TITLE: Pin$^+$ and Pin$^−$ structure for manifolds in any dimensions
QUESTION [7 upvotes]: For an oriented $d$-manifold $M$, we can ask whether the manifold
admits a Spin structure, say, if the transition functions for the
tangent bundle, which take values in $SO(d)$, can be lifted to $\operatorname{Spin}(d)$ while preserving the cocycle condition on triple overlaps of coordinate charts.
For an unoriented $d$-manifold $M$, we can ask whether the manifold
admits Pin$^+$ or Pin$^−$ structures (that is, lifts of transition functions to
either $\operatorname{Pin}^+(d)$ or $\operatorname{Pin}^−(d)$ from $O(d)$. This is analogous to lifting the transition functions to $\operatorname{Spin}(d)$ from $SO(d)$ for the spin manifold). 
If the manifold $M$ is orientable, then the conditions for Pin$^+$ or Pin$^−$ structures are the same. So it reduces to the old condition that $M$ admits a Spin structure or not.
Every 2-dimensional manifold admits a Pin$^−$ structure, but not necessarily a Pin$^+$ structure.
Every 3-dimensional manifold admits a Pin$^-$ structure, but not necessarily a Pin$^+$ structure.

Question: For some any other $d$, say $d = 0, 1, 4,$ etc. are there statements like: every $d$-dimensional manifold admits a Pin$^-$ structure? Or, every $d$-dimensional manifold admits a Pin$^+$ structure? Or are there useful conditions like SW classes $w_2+w_1^2=0$ like the case for $d = 2, 3$? What are these conditions in other dimensions?

P.S. The original post on MSE received almost no attention for a week.

REPLY [11 votes]: For $d \geq 4$, let $M_d = (S^1)^{d-4}\times\mathbb{CP}^2$. As tori are parallelisable, $w(M_d) = w(\mathbb{CP}^2)$, in particular $w_1(M_d) = 0$ and $w_2(M_d) \neq 0$, so $M_d$ does not admit a Spin, Pin$^+$, or Pin$^-$ structure. 
Therefore, there is no $d \geq 4$ such that every $d$-dimensional manifold admits a Pin$^+$/Pin$^-$ structure.