Integral representations of finite transformation groups. II: Non-simply- connected spaces
[This article is reviewed together with the preceding one (see Zbl 0705.57018.] Let R be a commutative ring with unit and let G be a finite group. By a theorem of \textit{L. G. Chouinard} [J. Pure Appl. Algebra 7, 287-302 (1976; Zbl 0327.20020)] a finitely generated RG-module M is projective if and only if M is RE-projective for all maximal p-elementary subgroups \(E\subseteq G.\) The main result of the first paper is a refinement of Chouinard's theorem for \(M=\bar H^*(X;R)\) where X is a connected G-space with the following properties: X is compact or has finite cohomological dimension mod p for all primes p dividing \(| G|\), and for every maximal p-elementary subgroup \(A\subseteq G\) the cohomology spectral sequence of \(E_{A\times A}X\to BA\) degenerates. The refinement consists in showing that it suffices to test projectivity of M over RC for cyclic subgroups C of prime order in order to obtain the projectivity of M as an RG-module. This is proved using Smith theory to show that the hypothesis of Chouinard's theorem is satisfied. The spectral sequence condition holds if X is a Moore space. The author presents evidence that probably it cannot be altogether dispensed with: For an arbitrary finite group G with non-periodic cohomology he constructs a \({\mathbb{Z}}\)-free, \({\mathbb{Z}}G\)-finitely generated, indecomposable \({\mathbb{Z}}G\)-module M with periodic cohomology which is \({\mathbb{Z}}C\)-projective for all prime order subgroups \(C\subseteq G\), but not \({\mathbb{Z}}G\)-projective. It remains open whether such a counterexample can be realized as the cohomology module of a G-space X; but by the main result it certainly is not isomorphic to the cohomology of a finite- dimensional Moore space with G-action. This provides a host of counterexamples to the generalized Steenrod problem beyond the counterexamples given by \textit{G. Carlsson} [Invent. Math. 64, 171-174 (1981; Zbl 0477.55007)] for \(G={\mathbb{Z}}_ p\times {\mathbb{Z}}_ p\). In fact, the assumption of finite dimension can be dropped, as is indicated in another paper of the author [Transformation groups. Proc. Symp., Poznań/Pol. 1985, Lect. Notes Math. 1217, 26-57 (1986; Zbl 0609.57021)]. Forthcoming papers with further applications are promised. In the second paper, the author extends the main result of the first paper to certain actions of discrete groups which may be infinite. Specifically, for \(\pi =\pi_ 1(X)\) he considers the (possibly infinite) extension \(1\to \pi \to \tilde G\to^{\vartheta}\to G\to 1\) such that the action of G on X lifts to an action of \(\tilde G\) on the universal covering space \(\tilde X\) of X. Let R be \({\mathbb{Z}}\) or the algebraic closure \(\overline{{\mathbb{F}}_ p}\) of a finite prime field, and assume that \(H_*(X;R\pi)\) is finitely presented and \(R\pi\)-projective and that for every maximal p-elementary subgroup \(A\subseteq G\) the Serre homology spectral sequence with coefficients in R collapses. Under these hypotheses the author proves that \(H_*(X;R\pi)\) is projective as an \(R\tilde G\)-module if and only if for all cyclic subgroups \(C\subseteq G\) of prime order (for \(R={\mathbb{Z}})\) or of order p (for \(R=\overline{{\mathbb{F}}_ p})\) \(H_*(X;R\pi)\) is projective over \(R\cdot \vartheta^{-1}(C)\). This is derived from the main result of the first paper by purely algebraic means. There are also relative versions. As an application the author deduces the following criterion for certain CW-complexes Z to be finitely dominated, using the finiteness conditions of \textit{C. T. C. Wall} [Proc. Roy. Soc., London, Ser. A 295, 129-139 (1966; Zbl 0152.219)]: Assume that \(\pi_ 1(Z)\) has a normal subgroup \(\pi\) of finite index. Then Z is finitely dominated if (and only if) the covering spaces of Z corresponding to \(\pi\) and to the inverse images \(\tilde C\subseteq \pi_ 1(Z)\) under \(\pi_ 1(Z)\to \pi_ 1(Z)/\pi\) of prime order subgroups C of \(\pi_ 1(Z)/\pi\) are finitely dominated. \{Reviewer's remark: In the author's wording, \(\pi\) is asked to be of finite order, but it seems that it is the finiteness of the index he wants.\}
scientific article; zbMATH DE number 4157506
scientific article