User:Adama44/complexspin

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

$1\to \mathbb {Z} _{2}\to \operatorname {Spin} ^{C}(n)\to \operatorname {SO} (n)\times \operatorname {U} (1)\to 1.$

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 $w_{2}\neq 0$, 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), Spin^C(4) admits the double-cover projection

$\operatorname {Spin} ^{C}(4)\to \operatorname {U} (1)\times \operatorname {SO} (4).$

References

• 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