Definitions

# Complex spin structure

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

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

On a 4-manifold M with a complete set of open neighborhoods {Ua}, the 2nd Stiefel-Whitney class $w_2 \left(T_M\right)in H^2 \left(M; mathbb\left\{Z\right\}_2\right)$ 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 $\left\{g_\left\{ab\right\}:U_a cup U_b to operatorname\left\{SO\right\}\left(4\right)\right\}$ to the simply-connected group Spin(4). These lifted cocycles (as well as the original cocycles) $h_\left\{ab\right\}$satisfy the cocycle condition,

$h_\left\{ab\right\}circ h_\left\{bc\right\} circ h_\left\{ca\right\}= 1.$

However, if $w_2neq 0$, the cocycle condition must be expanded to include the opposite 'orientation',

$h_\left\{ab\right\}circ h_\left\{bc\right\} circ h_\left\{ca\right\}= pm 1.$
In this case the concept of a spin structure must be generalized to a complex spin structure, and the original cocycles $g_\left\{ab\right\}$ must be lifted to this new structure. In four dimensions, this complex spin group can be formally defined as
$operatorname\left\{Spin\right\}^\left\{C\right\}\left(4\right)= operatorname\left\{U\right\}\left(1\right)timesoperatorname\left\{Spin\right\}\left(4\right) / pm 1.$

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

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

## References

Search another word or see Complex spin structureon Dictionary | Thesaurus |Spanish