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A Spin-c structure on an oriented rank-$n$ real vector bundle is a lift of its oriented frame bundle to ${\mathrm{Spin}}^c(n)=(\mathrm{Spin}(n)\times U(1))/\{(1,1),(-1,-1)\}$. 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: $⟨c_1(\mathfrak{s}),v⟩\equiv v\cdot v(\mathrm{mod}2)$.

A complex line bundle has a canonically oriented underlying real vector bundle of rank two. Its first Stiefel–Whitney class vanishes, and its second Stiefel–Whitney class is the mod-two reduction of its First Chern class.

The symplectic canonical class is the negative First Chern class of the tangent bundle for a compatible almost complex structure. The contractibility of compatible almost complex structures makes it independent of that choice. It is the class appearing in the symplectic adjunction formula.