Kwasik, Sławomir; Schultz, Reinhard Isolated fixed points of circle actions on 4-manifolds. (English) Zbl 0890.57017 Forum Math. 9, No. 4, 517-546 (1997). The paper studies similarities between topological and smooth actions of the circle \(S^1\) on 4-manifolds. For a smooth \(S^1\)-action on a 4-manifold with an isolated fixed point \(x\), there exists a neighborhood of \(x\) in which there are at most four orbit types. The main result of the paper asserts that the same theorem remains true for topological \(S^1\)-actions, under weak regularity conditions near \(x\). This allows the authors to give an upper estimate for the number of orbit types for a smooth \(S^1\)-action on \(S^3\times \mathbb{R}\) to the effect that there are at most three orbit types, and the action is equivariantly homotopy equivalent to a linear \(S^1\)-action on \(S^3\). Another consequence of the main result is a vanishing theorem for the signature of spin 4-manifolds with almost smoothable \(S^1\)-actions, a topological version of the well-known Atiyah-Hirzebruch theorem on vanishing of the \(\widehat{A}\)-genus. One more consequence presented by the authors is a characterization of homotopy types of closed simply connected 4-manifolds \(M\) equipped with almost smoothable \(S^1\)-actions. The characterization asserts that such a manifold \(M\) is homotopy equivalent to a connected sum of copies of \(S^2\times S^2\) and \(\pm \mathbb{C} \mathbb{P}^2\). Reviewer: K.Pawałowski (Poznań) MSC: 57M60 Group actions on manifolds and cell complexes in low dimensions 57P99 Generalized manifolds 57S10 Compact groups of homeomorphisms 57S25 Groups acting on specific manifolds Keywords:topological action; smooth action; circle; orbit types; signature of spin 4-manifolds; \(\widehat{A}\)-genus PDFBibTeX XMLCite \textit{S. Kwasik} and \textit{R. Schultz}, Forum Math. 9, No. 4, 517--546 (1997; Zbl 0890.57017) Full Text: DOI EuDML