Stiefel-Whitney classes are invertible and for $w$, the Stiefel-Whitney class of the tangent bundle of $M$, we have its inverse $\bar{w}$. I want to prove that if $n$ is not a power of 2 then the dual Stiefel-Whitney classes $\bar{w}_{n-1} = 0$.

For the Steenrod square $\mathrm{Sq} : H^\Pi (M) \rightarrow H^\Pi(M)$ we can define its inverse $\bar{Sq}$. Then we can see easily that $\langle \bar{Sq}(x), \mu \rangle = \langle \bar{w} \cup x , \mu \rangle$. Set $x$ as suitable element in $H^0(M)$ we can prove that $\bar{w}_n = 0$. And I also think of proving $\bar{w}_{n-1} = 0$ by letting $x$ be Poincaré dual to $\bar{w}_{n-1}$. However I can't check that $\langle \bar{Sq}(x) , \mu \rangle = 0$.


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