2
$\begingroup$

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?

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

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.

$\endgroup$
1
  • $\begingroup$ "The Thom complex of a trivial bundle is an iterated suspension of the base space." That is the key! Many thanks for the idea, JHF! $\endgroup$ – PhysicsMath Aug 2 '16 at 18:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.