Circle actions on 4-manifolds. II
The authors consider a closed, orientable topological 4-manifold \(M\) which admits a non-trivial \(S^1\)-action so that the following condition holds: (O) The number of orbit types near a fixed point is at most 4. This condition does hold for smooth actions, but it is unknown whether it holds for topological actions. It does not always hold for \(S^1\) actions on cohomology 4-manifolds. The authors show that when condition (O) holds, then the cup product form on \(H^2(M)/\)torsion decomposes into a direct sum of forms of rank 1 or 2. In particular, if the form is even, then its signature is zero. Under additional hypotheses, the result had been proved earlier by \textit{R. Fintushel} [Trans. Am. Math. Soc. 230, 147-171 (1977; Zbl 0362.57014); 242, 377-390 (1978; Zbl 0362.57015)], \textit{T. Yoshida} [Math. J. Okayama Univ. 20, 25-40 (1978; Zbl 0382.57021)] and \textit{W. Huck} [Manuscr. Math. 87, No. 1, 51-70 (1995; Zbl 0839.57024)].
scientific article; zbMATH DE number 1292440
scientific article