Periodic maps on \(R^7\) without fixed points (Q2777981)

Results of \textit{P. A. Smith} [Bull. Am. Math. Soc. 66, 401-415 (1960; Zbl 0096.37501)] imply that if \(f:\mathbb{R}^n\to\mathbb{R}^n\) is a smooth periodic map of period \(r>1\) and \(r\) is a prime power, then \(f\) has a fixed point. On the other hand, if \(r\) is not a prime power and \(n\geq 8\), then \textit{J. M. Kister} [Am. J. Math. 85, 316-319 (1963; Zbl 0119.18801)] constructed examples of smooth periodic maps \(f:\mathbb{R}^n\to\mathbb{R}^n\) of period \(r\) without fixed points. Kister's result cannot extend to dimensions \(n\leq 6\) because Smith's work also implies that, in those dimensions, for a smooth periodic map of period \(r\) there must be fixed point if \(r\) is divisible by no more than two primes. In this paper, the authors extend Kister's result to \(\mathbb{R}^7\). That is, they construct a smooth periodic map \(f:\mathbb{R}^7 \to\mathbb{R}^7\) of period \(r\), for any \(r\) not a prime power, such that \(f\) has no fixed points. Moreover, they show that, for any such \(r\), there are uncountably many topological equivalence classes of smooth periodic maps on \(\mathbb{R}^7\) without fixed points.