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.

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.

For an oriented d-manifold M, we can ask whether the manifold admits Spin structure, say, if the transition functions for the tangent bundle, which take values in SO(d), can be lifted to 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 Pin+(d) or Pin−(d) from O(d). This is analogous to lifting the transition functions to 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 in ME receives almost no attention for a week.

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.