Pin structures

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== Introduction ==
== 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>;
<wikitex>;
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
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.
+
+
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}}.
</wikitex>
</wikitex>
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$\RP^{4k}$ admits two $Pin^+$-structures and no $Pin^-$-structure.
$\RP^{4k}$ admits two $Pin^+$-structures and no $Pin^-$-structure.
+
$\RP^{4k+1}$ admits no $Pin^+$-structure 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.
$\RP^{4k+2}$ admits no $Pin^+$-structure and two $Pin^-$-structures.
+
$\RP^{4k+3}$ admits two $Pin^+$-structures and two $Pin^-$-structures.
$\RP^{4k+3}$ admits two $Pin^+$-structures and two $Pin^-$-structures.
</wikitex>
</wikitex>

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 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 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.

For more information on Pin-manifolds, including a computation of the low-dimensional bordism groups of Pin-manifolds, see [Kirby&Taylor1990].

2 Examples

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.


3 References

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