Pin structures
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The obstruction for existence of a $Pin^+$-structure on $M$ is the characteristic class $w_2(M)$. | 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 | + | 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 $w_2(M)+w_1(M)^2$. | 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 | + | 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. | A $Pin^\pm$-structure together with an orientation is equivalent to a $Spin$-structure. |
Revision as of 11:16, 1 April 2011
This page has not been refereed. The information given here might be incomplete or provisional. |
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 to the double (universal) cover . The non-connected group has two nontrivial central extensions (double covers) by with different group structures, denoted by and . A -structure on a manifold is thus a a reduction of the structure group of its tangent bundle from to , and similarly for -structures.
A -structure on is equivalent to a -structure on , a -structure on is equivalent to a -structure on .
The obstruction for existence of a -structure on is the characteristic class . If does admit structures, then the set of isomorphism classes of -structures on is acted upon freely and transitively by .
The obstruction for existence of a -structure on is the characteristic class . If does admit structures, then the set of isomorphism classes of -structures on is acted upon freely and transitively by .
A -structure together with an orientation is equivalent to a -structure.
For more information on -manifolds, including a computation of the low-dimensional bordism groups of -manifolds, see [Kirby&Taylor1990].
2 Examples
For :
admits two -structures and no -structure.
admits no -structure and no -structure.
admits no -structure and two -structures.
admits two -structures and two -structures.
3 References
- [Kirby&Taylor1990] R. C. Kirby and L. R. Taylor, 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