A splitting result for the free loop space of spheres and projective spaces

Abstract

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 ∗ (�X; F2) including the Steenrod algebra action. When X is a projective space CP n , HP n , the Cayley projective plane CaP 2 or a sphere S m we obtain a splitting result for integral and mod two cohomology of the suspension spectrum � ∞ (�X)+. The splitting is in terms of � ∞ X+ and the Thom spaces Th(q�), q ≥ 0 of the q-fold Whitney sums of the tangent bundle � over X.