Symmetrien von Mannigfaltigkeiten und rationale Homotopietheorie
It is a well known result that there exist closed compact manifolds having no symmetries \((=\) no non-trivial self maps of finite order); some earlier papers in this area are: \textit{E. M. Bloomberg} [Trans. Am. Math. Soc. 202, 67-78 (1975; Zbl 0261.57003]; \textit{P. E. Conner}, \textit{F. Raymond} and \textit{P. J. Weinberger} [Proc. second Conf. compact. Transform. Groups, Part II, Lect. Notes Math. 299, 81-108 (1972; Zbl 0274.57014)]; \textit{R. Schultz} [Topology, Proc. Symp., Siegen 1979, Lect. Notes Math. 788, 364-377 (1980; Zbl 0433.57015)]. The proofs of these results depend heavily on the non-triviality of the fundamental group of the manifolds involved. In this paper we show: Let \(M^ n\), \(n\geq 6\), be a simply connected manifold. Suppose \(\chi(M)=0\) and \(\sigma(M)=0\). Suppose further that \(M^ n\) is formal (in the sense of rational homotopy theory) and has cup length two. Then \(M^ n\) admits free \({\mathbb{Z}}/p^ n\)-actions for all but finitely many primes p and all \(n\geq 1\) if a certain condition on the distribution of the non-vanishing rational Pontryagin classes is guaranteed. Special cases of this result are: 1.2.a) Let \(M^ m_ 1\), \(M^ m_ 2\), m odd, be simply connected, rationally [m/3]-connected manifolds. Suppose \(M_ 1\) and \(M_ 2\) admit fixed point free \(S^ 1\)-actions. Then \(M_ 1\#M_ 2\) admits free \({\mathbb{Z}}/p^ n\)-actions for almost all primes and all \(n\geq 1\). 1.2.b) If \(M^ m\) is a manifold such that the rational Pontryagin classes \(p_ i(M)\) for \(i\geq [m/8]\) vanish (in particular if M is a (rational) \(\pi\)-manifold) then the condition on the distribution on the Pontryagin classes is always satisfied. 1.2.c) Let \(M^{8k-1}\) be a simply connected manifold such that \(H_ i(M_ i,{\mathbb{Q}})={\mathbb{Q}}\) if \(i=0\), \(8k-1, ={\mathbb{Q}}+{\mathbb{Q}}\) if \(i=4k- 1,\) 4k, and \(=0\) else. If \(p_ k(M)\neq 0\) then M does not admit a fixed point free \(S^ 1\)-action. If \(p_ k(M)=0\) then M admits free \({\mathbb{Z}}/p^ n\)-actions for almost all primes p.
existence of free \({\mathbb{Z}}\) mod \(p^ n\)-actions on formal simply connected manifolds
scientific article; zbMATH DE number 3866224
scientific article