Pin structures
(3 intermediate revisions by 2 users not shown) | |||
Line 1: | Line 1: | ||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
{{Stub}} | {{Stub}} | ||
== Introduction == | == Introduction == | ||
Line 25: | Line 13: | ||
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. | ||
+ | |||
+ | 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 | For more information on $Pin$-manifolds, including a computation of the low-dimensional bordism groups | ||
Line 39: | Line 32: | ||
== Examples == | == Examples == | ||
<wikitex>; | <wikitex>; | ||
+ | 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 $k\ge 1$: | For $k\ge 1$: | ||
Latest revision as of 21:22, 2 July 2011
This page has not been refereed. The information given here might be incomplete or provisional. |
[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 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.
There is also a group which is a central extension of by . -structures are obstructed by the integral characteristic class , and if they exist, isomorphism classes of -structures are in bijection with .
For more information on -manifolds, including a computation of the low-dimensional bordism groups of -manifolds, see [Kirby&Taylor1990].
[edit] 2 Examples
In dimension , all orientable surfaces admit a -structure, and hence both - and -structures. The nonorientable surface admits a -structure, but admits a -structure if and only if is even.
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.
[edit] 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