On the cohomology of loop spaces for some Thom spaces

In this paper we identify conditions under which the cohomology $H^*(\Omega M\xi;\k)$fortheloopspace$\Omega M\xi$oftheThomspace$M\xi$ of a spherical fibration $\xi \downarrow B$ can be a polynomial ring. We use the Eilenberg-Moore spectral sequence which has a particularly simple form when the Euler class $e(\xi )\in H^n(B;\text{k})$ vanishes, or equivalently when an orientation class for the Thom space has trivial square. As a consequence of our homological calculations we are able to show that the suspension spectrum ${\mathrm{\Sigma }}^{\mathrm{\infty }}\mathrm{\Omega }M\xi$ has a local splitting replacing the James splitting of $\mathrm{\Sigma }\mathrm{\Omega }M\xi$ when $M\xi$ is a suspension.