Actions of elementary Abelian p-groups
An n-dimensional near-manifold X is a locally compact topological space X such that for a closed subset \(S\subset X\), \(\dim S\leq n-2,\) X-S is a non-empty n-dimensional manifold. This definition has its usual generalizations and it is motivated by considering complex varieties with singular set S. The author continues the study of actions of \(G=({\mathbb{Z}}/p)^ r\) on near-manifolds, initially started by \textit{W. Browder} and \textit{N. M. Katz} [Current trends in algebraic topology, Semin. London/Ont. 1981, Can. Math. Soc. Conf. Proc. 2, Part 2, 23-33 (1982; Zbl 0551.57022)]. Let \(j^*: H^ n_ G(M)\to H^ n(M)\) be induced by the inclusion j: \(M\to E_ G\times_ GM\), and let \(E=| H^ n(M)/j^*H^ n_ G(M)|\) for an n-manifold. Theorem. \(\max_{x\in M}| G_ x| =| G| /E\). This is a consequence of a more general result about G-actions on near-manifolds. There are a number of interesting corollaries of this theorem and its method of proof. For example: Corollary. Let \(G=({\mathbb{Z}}/p)^ r\) act on a compact n-dimensional near-manifold X with a ``manifold fixed point'' \(x_ 0\in X^ G\) and \(n>0\). Then \(X^ G\neq \{x_ 0\}\). There are several applications to G-actions on projective varieties which provide examples of near-manifolds.
scientific article; zbMATH DE number 4096403