4-manifold with $w_1\neq 0$, $w_1^2=0$, $w_2\neq 0$ I wonder if there is a 4-manifold whose Stiefel-Whitney
classes satisfy $w_1\neq 0$, $w_2\neq 0$, and $w_1^2=0$?
There is no 3-manifold whose Stiefel-Whitney
classes are given by the above. For $\mathbb{R}P^4$, $w_1^2\neq 0$.

Let $M$ be the non-orientable $S^3$ bundle over $S^1$. Its $\Bbb Z/2$ Betti numbers are $b_1 = 1, b_2 = 0, b_3 = 1$. It's non-orientable, so $w_1 \neq 0$, but clearly $w_1^2 = 0$. Now take $M \# \Bbb{CP}^2$. $\Bbb{CP}^2$ has $w_1 = 0$ but $w_2 \neq 0$. 
