= Spin-c structure
{c}
{title2=$\operatorname{Spin}^c$}
A Spin-c structure on an oriented rank-$n$ real <vector bundle> is a lift of its oriented frame bundle to $\operatorname{Spin}^c(n)=(\operatorname{Spin}(n)\times U(1))/\{(1,1),(-1,-1)\}$. Its <determinant line bundle> is associated with $[s,z]\mapsto z^2$; its <First Chern class> reduces modulo two to the second <Stiefel–Whitney class>. When such structures exist, their isomorphism classes form an affine space over $H^2$; tensoring by a complex line bundle of first Chern class $h$ changes the determinant class by $2h$. On a closed oriented three-manifold they always exist. On a four-manifold, the determinant class is characteristic: $\langle c_1(\mathfrak s),v\rangle\equiv v\cdot v\pmod2$.
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