Smooth \(S^ 1\) actions on homotopy \({\mathbb{C}}P^ 4\)'s. (Q1081901)

Suppose X is a closed smooth homotopy \({\mathbb{C}}P(n)\) and \(c\in H^ 2({\mathbb{C}}P(n); {\mathbb{Z}})\) is a generator. The following is called the Petrie conjecture: If X admits a smooth \(S^ 1\)-action, then \(P(X)=(1+c^ 2)^{n+1}\), P(X) the total Pontryagin class [cf. \textit{T. Petrie}, Bull. Am. Math. Soc. 78, 105-153 (1972; Zbl 0247.57010)]. This has been verified for \(n=3\) by \textit{Í. J. Dejter} [Mich. Math. J. 23, 83-95 (1976; Zbl 0315.55022)] and by a lot of other authors in various special cases. Here the author shows that if X is a homotopy \({\mathbb{C}}P(4)\) which has a non-trivial \(S^ 1\)-action then \(\hat A(X)=\hat A({\mathbb{C}}P(4))\).