#### Volume 9, issue 1 (2005)

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A stable classification of Lefschetz fibrations

### Denis Auroux

Geometry & Topology 9 (2005) 203–217
 arXiv: math.GT/0412120
##### Abstract

We study the classification of Lefschetz fibrations up to stabilization by fiber sum operations. We show that for each genus there is a “universal” fibration ${f}_{g}^{0}$ with the property that, if two Lefschetz fibrations over ${S}^{2}$ have the same Euler–Poincaré characteristic and signature, the same numbers of reducible singular fibers of each type, and admit sections with the same self-intersection, then after repeatedly fiber summing with ${f}_{g}^{0}$ they become isomorphic. As a consequence, any two compact integral symplectic 4–manifolds with the same values of $\left({c}_{1}^{2},{c}_{2},{c}_{1}\cdot \left[\omega \right],{\left[\omega \right]}^{2}\right)$ become symplectomorphic after blowups and symplectic sums with ${f}_{g}^{0}$.

##### Keywords
symplectic 4–manifolds, Lefschetz fibrations, fiber sums, mapping class group factorizations
Primary: 57R17
Secondary: 53D35