= Smooth exponential sequence
On a <smooth manifold>, the sequence of <sheaves of abelian groups>
$$
0\to\underline{\mathbb Z}\to\mathcal C^\infty_{\mathbb C}\xrightarrow{\exp(2\pi i\,\cdot)}\mathcal C^{\infty,*}_{\mathbb C}\to1
$$
is exact. The additive smooth-function sheaf is a <fine sheaf>, so the connecting map identifies $H^1(X,\mathcal C^{\infty,*}_{\mathbb C})$ with $H^2(X,\mathbb Z)$. The former classifies smooth <complex line bundles> and the connecting class is their <First Chern class>.
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