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Generalized symplectic rational blowdowns. (English) Zbl 0991.57027

The rational blowdown surgery is to remove a neighborhood of a linear chain of embedded spheres whose boundary is the lense space \(L(n^2,n-1)\), \(n \geq 2\), and replace it with a rational ball with boundary \(L(n^2,n-1)\). Fintushel and Stern use the rational blowdown surgery on smooth 4-manifolds to obtain an infinite family of exotic \(K3\) surfaces which are homeomorphic but not diffeomorphic to a degree 4 complex surface in \(\mathbb{C}\mathbb{P}^3\), all admit symplectic structures. If \(n, m \geq 1\) and relatively prime, the lens spaces \(L(n^2,nm-1)\) bound rational balls \(B_{m.n}\). We can define a broader class of rational blowdown, called generalized rational blowndown. The author shows that the generalized rational blowdown of a symplectic 4-manifold along symplectic spheres admits a symplectic structure induced from the symplectic structures on \(M\) and \(B_{m.n}\) which is considered as the total space of a singular Lagrangian fibration.

MSC:

57R17 Symplectic and contact topology in high or arbitrary dimension
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
57R50 Differential topological aspects of diffeomorphisms

Citations:

Zbl 0991.57038
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References:

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