Journal of the
Korean Mathematical Society JKMS

Let $P \xrightarrow{\pi} M$ be an oriented $S^2$-fiber bundle over a closed manifold $M$ and let $Q$ be its associated $SO(3)$-bundle, then we investigate the ring structure of the cohomology of the total space $P$ by constructing the coupling form $\tau_A$ induced from an $SO(3)$ connection $A$. We show that the cohomology ring of total space splits into those of the base space and the fiber space if and only if the Pontrjagin class $p_1(Q) \in H^4(M;\mathbb{Z})$ vanishes. We apply this result to the twistor spaces of 4-manifolds.

Keywords: $S^2$-fiber bundle, coupling 2-form, twistor space

MSC numbers: 53D05