Stiefel-Whitney and Wu classes of $d$-manifold

Question

Is it true that Stiefel-Whitney $w_i$ and Wu classes $u_i$ of $d$-manifold, we always have the following: $$ u_{d-1}=0, \tag{1} $$ $$ u_d=0, \tag{2} $$ $$ Sq^1(u_{d-1})=0. \tag{3} $$ in any dimensions $d$?

• Why are the above conditions true?

• Do we have more conditions than the above in some dimensions $d$?

Answer 1

Let $M$ be a closed $d$-dimensional manifold. Recall that the Wu classes $u_i$ satisfy $\operatorname{Sq}^i(x) = u_i\cup x$ for all $x \in H^{d-i}(M; \mathbb{Z}_2)$. One of the properties of Steenrod squares is that $\operatorname{Sq}^i(x) = 0$ for $i > \deg x$. In particular, for $i > d - i$ (i.e. $i > \frac{d}{2}$), we have $0 = \operatorname{Sq}^i(x) = u_i\cup x$ for all $x \in H^{d-i}(M; \mathbb{Z}_2)$. By Poincaré duality, we see that $u_i = 0$.