Let $E$ be a vector bundle over a simplicial space $B$.

Let $$\mathfrak{o}_k\in H^k(B,\{\pi_{k-1}(V(n-k+1)\})$$ be the $k$th obstruction to a $k$ independent vector fields. ($V(n,k)$ the Stiefel manifold.)

Then the even Stiefel-Whitney classes are equal to these obstructions: $$w_{2k}=\mathfrak{o}_{2k}$$

and the odd are the mod $2$ reductions, $$w_{2k+1}=\mathfrak{o}_{2k+1} \mod 2$$

Steenrod has shown

$$\beta^*(\mathfrak{o}_{2k} )=\mathfrak{o}_{2k+1}.$$

Where $\beta$ is the Bockstein operator associated to the mod 2 reduction $\mathbb{Z}\rightarrow \mathbb{Z}_2$.

Does it hold in general that

$$\mathfrak{o}_{2k+1}=0\Leftrightarrow w_{2k+1}=0 \ \ ? $$

Let $E$ be a vector bundle over a simplicial space $B$.

Let $$\mathfrak{o}_k\in H^k(B,\{\pi_{k-1}(V(n,n-k+1))\})$$ be the $k$th obstruction to $k$ independent sections. ($V(n,k)$ the Stiefel manifold.)

Then the even Stiefel-Whitney classes are equal to these obstructions: $$w_{2k}=\mathfrak{o}_{2k}$$

and the odd are the mod $2$ reductions, $$w_{2k+1}=\mathfrak{o}_{2k+1} \mod 2$$

Steenrod has shown

$$\beta^*(\mathfrak{o}_{2k} )=\mathfrak{o}_{2k+1}.$$

Where $\beta$ is the Bockstein operator associated to the mod 2 reduction $\mathbb{Z}\rightarrow \mathbb{Z}_2$.

Does it hold in general that

$$\mathfrak{o}_{2k+1}=0\Leftrightarrow w_{2k+1}=0 \ \ ? $$