Integral representations of finite transformation groups. III: Simply- connected four-manifolds
For a simply-connected 4-manifold X with an action of a finite group G, the author studies the G-module structure of \(H_ 2(X)\) and its relationship to the topology of the G-action on X [cf. the previous parts, ibid. 59, 215-226, 227-236 (1989; Zbl 0705.57018 and Zbl 0705.57019)]. For \(G={\mathbb{Z}}_ p\), the author shows that \(H_ 2(X)\) and the singular set of the action determine each other in the context of regular Poincaré G-complexes X [cf. \textit{J. Hambleton} and \textit{M. Kreck}, Math. Ann. 280, No.1, 85-104 (1988; Zbl 0616.57009)]. Then, the author applies this information to determine the G-module \(H_ 2(X)\) for more general finite groups; e.g., for p-elementary abelian groups. He presents also a necessary and sufficient condition for an infinite- dimensional free G-complex to be G-homotopy equivalent to a finitely dominated Poincaré G-complex.
scientific article; zbMATH DE number 4173930
scientific article