 1980
Published: 1 May 1983
Ib Henning Madsen
Charles B. Thomas
C. Terence C. (Charles) (Clegg) Wall