User:Adama44/complexspin

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.

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

On a 4-manifold M with a complete set of open neighborhoods {Ua}, 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 $$\{g_{ab}:U_a \cup U_b \to \operatorname{SO}(4)\}$$ 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 $$w_2\neq 0$$, 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
 * $$\operatorname{Spin}^{C}(4)= \operatorname{U}(1)\times\operatorname{Spin}(4) / \pm 1.$$

In the same manner that Spin(4) is a double cover of SO(4), SpinC(4) admits the double-cover projection
 * $$\operatorname{Spin}^{C}(4)\to\operatorname{U}(1)\times\operatorname{SO}(4).$$