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) for the loop space Omega M xi of the Thom space M xi of a spherical fibration xi down arrow 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) epsilon H-n (B; 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 Sigma(infinity)Omega M xi has a local splitting replacing the James splitting of Sigma Omega M xi when M xi is a suspension.