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)$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}SO(n)$ to the double (universal) cover $Spin(n)$$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^+$$Pin^+$-structure on $M$$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.

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