Topological spherical space form problem. III. Dimensional bounds and smoothing
Ib Madsen, Charles R. Jr. Thomas, C. T. C. Wall · Pacific Journal of Mathematics · 1983
In the two earlier papers in this series we showed, that if the finite group 7r has period 2 din cohomology, and if for all primesp a subgroup of order 2p is cyclic, then there exists a free topological action by π on a sphere of dimension S 2nd ~\ 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 TΓ on s 2e ^~ι 9 the sphere with the standard differentiate structure.Here e{iτ) is the Artin-Lam induction exponent, that is, the least positive integer such that e(π)\ 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(m\ and that our result is geometrically the best possible, except for one class of groups.