Pin structures

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1 Introduction

For an oriented manifold a spin structure is a reduction of the structure group of its tangent bundle from the connected topological group $SO(n)$${{Stub}} == Introduction == <wikitex>; For an oriented manifold a [[Wikipedia:Spin_structure|spin structure]] is a reduction of the structure group of its tangent bundle from the connected topological group $SO(n)$ to the double (universal) cover $Spin(n)$. The non-connected group $O(n)$ has two nontrivial central extensions (double covers) by $\Zz_2$ with different group structures, denoted by $Pin^+(n)$ and $Pin^-(n)$. A $Pin^+$-structure on a manifold is thus a a reduction of the structure group of its tangent bundle from $O(n)$ to $Pin^+(n)$, and similarly for $Pin^-$-structures. A $Pin^+$-structure on $M$ is equivalent to a $Spin$-structure on $TM\oplus 3 det(TM)$, a $Pin^-$-structure on $M$ is equivalent to a $Spin$-structure on $TM\oplus det(TM)$. The obstruction for existence of a $Pin^+$-structure on $M$ is the characteristic class $w_2(M)$. If $M$ does admit $Pin$ structures, then the set of isomorphism classes of $Pin^+$-structures on $M$ is acted upon freely and transitively by $H^1(M;\Zz_2)$. The obstruction for existence of a $Pin^-$-structure on $M$ is the characteristic class $w_2(M)+w_1(M)^2$. If $M$ does admit $Pin$ structures, then the set of isomorphism classes of $Pin^-$-structures on $M$ is acted upon freely and transitively by $H^1(M;\Zz_2)$. A $Pin^\pm$-structure together with an orientation is equivalent to a $Spin$-structure. There is also a group $Pin^c$ which is a central extension of $O(n)$ by $S^1$. $Pin^c$-structures are obstructed by the integral characteristic class $W_3(M)$, and if they exist, isomorphism classes of $Pin^c$-structures are in bijection with $H^2(M)$. For more information on $Pin$-manifolds, including a computation of the low-dimensional bordism groups of $Pin$-manifolds, see {{cite|Kirby&Taylor1990}}. </wikitex> == Examples == <wikitex>; For $k\ge 1$: $\RP^{4k}$ admits two $Pin^+$-structures and no $Pin^-$-structure. $\RP^{4k+1}$ admits no $Pin^+$-structure and no $Pin^-$-structure. $\RP^{4k+2}$ admits no $Pin^+$-structure and two $Pin^-$-structures. $\RP^{4k+3}$ admits two $Pin^+$-structures and two $Pin^-$-structures. </wikitex> == References == {{#RefList:}} [[Category:Theory]]SO(n)$ to the double (universal) cover $Spin(n)$$Spin(n)$. The non-connected group $O(n)$$O(n)$ has two nontrivial central extensions (double covers) by $\Zz_2$$\Zz_2$ with different group structures, denoted by $Pin^+(n)$$Pin^+(n)$ and $Pin^-(n)$$Pin^-(n)$. A $Pin^+$$Pin^+$-structure on a manifold is thus a a reduction of the structure group of its tangent bundle from $O(n)$$O(n)$ to $Pin^+(n)$$Pin^+(n)$, and similarly for $Pin^-$$Pin^-$-structures.

A $Pin^+$$Pin^+$-structure on

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

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$M$ 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

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$M$ is the characteristic class $w_2(M)$$w_2(M)$. If

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$M$ does admit $Pin$$Pin$ structures, then the set of isomorphism classes of $Pin^+$$Pin^+$-structures on

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$M$ 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

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$M$ is the characteristic class $w_2(M)+w_1(M)^2$$w_2(M)+w_1(M)^2$. If

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$M$ does admit $Pin$$Pin$ structures, then the set of isomorphism classes of $Pin^-$$Pin^-$-structures on

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$M$ 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

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