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- Jun 20, 2014

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Suppose $M$ is a smooth path-connected manifold. Consider the differential form

$$\nu = \Re\left\{\frac{1}{2\pi i} \frac{dz}{z}\right\}$$

which generates $H^1_{dR}(\Bbb C^\times)$, the first de Rham cohomology of $\Bbb C^\times$. Show that every smooth map $f : M \to \Bbb C^\times$ can be lifted to smooth map $M\to \Bbb C$ via the exponential map, provided that the image of $\nu$ under $f^* : H^1_{dR}(\Bbb C^\times) \to H^1_{dR}(M)$ is zero.

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