# Manifolds with $w_1(TM)\cup w_1(TM)=0$ and $w_2(TM)=0$ but $w_1(TM)\neq 0$

For a generic dimension $$d$$, is there an nonorientable manifold $$M$$ (i.e. $$w_1(TM)\neq 0$$) with vanishing $$w_1(TM)\cup w_1(TM)$$ and $$w_2(TM)$$, i.e.,

$$w_1(TM)\cup w_1(TM)=0, ~~~~~ w_2(TM)=0, ~~~~~w_1(TM)\neq 0?$$

Here $$w_i(TM)$$ is the $$i^{\text{th}}$$ Stiefel-Whitney class of the tangent bundle of the manifold $$M$$. For $$d=2$$, the Klein bottle is an example.

If such manifolds exist, what kind of structure do they carry? For example, if $$w_1(TM)=0$$, and $$w_2(TM)=0$$, then the manifold can be equipped with a spin structure, and we say it is a spin manifold. I'd like to see what is the corresponding structure in the above more complicated case.

A smooth manifold $$M$$ admits a pin$$^+$$ structure if and only if $$w_2(M) = 0$$, and a pin$$^-$$ structure if and only if $$w_1(M)^2 + w_2(M) = 0$$; see this page for some information on pin structures. The manifolds you are enquiring about satisfy both conditions and hence admit pin$$^+$$ and pin$$^-$$ structures. However, as $$w_1(M) \neq 0$$, they are non-orientable, so they do not admit spin structures.
There exist manifolds of arbitrary dimension which satisfy your requirements. For example, $$K\times S^n$$ where $$K$$ is the Klein bottle.