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Surgery theory and geometry of representations. (English) Zbl 0654.57001

DMV Seminar, 11. Basel etc.: Birkhäuser Verlag. vii, 115 p. DM 39.80 (1988).
The book consists of two independent parts. The first part written by Tammo tom Dieck deals with applications of surgery theory to specific problems in transformation groups while the second part written by Ian Hambleton gives a systematic introduction to some parts of surgery theory.
In the first part of the book, the applications presented by tom Dieck are concerned with homotopy representations of a finite group \(G\) introduced by T. tom Dieck and T. Petrie [Homotopy representations of finite groups, Publ. Math., Inst. Hautes Étud. Sci. 56, 129–170 (1982; Zbl 0507.57025)]. A basic problem discussed here reads as follows. Which homotopy representations of \(G\) have the \(G\)-homotopy type of smooth representation forms of \(G\); i.e., of smooth \(G\)-manifolds \(M\) such that for each subgroup \(H\) of \(G\), the fixed point set \(M^ H\) is homeomorphic to a sphere (or empty)? The author shows that the geometry of smooth representation forms imposes several restrictions so that not every \(G\)-homotopy type of homotopy representations has its smooth representant. On the other hand, the author proves that if \(G\) is a cyclic \(p\)-group, then a homotopy representation of \(G\) has the \(G\)-homotopy type of a representation sphere of \(G\). This is not true for cyclic groups of order \(pq\), \(p\neq q\) both odd primes. However, as the author shows, each homotopy representation \(X\) of a cyclic group \(G\) is stably linear; i.e., there exists a representation sphere \(SV\) of \(G\) such that the join \(X^*SV\) has the \(G\)-homotopy type of a representation sphere of \(G\). There is a basic invariant of the \(G\)-homotopy type of a homotopy representation of \(G\), namely, its dimension function. The author discusses the classification of homotopy representations with the same dimension function. The related material is collected here mostly from earlier works of tom Dieck. The author reproves also the result of Milnor that the dihedral groups \(D_{2m}\) cannot act freely on spheres and he studies more closely the geometry of representation forms for \(D_{2m}\). Moreover, for representation forms \(X\) of a finite group \(G\) and two subgroups \(H\) and \(K\) of \(G\), the author deals with the linking numbers of \(X^H\) and \(X^K\) in \(X\). In particular, he converts Brieskorn varieties with smooth \(G\)-actions into homotopy spheres with smooth \(G\)-actions and prescribed linking numbers when \(G=D_{2m}\), \(H\neq K\) both isomorphic to \(\mathbb Z/2\), and when \(G=H\times K=\mathbb Z/2\times\mathbb Z/2\). In the first part of the book, the actions are constructed using surgery theory. The method of surgery theory uses normal maps, occurring here as “tangential structures” on homotopy representations \(X\) of \(G\), and is concerned with the problem of changing the \(G\)-map \(f:M\to X\) in the tangential structure into a \(G\)-homotopy equivalence. The author describes the following three methods for the construction of manifolds \(M\) in tangential structures on \(X\): construction of manifolds from representations, manifolds given by algebraic varieties, and construction of manifolds by transversality.
In the second part of the book, Hambleton surveys some of the methods for determining surgery obstructions in surgery problems with finite fundamental groups. He begins with describing the geometrical setting for surgery; the standard notions occur including elementary surgery of type \((k,n-k)\) and its trace, Poincaré duality space \(X\) of formal dimension \(n\), the Spivak normal fibre space of \(X\), degree 1 normal map, surgery obstruction groups, and surgery exact sequence. The discussion of the algebraic surgery theory the author starts from the notion of a symmetric Poincaré complex due to R. Ranicki [The algebraic theory of surgery I. Foundations, and II. Applications to topology, Proc. Lond. Math. Soc., III. Ser. 40, 87–192 (1980; Zbl 0471.57010) and 193–283 (1980; Zbl 0471.57011)]. Then he emphasizes the role of the description of the \(L\)-group \(L_n(A,\varepsilon)\) as the cobordism group of \(n\)-dimensional quadratic Poincaré complexes of free \(A\)-modules for any ring \(A\) with involution \(\varepsilon:A\to A\), or even for the more general notion of a ring with antistructure. The cobordism description allows to derive exact sequences useful for calculations. The author gives examples of such exact sequences and using them he relates the computation of surgery obstruction groups to the \(L\)-theory of rings. He shows that the \(L\)-groups are determined only up to extensions and the calculations can be given in terms of ideal class groups. Then the author concentrates on the problem of determining whether a surgery obstruction is zero or non-zero, and discusses two advances. The first is an improvement of the Dress induction theorem and the second result concerns the detection of surgery obstructions arising from a degree 1 normal map. Finally, he gives an application of the techniques in studying semi-free topological actions of a finite group on \(\mathbb R^{n+k}\) with fixed point set \(\mathbb R^k\).
Both parts of the book contain material not presented in any textbook and provide interesting applications of surgery theory to transformation groups, as well as give suggestions for reading the literature and studying the related problems.
Reviewer: K.Pawałowski

MSC:

57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
57S17 Finite transformation groups
57R67 Surgery obstructions, Wall groups
57S25 Groups acting on specific manifolds
57R65 Surgery and handlebodies
57R91 Equivariant algebraic topology of manifolds
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