# at.algebraic topology – Even, non liftable Stiefel-Whitney class

Let $$M$$ be a smooth manifold and $$E$$ a smooth real vector bundle of even rank over $$M$$.
If $$E$$ admits of a complex vector bundle structure $$mathcal E$$ ($$mathcal E_mathbb R=E$$) then all odd Stiefel-Whitney classes of $$E$$ vanish: $$w_{2i+1}(E)=0$$
Moreover the even Stiefel-Whitney classes of $$E$$ are the images under the reduction morphism $$operatorname {red}^{2i}:H^{2i}(M,mathbb Z)to H^{2i}(M,mathbb F_2)$$ of its Chern classes, namely $$w_{2i}(E)=operatorname {red}^{2i}(c_i(mathcal E))$$
My question
Is there a real vector bundle of even rank $$E$$ with all odd $$w_{2i+1}(E)=0$$ that nevertheless cannot be endowed with a complex structure just because some $$w_{2i}(E)in H^{2i}(M,mathbb F_2)$$ cannot be lifted to $$mathbb Z$$?
Explicitly, the equation $$operatorname {red}^{2i}(c_i)=w_{2i}(E)in H^{2i}(M,mathbb F_2)$$ has no solution $$c_iin H^{2i}(M,mathbb Z)$$ .
(The answer is probably quite easy but not for me classical algebraic geometer very unexperienced with real vector bundles.)