O(2) actions on the 5-sphere (Q1819420)

A smooth \(S^ 1\)-action is called pseudofree, if it is free except for finitely many exceptional orbits. The authors study the following question, which they call Montgomery-Yang problem: Must a pseudofree \(S^ 1\)-action on \(S^ 5\) have \(\leq 3\) exceptional orbits ? The answer is ''yes'' under the added hypothesis that the pseudofree \(S^ 1\)-action extends to an action of O(2). For a proof assume that the pseudofree \(S^ 1\)-action on \(S^ 5\) has n exceptional orbits. In a former paper [Ann. Math., II. Ser. 122, 335-364 (1985; Zbl 0602.57013)] the authors had associated an integral invariant R(X) to the orbit space \(X=S^ 5/S^ 1\) and proved that \(R(X)<0\). The proof used techniques of gauge theory. Again using techniques of gauge theory now it is proved that \(2n-9-R(X)<0\). Because R(X) is negative, this is only possible for \(n\leq 3\). Finally the authors construct pseudofree \(S^ 1\)-actions on rational homology 5-spheres from given orbit spaces.