Complex spin structure

In mathematics a complex spin group Spin^"C"("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.Scorpan, A., 2005 The Wild World of 4 Manifolds]

The complex spin group can be defined by the exact sequence :$$1 o mathbb{Z}_2 o operatorname{Spin}^{C}(n) o operatorname{SO}(n) imesoperatorname{U}(1) o 1.

On a 4-manifold "M" with a complete set of open neighborhoods {"U"_"a"}, the 2nd Stiefel-Whitney class $$w_2 (T_M)in H^2 (M; mathbb{Z}_2) is the obstruction to finding a global spin structure. In other words, if w₂=0 then one can find a global spin structure Spin(4) by lifting a cocycle $${g_{ab}:U_a cup U_b o operatorname{SO}(4)}to the simply-connected group Spin(4). These lifted cocycles (as well as the original cocycles) $$h_{ab}satisfy the cocycle condition, :$$h_{ab}circ h_{bc} circ h_{ca}= 1.

However, if $$w_2 eq 0, the cocycle condition must be expanded to include the opposite 'orientation',:$$h_{ab}circ h_{bc} circ h_{ca}= pm 1.In this case the concept of a spin structure must be generalized to a complex spin structure, and the original cocycles $$g_{ab} must be lifted to this new structure. In four dimensions, this complex spin group can be formally defined as:$$operatorname{Spin}^{C}(4)= operatorname{U}(1) imesoperatorname{Spin}(4) / pm 1.

In the same manner that Spin(4) is a double cover of SO(4), Spin^C(4) admits the double-cover projection:$$operatorname{Spin}^{C}(4) ooperatorname{U}(1) imesoperatorname{SO}(4).

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