Symmetries of homology complex projective planes (Q916161)

We consider here the question of which finite and compact Lie groups occur as groups of locally smooth symmetries of topological four- manifolds having the same integral homology as \({\mathbb{C}}P^ 2\), the complex projective plane. Our main result is that all such groups have a faithful projective representation of complex dimension 3 and therefore they also act linearly on \({\mathbb{C}}P^ 2\). In the case of homologically trivial actions this result has been previously shown by \textit{D. M. Wilczyński} [Ph. D. Thesis Indiana Univ. (1987)] and \textit{I. Hambleton} and \textit{R. Lee} [J. Algebra 116, 227-242 (1988; Zbl 0659.20041)]. The general principle behind recent results concerning locally smooth actions on homology \({\mathbb{C}}P^ 2\)'s is that the rigid topology of these manifolds forces the action to have the same local invariants as those of some linear action on \({\mathbb{C}}P^ 2\). This means that both actions have the same fixed points (if any), singular sets, local representations at the fixed points, etc.