Pin structures

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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 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



==

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.

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.



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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