Past exam of the lie Groups and Differentiable Manifolds course of the University of Oxford / 2008 / B / 6 / Solution

In a coordinate chart, define the exterior derivative byThe chain rule makes this independent of coordinates. It is real linear, increases form degree by one, agrees with the differential on functions, satisfies $d^2=0$, and obeys the graded Leibniz ruleIt also commutes with pullback by a smooth map. These properties characterize it locally and hence globally.

de Rham cohomology is the quotient of closed forms by exact forms. If $d\alpha =d\beta =0$, Leibniz gives $d(\alpha \wedge \beta )=0$. Changing $\alpha$ to $\alpha +d\eta$ changes the product by $d(\eta \wedge \beta )$; changing $\beta$ to $\beta +d\xi$ changes it by $(-1{)}^{r}d(\alpha \wedge \xi )$. Thusis independent of representatives. This gives the graded-commutative de Rham cohomology ring.