User:Adama44/complexspin
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In mathematics a complex spin group SpinC(n) is a generalized form of a spin group. Although not all manifolds admit a spin group, all 4-manifolds admit a complex spin group.[1]
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 is the obstruction to finding a global spin structure. In other words, if w2=0 then one can find a global spin structure Spin(4) by lifting a cocycle to the simply-connected group Spin(4). These lifted cocycles (as well as the original cocycles) satisfy the cocycle condition,
However, if , the cocycle condition must be expanded to include the opposite 'orientation',
In this case the concept of a spin structure must be generalized to a complex spin structure, and the original cocycles must be lifted to this new structure. In four dimensions, this complex spin group can be formally defined as
In the same manner that Spin(4) is a double cover of SO(4), SpinC(4) admits the double-cover projection
Notes
[edit]- ^ Scorpan, A., 2005 The Wild World of 4 Manifolds
References
[edit]- Scorpan, Alexandru (2005), The Wild World of 4-Manifolds, Providence, Rhode Island: American Mathematical Society
- Asselmeyer-Maluga, Torsten; Brans, Carl H (2007), Exotic Smoothness and Physics: Differential Topology and Spacetime Models, New Jersey, London: World Scientific