A note on Brieskorn spheres and the generalized Smith conjecture (Q5954532)

The Smith conjecture, proved for periodic diffeomorphisms, asserts that no periodic transformation of the 3-sphere \(S^3\) can have a tame knotted circle \(S^1\) as its fixed point set [\textit{J. W. Morgan} and \textit{H. Bass} (eds.), The Smith conjecture, Pure Appl. Math 112 (1984; Zbl 0599.57001)]. Similarly, the generalized Smith conjecture asserts that no periodic transformation of the \(n\)-sphere \(S^n\) can have the tame knotted \(S^{n-2}\) as its fixed point set, where \(n\geq 4\). As C. H. Giffen, D. W. Summers, and C. M. Gordon have shown, the generalized Smith conjecture is false also for periodic diffeomorphisms.NEWLINENEWLINENEWLINEIn the paper under review, by using periodic diffeomorphisms of Brieskorn manifolds for any period \(m\geq 2\), the author gives new explicit counterexamples to the generalized Smith conjecture. The author's method differs from those already known and has an algebraic nature. In particular, the author gives a simple method of determining whether a Brieskorn manifold is a topological sphere.