On the cohomology of loop spaces for some Thom spaces

In this paper we identify conditions under which the cohomology H∗(ΩMξ; k) for the loop space ΩMξ of the Thom space Mξ of a spherical fibration ξ ↓ B can be a polynomial ring. We use the Eilenberg–Moore spectral sequence which has a particularly simple form when the Euler class e(ξ) ∈ H(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 Σ∞ΩMξ has a local splitting replacing the James splitting of ΣΩMξ when Mξ is a suspension.