# Stiefel-Whitney Classes and Obstructions

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

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

• I believe this is answered affirmatively by Problem 100 (+ comments) of Prasolov (2007). Mar 5, 2017 at 13:29
• @FrancoisZiegler Yeah, thats actually the book I am reading. He seems to think that the property $2\mathfrak{o}_{2k+1}=0$ implies the conclusion in my question. In fact on page 143 he says "$\mathfrak{o}_1=0$ and $w_1=0$ are equivalent because $2\mathfrak{o}_{1}=0$". Am I crazy but I just dont see this. Mar 5, 2017 at 13:37
• Look at $\mathbb{Z}_4$ let $c=2$ then $c\equiv 0 \mod 2\mathbb{Z}_4$ and $2c=0$ but $c\neq 0$. Mar 5, 2017 at 13:39
• By the way, another name for $\mathfrak{o}_{2k + 1}$ is $W_{2k + 1}$ which is an integral Stiefel-Whitney class. Jun 12, 2018 at 15:37