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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

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

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    $\begingroup$ I believe this is answered affirmatively by Problem 100 (+ comments) of Prasolov (2007). $\endgroup$
    – Francois Ziegler
    Commented Mar 5, 2017 at 13:29
  • $\begingroup$ @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. $\endgroup$
    – Rene Schipperus
    Commented Mar 5, 2017 at 13:37
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    $\begingroup$ Look at $\mathbb{Z}_4$ let $c=2$ then $c\equiv 0 \mod 2\mathbb{Z}_4$ and $2c=0$ but $c\neq 0$. $\endgroup$
    – Rene Schipperus
    Commented Mar 5, 2017 at 13:39
  • $\begingroup$ By the way, another name for $\mathfrak{o}_{2k + 1}$ is $W_{2k + 1}$ which is an integral Stiefel-Whitney class. $\endgroup$
    – Michael Albanese
    Commented Jun 12, 2018 at 15:37

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