Pin structures
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== Introduction == | == Introduction == | ||
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For an oriented manifold a [[Wikipedia:Spin_structure|spin structure]] is a reduction of the structure group of its | For an oriented manifold a [[Wikipedia:Spin_structure|spin structure]] is a reduction of the structure group of its | ||
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tangent bundle from the connected topological group $SO(n)$ to the double (universal) cover $Spin(n)$. | 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 $\Zz_2$ | The non-connected group $O(n)$ has two nontrivial central extensions (double covers) by $\Zz_2$ | ||
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A $Pin^+$-structure on a manifold is thus a a reduction of the structure group of its tangent bundle | 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. | from $O(n)$ to $Pin^+(n)$, and similarly for $Pin^-$-structures. | ||
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+ | 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 $w_2(M)$. | The obstruction for existence of a $Pin^+$-structure on $M$ is the characteristic class $w_2(M)$. | ||
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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. | ||
− | + | For more information on $Pin$-manifolds, including a computation of the low-dimensional bordism groups | |
+ | of $Pin$-manifolds, see {{cite|Kirby&Taylor1990}}. | ||
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$\RP^{4k}$ admits two $Pin^+$-structures and no $Pin^-$-structure. | $\RP^{4k}$ admits two $Pin^+$-structures and no $Pin^-$-structure. | ||
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$\RP^{4k+1}$ admits no $Pin^+$-structure and no $Pin^-$-structure. | $\RP^{4k+1}$ admits no $Pin^+$-structure and no $Pin^-$-structure. | ||
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$\RP^{4k+2}$ admits no $Pin^+$-structure and two $Pin^-$-structures. | $\RP^{4k+2}$ admits no $Pin^+$-structure and two $Pin^-$-structures. | ||
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$\RP^{4k+3}$ admits two $Pin^+$-structures and two $Pin^-$-structures. | $\RP^{4k+3}$ admits two $Pin^+$-structures and two $Pin^-$-structures. | ||
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Revision as of 10:09, 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 in bijection with .
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 in bijection with .
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