Rigid versus non-rigid cyclic actions
The authors study locally smooth \({\mathbb{Z}}_ p\)-actions (p odd prime) on homotopy \({\mathbb{C}}P^ n\)'s with fixed point-set a codimension two component and an isolated point (Type \(II_ 0)\). They treat the question of algebraic standardness as introduced by \textit{K. H. Dovermann} [Math. Proc. Camb. Philos. Soc. 101, 487-507 (1987; Zbl 0629.57025)]. Theorem A: Let X be a cohomology \({\mathbb{C}}P^ n\). Then there is a constant \(c_ x\), which depends only on the Pontryagin class of X, such that any locally smooth \({\mathbb{Z}}_ p\) action of Type \(II_ 0\) is algebraically standard if \(p\geq c_ x.\) If \(n\gg 2p\) they in contrast obtain also that there are infinitely many homotopy \({\mathbb{C}}P^ n\)'s of Type \(II_ 0\) with nonstandard Pontryagin class (Theorem B). They conjecture: There exists a function d(n) such that Type \(II_ 0\) actions on cohomology \({\mathbb{C}}P^ n\)'s are algebraically standard if \(p\geq d(n)\). The proofs make use of careful estimates of terms supplied by the G-signature theorem and surgery-theory.
locally smooth \({\mathbb{Z}}_ p\)-actions on homotopy \({\mathbb{C}}P^ n\prime s\)
scientific article; zbMATH DE number 4137030