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.

The complex spin group can be defined by the exact sequence

$1\; to\; mathbb\{Z\}\_2\; to\; operatorname\{Spin\}^\{C\}(n)\; to\; operatorname\{SO\}(n)timesoperatorname\{U\}(1)\; to\; 1.$

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 $\{g\_\{ab\}:U\_a\; cup\; U\_b\; to\; operatorname\{SO\}(4)\}$ 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 $w\_2neq\; 0$, 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

$operatorname\{Spin\}^\{C\}(4)=\; operatorname\{U\}(1)timesoperatorname\{Spin\}(4)\; /\; pm\; 1.$

In the same manner that Spin(4) is a double cover of SO(4), Spin^C(4) admits the double-cover projection

$operatorname\{Spin\}^\{C\}(4)tooperatorname\{U\}(1)timesoperatorname\{SO\}(4).$

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