Finite group actions on a homotopy complex projective spaces
A homotopy \(CP^ n\) is a 2n-dimensional, smooth, closed, manifold which is homotopy equivalent to complex projective space \(CP^ n\). For \(n\geq 3\), it is known that there are infinitely many differentiably distinct homotopy \(CP^ n\)'s. The main result of this paper is that if G is an odd order group, or if G is 2-Sylow subgroup which is cyclic or \(Z_ 2\oplus Z_ 2\), or if G is a finite product of \(Z_ 2\)'s, then G acts smoothly and effectively on infinitely many differentiably distinct homotopy \(CP^ n\)'s, for some n. This is in contrast to the rigidity of Pontryagin classes conjectured for \(S^ 1\)-actions by \textit{T. Petrie} in his 1972 paper [Bull. Am. Math. Soc. 78, 105-153 (1972; Zbl 0247.57010)]. The major tool involved in the proof is equivariant surgery theory. The normal maps on which surgery is performed are endowed with a G-equivariant, orientation reversing involution thereby ensuring that the relevant equivariant signatures vanish.
infinitely many differentiably distinct homotopy \(CP^ n\prime s\)
scientific article; zbMATH DE number 4028441
scientific article