Group actions and codes (Q2715668)

The paper is inspired by the following statement and a question due to F. Raymond and R. Schultz: ``It is generally felt that a manifold `chosen at random' will have very little symmetry. Can this intuitive notion be made more precise?'' The author has studied this question in his previous work [\textit{V. Puppe}, Ann. Math. Stud. 138, 283-302 (1995; Zbl 0929.57026)]. The results of the paper under review yield evidence for a positive answer to the question in the case of involutions with ``maximal number of isolated fixed points''. An involution is said to have a maximal number of isolated fixed points if all the fixed points of it are isolated and \(\dim_{\mathbb{F}_2}(\bigoplus _iH^i(M;{\mathbb{F}}_2))=|M^{\mathbb{Z} _2}|\). An action with a maximal number of isolated fixed points on a 3-dimensional, closed manifold determines a binary self-dual code of lengthh \(|M^{\mathbb{Z}_2}|\). Such a code determines the cohomology algebra \(H^*(M;\mathbb{F}_2)\) and the equivariant cohomology \(H^*_{\mathbb{Z}_2}(M;\mathbb{F}_2)\). By using results on the classification of binary self-dual binary codes the author obtains asymptotic inequalities which can be interpreted by saying that ``most'' \(\mathbb{F}_2\)-cohomology types of closed 3-manifolds do not admit involutions with a maximal number of isolated fixed points.