On the topological classification of pseudofree group actions on 4-manifolds: II
For Part I see [the second named author, Math. Z. 217, 335-366 (1994; Zbl 0813.57030)]. Let \(G\) be a finite cyclic group which acts (orientation preserving) locally linearly and pseudofreely on a closed oriented simply connected 4-manifold \(M\) with a nonempty fixed point set. Let \(O(\lambda)\) be the quotient of \(\Aut (H_2(M), \lambda)\) modulo those automorphisms inducing the identity on the Tate cohomology group \(\widehat H^0 (G;H_2(M))\). Similarly defined are the groups \(O(\lambda_r)\) for \(\lambda_r= \lambda\oplus rH\), where \(H\) is the hyperbolic form on \(\mathbb{Z}[G]^2\) and \(O_{st}(\lambda)= O(\lambda_r)\) with \(r\gg 0\). Then it follows that \[ O(\lambda) \subseteq O(\lambda_1) \subseteq O_{st} (\lambda) \subseteq O_*(\lambda), \] where the stable \(O\)-group \(O_*(\lambda)\) is the group of determinant \(\pm 1\) isometries of the associated quadratic form on \(\widehat H^0 (G;H_2 (M))\). The authors prove the following stability result of the equivariant intersection form \(\lambda\) of the \(G\)-action on \(M\): \[ O(\lambda_1)= O_{st} (\lambda)= O_*(\lambda). \] Then they obtain a topological rigidity theorem which states that two locally linear, pseudofree actions on \(M\), with the equivariant intersection forms indefinite and of rank at least 3 at each irreducible character, are topologically conjugate by an orientation preserving homeomorphism if and only if their oriented local representations at the corresponding fixed points are linearly equivalent.
scientific article; zbMATH DE number 929693
scientific article