On the cohomology of loop spaces for some Thom spaces

摘要

In this paper we identify conditions under which the cohomologyH*(ΩMξ;k) for the loop space ΩMξ of theThom space Mξ of a spherical fibration ξdownarrow Bcan be a polynomial ring. We use the Eilenberg-Moore spectralsequence which has a particularly simple form when the Eulerclass e(ξ)∈ Hn(B;k) vanishes, or equivalently when anorientation class for the Thom space has trivial square. Asa consequence of our homological calculations we are able toshow that the suspension spectrum Σ∞ΩMξhas a local splitting replacing the James splitting ofΣΩMξ when Mξ is a suspension.