2 6 N ov 2 00 4 A splitting result for the free loop space of spheres and projective spaces

Let X be a 1-connected compact space such that the algebra H(X;F2) is generated by one single element. We compute the cohomology of the free loop space H(ΛX;F2) including the Steenrod algebra action. When X is a projective space CP, HP, the Cayley projective plane CaP or a sphere Sm we obtain a splitting result for integral and mod two cohomology of the suspension spectrum Σ(ΛX)+. The splitting is in terms of Σ X+ and the Thom spaces Th(qτ), q ≥ 0 of the q-fold Whitney sums of the tangent bundle τ over X.