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An orientable smooth manifold $X$ admits a $\text{Spin}^c$ structure iff its second Stiefel-Whitney class $w_2 \in H^2(X, \mathbb{F}_2)$ is the reduction of a class $c_1 \in H^2(X, \mathbb{Z})$. This condition is equivalent to the condition that the third integral Stiefel-Whitney class $W_3 = \beta w_2 \in H^3(X, \mathbb{Z})$ vanishes, and I guess this is ...



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