Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 115 / 2

The integral of an exact differential form over the closed curve $S^1$ is zero by Stokes theorem, so the map is well defined on de Rham cohomology. It is surjective because the angular form $d\theta$ has integral $2\pi$. If a closed one-form $\alpha$ has zero integral, defineThe zero period makes this definition $2\pi$-periodic, and $dF=\alpha$. Thus the kernel is zero andis an isomorphism.

The overlap $U_0\cap U_1$ is ${\mathbb{C}}^{\ast }$ in the coordinate $z$, and radial projection $z\mapsto z/|z|$ is a deformation retraction onto $S^1$. The homotopy invariance of de Rham cohomology identifies its first de Rham cohomology with that of $S^1$, and the identification is compatible with integration around the unit circle. Hence the same integral map is an isomorphism.

Each $U_j$ is isomorphic to $\mathbb{C}$ and is therefore contractible. Every complex line bundle over a contractible paracompact space is trivial, so $L{|}_{U_0}$ and $L{|}_{U_1}$ admit the required trivializations.

Since $\beta =f^{-1}df$,The fundamental theorem of calculus and the chain rule then giveThus $F$ is constant.

Parametrize the unit circle by a loop $\gamma :[0,2\pi ]\to {\mathbb{C}}^{\ast }$. Part (d) and $f(\gamma (0))=f(\gamma (2\pi ))$ implyThe kernel of the complex exponential function is $2\pi i\mathbb{Z}$, sofor some $k\in \mathbb{Z}$. This integer is the winding number of $f{|}_{S^1}$.

For $h=z^{-k}f$,The scalar-valued form $h^{-1}dh$ is closed, andBy the period isomorphism from part (b), $[\alpha ]=0$ in complexified de Rham cohomology. Hence there is a smooth complex-valued function $\phi$ on $U_0\cap U_1$ with $\alpha =d\phi$.

Using $d\phi =h^{-1}dh$,The overlap ${\mathbb{C}}^{\ast }$ is connected, so $h{e}^{-\phi }=c$ for some $c\in {\mathbb{C}}^{\ast }$. Choose $C\in \mathbb{C}$ with $e^C=c$ and replace $\phi$ by $\phi +C$. Then $h=e^{\phi }$.

Let $({\rho }_0,{\rho }_1)$ be a smooth partition of unity subordinate to $(U_0,U_1)$. On $U_0$, extend ${\rho }_1\phi$ by zero away from the overlap, and on $U_1$ extend ${\rho }_0\phi$ similarly. DefineOn the overlap,and therefore

Changing the two local frames by the nowhere-zero functions ${\psi }_j$ replaces the transition function $f$ byWith the standard convention that the tautological bundle ${\mathcal{O}}_{{\mathbb{CP}}^1}(-1)$ has transition function $z$, its $k$th tensor power ${\mathcal{O}}_{{\mathbb{CP}}^1}(-k)$ has transition function $z^k$. ThusThis is the smooth classification of complex line bundles on the complex projective line by their winding number.