Pin structures
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[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