Pin structures

Latest revision as of 21:22, 2 July 2011

[edit] 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 $<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> {{Stub}} == Introduction == <wikitex>; For an oriented manifold a [[Wikipedia:Spin_structure|spin structure]] is a reduction of the structure group of the tangent bundle from the connected topological group $SO(n)$ to its universal double cover $Spin(n)$. The non-connected group $O(n)$ has two nontrivial central extensions (double covers) by $\Zz_2$ </wikitex> == <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. 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 in bijection with $H^1(M;\Zz_2)$. 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 in bijection with $H^1(M;\Zz_2)$. A $Pin^\pm$-structure together with an orientation is equivalent to a $Spin$-structure. </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 $Spin(n)$. 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 [Kirby&Taylor1990].

[edit] 2 Examples

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

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.

[edit] 3 References

• [Kirby&Taylor1990] R. C. Kirby and L. R. Taylor, $Pin$ 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