Cyclic group actions on manifolds from deformations of rational homotopy types (Q1298114)

This paper finishes the series [\textit{P. Löffler} and the author, ibid. 271, 549-576 (1985; Zbl 0544.57018); the author, Lect. Notes Math. 1509, 313-325 (1992; Zbl 0752.57016); Trans. Am. Math. Soc. 347, No. 1, 137-153 (1995; Zbl 0865.57036)] of papers that aim to show the existence of non-trivial group actions (``symmetries'') on certain classes of manifolds. More specifically, the author asks whether there is a semifree smooth action of the circle group \(T=S^1\) -- resp., a non-trivial action of a cyclic group \(\mathbb{Z}/p\), \(p\) a prime -- on a given manifold \(X\) with a fixed point set of a given rational homotopy type \(F\). The author assumes that the rational homotopy types of \(X\) and \(F\) are related by a deformation in the sense of [\textit{C. Allday}, Topology 17, 95-100 (1978; Zbl 0421.57012)] between their (Sullivan) graded differential algebra models (cf. [\textit{D. Sullivan}, Publ. Math., Inst.Hautes Étud. Sci. 47(1977), 269-331 (1978; Zbl 0374.57002); \textit{S. Halperin}, Lectures on minimal models, Mém. Soc. Math. Fr., Nouv. Sér. 9-10 (1983; Zbl 0536.55003)]): Roughly speaking, the author assumes the conclusion of the Borel localization theorem [\textit{W. Y. Hsiang}, Cohomology theory of topological transformation groups, Ergeb. Math. Grenzgeb., Bd. 85 (1975; Zbl 0429.57011); \textit{C. Allday} and \textit{V. Puppe}, Cohomological methods in transformation groups, Camb. Stud. Adv. Math. 32 (1993; Zbl 0799.55001)] on the rational homotopy level. Under certain additional assumptions, the author proves a converse of that theorem: he shows that there is a semifree smooth \(T\)-action on a manifold \(Y\) rationally homotopy equivalent to \(X\) with fixed point set \(Y^T\) rationally homotopy equivalent to \(F\). Moreover, for all but finitely many primes \(p\), he finds non-trivial smooth actions of \(\mathbb{Z}/p\) on \(X\) itself with fixed point set rationally homotopy equivalent to \(F\).