based on the proof of Stasheff’s theorem on the classification of Hurewicz fibrations [@stasheff]. More general versions of this theorem also appear in [@french] and [@waner2].

Also note that, as in the case of orientable vector bundle theory, we can give an orientable version of the classification of $H$-fibrations over a trivial $H$-space $B$. If $p:E \to B$ is an $H$-fibration with fiber $F_H$ over a trivial $H$-space, then there is a natural group homomorphism $\chi: \pi_1 (B)\to \pi_0 (\operatorname{Aut}_H (F_H))=\mathcal{E}_H (F_H)$ where $\mathcal{E}_H(F_H)$ denotes the group of homotopy classes of self $H$-homotopy equivalences of $F_H$. If this homomorphism is trivial, then we call the $H$-fibration $p$ a [*homotopy orientable*]{} $H$-fibration. This notion of orientability is stronger than usual notion of orientable fibration where one only requires the action on the homology of $F_H$ to be trivial (see [@ehrlich]). Note that an $H$-fibration is homotopy orientable if and only if its classifying map $f: B \to B\operatorname{Aut}_H (F_H)$ lifts to a map $\tilde f : B \to B \operatorname{Aut}^{I} _H (F_H )$. Also note that homotopy orientable fibrations are classified (as homotopy orientable fibrations) by the homotopy classes of maps $[B , B\operatorname{Aut}_H ^I (F_H)]$. We will be using these 