Topics in transformation groups (Q2776329)

The paper under review is an excellent survey on some basic topics of finite transformation groups. The paper consists of three sections. In the first one the authors give the basic notions, examples and some fundamental results of transformation groups. Among the key results are the slice theorem, the \(G\)-tubular neighborhood theorem and the orbit theorem. In this section, also the notion of a \(G\)-CW complex is introduced and the Equivariant Whitehead theorem for \(G\)-CW complexes is stated. Illman's recent results on equivariant triangulability of proper smooth \(G\)-manifolds are also mentioned. NEWLINENEWLINENEWLINEIn Section 2 the authors summarize some of the basic techniques and results on the algebraic side of the theory of finite transformation groups. The necessary definitions and ideas have been made available here, and for the details the reader is referred to the abundant list of carefully chosen references. This section is devoted basically to cohomological aspects of finite transformation groups. Cohomological ideas connect the geometry of group actions to algebraic invariants of finite groups. After outlining the basic tools in the subject, the authors describe the most important results here and then provide selected topics where these ideas and closely related notions can be applied. The texts by \textit{C. Allday} and \textit{V. Puppe} [Cohomological methods in transformation groups, Camb. Stud. Adv. Math. 32 (1993; Zbl 0799.55001)], \textit{G. E. Bredon} [Introduction to compact transformation groups, Pure Appl. Math., Academic Press, 46 (1972; Zbl 0246.57017)]and \textit{T. tom Dieck} [Transformation groups, De Gruyter Stud. Math. 8 (1987; Zbl 0611.57002)] are recommended as background references. The titles of the corresponding subsections give enough information about the content of Section 2. They are: 2.1 Introduction; 2.2 Universal \(G\)-spaces and the Borel construction; 2.3 Free group actions of spheres; 2.4 Actions of elementary abelian groups and the localization theorem; 2.5 The structure of equivariant cohomology; 2.6 Tate cohomology, exponents and group actions; 2.7 Acyclic complexes and the Conner conjecture; 2.8 Subgroup complexes and homotopy approximations to classifying spaces; 2.9 Group actions and discrete groups; 2.10 Equivariant \(K\)-theory, 2.11 Equivariant stable homotopy theory; 2.12 Miscellaneous problems. NEWLINENEWLINENEWLINESection 3 is devoted to geometric methods in transformation groups. The authors start this section with a discussion of five well-known open problems, the solution of which would lead to clear advances in transformation group theory. The first one is the Borel conjecture stating that if a discrete group \(\Gamma\) acts freely and properly on contractible manifolds \(M\) and \(N\) with compact quotients, then the quotients are homeomorphic. Homotopy-theoretic and group-theoretic motivations of the Borel conjecture are also discussed. There is a sharper form of the Borel conjecture: a homotopy equivalence between closed, aspherical manifolds is homotopic to a homeomorphism. There is also a reasonable version of the Borel conjecture for manifolds with boundary: a homotopy equivalence between compact, aspherical manifolds which is a homeomorphism on the boundary is homotopic, relative to the boundary, to a homeomorphism. NEWLINENEWLINENEWLINEThe second famous open problem in this section states that any smooth action of a finite group on \(\mathbb S^3\) is equivalent to a linear action. A key case here is resolved: P. A. Smith showed that for a prime \(p\), if \(\mathbb Z/p\) acts smoothly, preserving orientation on \(\mathbb S^3\) with a non-empty fixed point set, then the fixed point set is an embedded circle. He conjectured that the fixed point set is always unknotted. In the book edited by \textit{J. W. Morgan} and \textit{H. Bass} [The Smith conjecture, Pure Appl. Math., Academic Press, 112 (1979; Zbl 0599.57001)] it was proven that such an action is equivariantly diffeomorphic to a linear action, giving the Smith conjecture. The linearization question for general actions of finite groups is yet unresolved. The authors formulate also the following generalization of the Poincaré conjecture: a closed \(3\)-manifold with finite fundamental group is diffeomorphic to a linear spherical space form \(\mathbb S^3/G\). As concerns the actions of finite groups on \(\mathbb S^n\), \(n\geq 4\), they are reasonably well understood and need not be equivalent to linear actions. NEWLINENEWLINENEWLINEThe third open problem is the Hilbert-Smith conjecture claiming that a locally compact topological group \(G\) acting effectively on a connected manifold \(M\) must be a Lie group. The following equivalent form of the problem is well known: the additive group of the \(p\)-adic integers can not act effectively on a connected manifold. The authors list specific known cases where the conjecture is resolved. The reviewer would like to mention also a recent short proof of the Hilbert-Smith conjecture for the case of free Lipschitz actions of compact groups, given by \textit{E. V. Shchepin} [Mat. Notes 65, No. 3, 381-385 (1999); translation from Mat. Zametki 65, No. 3, 457-463 (1999; Zbl 0962.37008)]. NEWLINENEWLINENEWLINEThe fourth open problem concerns finite group actions on a product of spheres. It is evident that any finite group acts freely on a product of spheres. The main problem here, which remains unsolved, is to show that the number of spheres with any given free \(G\)-action will bound the rank of the elementary abelian subgroups in the group. More concretely, the following questions still remain unanswered: if \(G\) is a finite group of rank \(k>1\), does \(G\) act freely on a finite dimensional CW complex homotopy equivalent to a product of \(k\) spheres? If so, does \(G\) act freely on a product of \(k\) spheres? NEWLINENEWLINENEWLINEFinally, the fifth conjecture concerns asymmetrical manifolds and it claims that there is a closed, simply connected manifold which does not admit an effective action of a finite group. NEWLINENEWLINENEWLINEIn the remaining part of Section 3 the authors introduce some geometric methods and techniques involved in finite group actions. These include the finiteness obstruction and \(K_0\), simple homotopy theory and \(K_1\), and surgery theory. Rather than to introduce these topics abstractly, the authors do this through a concrete geometric situation, the study of topological spherical space forms, manifolds whose universal cover is a sphere. NEWLINENEWLINENEWLINEIn the bibliography the authors list a number of carefully chosen references on several important topics of transformation groups which are not discussed in the present survey. These topics include compact Lie group actions, equivariant bordism, group actions on \(4\)-manifolds, group actions on knot complements, etc. The survey under review could serve as an inseparable guide for researchers in the group actions. The accessible form of the presentation makes it attractive also for beginners.NEWLINENEWLINEFor the entire collection see [Zbl 0977.00029].