Secondary Stiefel-Whitney class and diffeomorphisms of rational and ruled symplectic 4-manifolds

Details Secondary Stiefel-Whitney class and diffeomorphisms of rational and ruled symplectic 4-manifolds Secondary Stiefel-Whitney class and diffeomorphisms of rational and ruled symplectic 4-manifolds

2009-04-02

arxiv.org/abs/0904.0283v2

Mathematics

Symplectic Geometry

18 pages

Scientific paper

We introduce the secondary Stiefel-Whitney class $\tilde w_2$ of homotopically trivial diffeomorphisms and show that a homotopically trivial symplectomorphism of a ruled 4-manifold is isotopic to identity if and only if the class $\tilde w_2$ vanishes. Using this, we give a detailed description of the combinatorial structure of the diffeotopy group of ruled symplectic 4-manifolds $X$, either minimal or blown-up, and its action on the homology and homotopy groups $H_2(X,\zz)$, $\pi_1(X)$, and $\pi_2(X)$.

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