Talk:Does the existence of a string structure depend on a spin structure ? (and a generalization)
From Manifold Atlas
Re Answers: do we know the positive answer for all relevant , or just some. i.e. if with can the obstruction to lifting a structure to depend upon the choice of structure?
This is answered now too by a mod 2 consideration and Stong's paper. I believe the mod 3 consideration works for all multiples of 4. Interestingly, modulo bigger primes seems it seems to hold that the obstructions to lifting from to are equal if the -structure is the same, but can be different if one only knows that the -structure is the same.