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Isolated fixed points of circle actions on 4-manifolds. (English) Zbl 0890.57017

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 \(\pm \mathbb{C} \mathbb{P}^2\).

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
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