# Pin structures

## 1 Introduction


A $Pin^+$$Pin^+$-structure on $M$$M$ is equivalent to a $Spin$$Spin$-structure on $TM\oplus 3 det(TM)$$TM\oplus 3 det(TM)$, a $Pin^-$$Pin^-$-structure on $M$$M$ is equivalent to a $Spin$$Spin$-structure on $TM\oplus det(TM)$$TM\oplus det(TM)$.

The obstruction for existence of a $Pin^+$$Pin^+$-structure on $M$$M$ is the characteristic class $w_2(M)$$w_2(M)$. If $M$$M$ does admit $Pin$$Pin$ structures, then the set of isomorphism classes of $Pin^+$$Pin^+$-structures on $M$$M$ is acted upon freely and transitively by $H^1(M;\Zz_2)$$H^1(M;\Zz_2)$.

The obstruction for existence of a $Pin^-$$Pin^-$-structure on $M$$M$ is the characteristic class $w_2(M)+w_1(M)^2$$w_2(M)+w_1(M)^2$. If $M$$M$ does admit $Pin$$Pin$ structures, then the set of isomorphism classes of $Pin^-$$Pin^-$-structures on $M$$M$ is acted upon freely and transitively by $H^1(M;\Zz_2)$$H^1(M;\Zz_2)$.

A $Pin^\pm$$Pin^\pm$-structure together with an orientation is equivalent to a $Spin$$Spin$-structure.

There is also a group $Pin^c$$Pin^c$ which is a central extension of $O(n)$$O(n)$ by $S^1$$S^1$. $Pin^c$$Pin^c$-structures are obstructed by the integral characteristic class $W_3(M)$$W_3(M)$, and if they exist, isomorphism classes of $Pin^c$$Pin^c$-structures are in bijection with $H^2(M)$$H^2(M)$.

For more information on $Pin$$Pin$-manifolds, including a computation of the low-dimensional bordism groups of $Pin$$Pin$-manifolds, see [Kirby&Taylor1990].

## 2 Examples

In dimension $2$$2$, all orientable surfaces admit a $Spin$$Spin$-structure, and hence both $Pin^+$$Pin^+$- and $Pin^-$$Pin^-$-structures. The nonorientable surface $N_n := \#^n \RP^2$$N_n := \#^n \RP^2$ admits a $Pin^-$$Pin^-$-structure, but admits a $Pin^+$$Pin^+$-structure if and only if $n$$n$ is even.

For $k\ge 1$$k\ge 1$:

$\RP^{4k}$$\RP^{4k}$ admits two $Pin^+$$Pin^+$-structures and no $Pin^-$$Pin^-$-structure.

$\RP^{4k+1}$$\RP^{4k+1}$ admits no $Pin^+$$Pin^+$-structure and no $Pin^-$$Pin^-$-structure.

$\RP^{4k+2}$$\RP^{4k+2}$ admits no $Pin^+$$Pin^+$-structure and two $Pin^-$$Pin^-$-structures.

$\RP^{4k+3}$$\RP^{4k+3}$ admits two $Pin^+$$Pin^+$-structures and two $Pin^-$$Pin^-$-structures.

## 3 References

• [Kirby&Taylor1990] R. C. Kirby and L. R. Taylor, $Pin$$Pin$ structures on low-dimensional manifolds, Geometry of low-dimensional manifolds, 2 (Durham, 1989), Cambridge Univ. Press (1990), 177–242. MR1171915 (94b:57031) Zbl 0754.57020