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Modifying surfaces in 4-manifolds by twist spinning. (English) Zbl 1104.57018
Let \(X\) be a smooth 4-dimensional manifold and \(\Sigma\subset X\) be a smooth surface of positive genus. Given a knot \(K\) in \(S^3\) and an integer \(m\), the author constructs a new surface \(\Sigma_K(m)\) in \(X\) obtained from \(\Sigma\) by a so-called \(m\)-twist rim surgery. This surgery is a “twisted version” of the rim surgery introduced by R.Fintushel and R. Stern [Math. Res. Lett. 4, No.6, 907–914 (1997; Zbl 0894.57014)], this is where the name comes from. The author finds some topological conditions under which the pair \((X,\Sigma_K(m))\) is homeomorphic to the pair \((X, \Sigma)\). As an application, the author gets examples of smoothly knotted surfaces in \({\mathbb C}{\mathbb P}^2\) that are topologically unknotted, and these surfaces are not isotopic to complex curves.

MSC:
57R57 Applications of global analysis to structures on manifolds
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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