Spherical bundles adapted to a $G$-fibration

A spherical fibration $p:E \to B$ is said to be adapted to a $G$-fibration $\pi :E \to E/G$ if there is a fibration $q:E/G \to B$ with fibre the quotient of a sphere by a free $G$-action and such that the composition $q \circ \pi = p$. In this paper it is shown that for spherical bundles in the PL, TOP or Homotopy categories that are adapted to ${Z_2}$- or ${S^1}$-fibrations there is a procedure analogous to the splitting principle for vector bundles that enables one to define characteristic classes for these fibrations and to relate them to the usual characteristic classes. The methods are applied to show that a spherical fibration over a $4$-connected base which is adapted to an ${S^1}$-fibration admits a PL structure.