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.

closedthree-manifold admits a Pin$^-$ structure. $\endgroup$ – Michael Albanese Dec 24 '16 at 0:11