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Model aspherical manifolds with no periodic maps. (English) Zbl 0915.57017
An aspherical manifold \(M\) is a closed, connected manifold whose universal covering is contractible. The starting point for the study of these spaces was the work of W. Hurewicz [Proc. Akad. Wet. Amsterdam 39, 215-224 (1936; Zbl 0013.28303)]. An aspherical manifold is a \( K(E, 1)\)-space: all higher homotopy groups are trivial, but the fundamental group \( \pi _1 (M) \approx E. \) An interesting class of aspherical manifolds are those arising from Seifert fiber space constructions, sometimes called model aspherical manifolds. The main aspects of this method are reviewed in Chapter 2 of this paper [see also Y. Kamishima, K. B. Lee and F. Raymond, Q. J. Math., Oxf. II. Ser. 34, 433-452 (1983; Zbl 0542.57013); the author, Manuscr. Math. 90, No. 1, 63-83 (1996; Zbl 0859.57038)]. A. Borel proved that, if the fundamental group \( E \) of an aspherical manifold \( M \) is centerless and the outer automorphism group \(\operatorname{Aut}(E)/\text{Inn}(E) \) of \( E \) is torsion-free, then \( M \) admits no periodic maps, or equivalently, there are no non-trivial finite groups of homeomorphisms acting effectively on \( M\). Starting from this result, several examples of rather complex aspherical manifolds exhibiting this total lack of periodic maps have been presented [see F. Raymond and J. L. Tollefson, Trans. Am. Math. Soc. 221, 403-418 (1976; Zbl 0333.57002), Corr. in 272, 803-807 (1982; Zbl 0502.57016); P. E. Conner, F. Raymond and P. J. Weinberger, Proc. second Conf. compact Transform. Groups 2, Massachusetts, Amherst 1971, Lect. Notes Math. 299, 81-108 (1972; Zbl 0274.57014)]. All these examples have solvable fundamental groups. The author investigates to what extent the converse of Borel’s theorem holds for model aspherical manifolds. In particular, for e.g. flat Riemannian manifolds, infra-nilmanifolds and infra-solvmanifolds of type \( (R), \) it turns out that having a centerless fundamental group with torsion free outer automorphism group is also necessary to conclude that all finite groups of affine diffeomorphisms acting effectively on the manifold are trivial. The problem of finding less complex examples of such aspherical manifolds with no periodic maps is discussed. Some open problems are pointed out.

MSC:
57S25 Groups acting on specific manifolds
20H15 Other geometric groups, including crystallographic groups
57S17 Finite transformation groups
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
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