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Complex spin structure

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  1. In mathematics a complex spin group SpinC(n) is a generalized form of a spin group

    Source: Wikipedia Unhappy with this fact? more info
    created by user picturefactobot on March 2, 2009
  2. Although not all manifolds admit a spin group, all 4-manifolds admit a complex spin group

    Source: Wikipedia Unhappy with this fact? more info
    created by user picturefactobot on March 2, 2009
  3. The complex spin group can be defined by the exact sequence On a 4-manifold M with a complete set of open neighborhoods Ua, the 2nd Stiefel-Whitney class w₂ (TM)∈ H² (M; \mathbbZ₂) is the obstruction to finding a global spin structure

    Source: Wikipedia Unhappy with this fact? more info
    created by user picturefactobot on March 2, 2009
  4. In other words, if w2=0 then one can find a global spin structure Spin(4) by lifting a cocycle \gₐb:Uₐ ∪ Ub → SO(4)\ to the simply-connected group Spin(4)

    Source: Wikipedia Unhappy with this fact? more info
    created by user picturefactobot on March 2, 2009
  5. These lifted cocycles (as well as the original cocycles) hₐbsatisfy the cocycle condition, However, if w₂≠ 0, the cocycle condition must be expanded to include the opposite 'orientation', In this case the concept of a spin structure must be gen

    Source: Wikipedia Unhappy with this fact? more info
    created by user picturefactobot on March 2, 2009
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