# The Thom spectrum of the stable framing tangential structure is the sphere spectrum

where a stable framing is a stable tangential structure $\mathcal{X} = EO \to BO$ (ref. Dan Freed's notes Exercise 9.50). This is Exercise 10.32 in Dan Freed's notes and I have no idea to get started with the proof. Could somebody sketch the main ideas and steps?

• You should ask this question at Mathoverflow. – Moishe Kohan Aug 1 '16 at 2:44
• @studiosus: why? My recent posts were knocked down by the mathematicians there hardly so I dare not to post questions there. – PhysicsMath Aug 2 '16 at 18:28

The "Thom prespectrum" consists of spaces like $EO(n)^{S(n)}$ in each level. Each base space $EO(n)$ is contractible, so any vector bundle over it is trivial. The Thom complex of a trivial bundle is an iterated suspension of the base space. This gives you spheres (up to homotopy) in each level. Then it's just a matter of checking that the bonding maps are what you'd expect them to be.