I want to show that for even dimensional spheres, there does not exist a nowhere vanishing vector field, namely a non-trivial cross section of its tangent bundle. I am wondering how elementary the proof could be. While I know this should follow from Lefschetz fixed-point theorem (I have seen the proof when $n=2$ a long time ago), I believe there should be easier proofs available.
It seems this theorem follows from standard differential topology tools (like Poincare-Hopf theorem and computing the Euler characteristic). And similarly one may use tools like Euler class. But is there a more elementary proof available (at the level of singular homology and cohomology)?
Here is a "cyclic type argument". Assuming I can show that if a nonwhere vanshing vector field exists, then the antipodal map must be homotopically equivalent to the identity map, then I would be able to prove the original statement since the antipodal map for even dimensional sphere has degree $-1$. However, I do not know how to prove the first statement without passing to even and odd cases. So this does not really help very much.
Sorry if this question is too trivial. I am looking for a hint (not an answer).
Reference: Milnor&Stasheff, Problem 2-B