Exercises on characteristic classes

1. a) Compute the Stiefel-Whitney classes of the tangent bundle of RP . (Use the method from class for the tangent Chern classes of complex projectives spaces.) b) Conclude that if the tangent bundle is trivial, then n = 2 − 1 for some m. (In fact n must be 0, 1, 3, 7, but this is much harder to prove; one proof uses the Bott periodicity theorem.) c) Deduce (very easily!) a complete characterization of which real and complex projective spaces admit a spin structure. Recall that having a spin structure is equivalent to w1(τ) = w2(τ) = 0, where τ is the tangent bundle.