Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 115 / 2

In local coordinates, forthe exterior derivative isApplying $d$ again gives symmetric second partial derivatives contracted with the antisymmetric wedge $d{x}^k\wedge d{x}^j$, hence $d^2=0$. Expanding coefficients and moving $d\alpha$ past a degree-$r$ form gives the graded Leibniz rule

For a function $f$ and smooth $F:X\to Y$, the chain rule givesEvery differential form is locally a sum of products $f_{0}d{f}_1\wedge \cdots \wedge d{f}_r$. Since pullback preserves products and wedge products, the function case and the graded Leibniz rule implyfor every form.

The de Rham cohomology of $X$ isBecause pullback commutes with $d$, it sends closed forms to closed forms and exact forms to exact forms. Thus a smooth map induces

Let $H:X\times I\to Y$ be a smooth homotopy and write $H_t(x)=H(x,t)$. If $V={\partial }_t$, Cartan's magic formula givesIntegrating defines a degree-minus-one operator $K$ satisfyingFor closed $\omega$, the difference is exact, so smoothly homotopic maps induce the same map on de Rham cohomology.

If $F:X\to Y$ is a homotopy equivalence with inverse up to homotopy $G$, functoriality and homotopy invariance giveHence $F^{\ast }$ is an isomorphism.

We induct on $n$. The claim is immediate for ${\mathbb{CP}}^0$. In the stated cover, $U=U_0\cong {\mathbb{C}}^n$ is contractible. The homotopydeformation retracts $V$ onto the hyperplane $z_0=0$, which is ${\mathbb{CP}}^{n-1}$. Moreover,deformation retracts onto $S^{2n-1}$.

The Mayer--Vietoris sequence for the de Rham complex containsFor odd $i>1$, the group on the right vanishes by contractibility and the induction hypothesis, while the group on the left vanishes because $i-1$ is positive and even and is neither $0$ nor $2n-1$. Thus the middle group vanishes. For $i=1$, the preceding mapis surjective because all three spaces are connected, so the connecting map into $H^1({\mathbb{CP}}^n)$ is zero. Therefore all odd de Rham cohomology groups of ${\mathbb{CP}}^n$ vanish.