THE TOPOLOGY OF S2-FIBER BUNDLES

Let$P{\rightarrow}M$ 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 TA 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 Pontrjangin class $p_1(Q)\;{\in}\;H^4(M;\mathbb{Z})$ vanishes. We apply this result to the twistor spaces of 4-manifolds.