Pin structures
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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 to its universal double cover . The non-connected group has two nontrivial central extensions (double covers) by
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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.
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.
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.