acts as the identity on homology is isotopic to the identity*]{}.

\[ex:ruled\]Let $M$ be an orientable smooth four-manifold that admits the structure of a fibration over a closed orientable surface $\Sigma$ of positive genus with fibers diffeomorphic to the $2$-sphere: $$S^2{\ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow}}M \stackrel{\pi}{\longrightarrow} \Sigma.$$ Fix an orientation of $M$ and an orientation of the fibers. Let $F\in H_2(M;{{\mathbb Z}})$ be the homology class of the fiber. Call a symplectic form ${{\omega}}$ on $M$ [**compatible with the fibration**]{} if it restricts to a symplectic form on each fiber. Call it [**compatible with the orientations**]{} if its cohomology class $a$ satisfies $$a^2>0,\qquad {\langle a, F\rangle}>0.$$ Here are some basic facts.

[**1.**]{} By Seiberg–Witten theory [@TAUBES1; @TAUBES2] a cohomology class $a\in H^2(M;{{\mathbb R}})$ is represented by a symplectic form if and only if $a^2\ne0$ and ${\langle a, F\rangle}\ne 0$. Any such cohomology class is uniquely determined by the numbers $a^2$ and ${\langle a, F\rangle}$.

[**2.**]{} A theorem of McDuff [@MCDUFF1] (see also [@MS2 Theorem 9.4.1]) shows that every symplectic form on $M$ that admits a symplectically embedded two-sphere in the homology class $F$ or $-F$ is diffeomorphic to one that is compatible with the fibration. The existence of the required symplectic sphere follows from 