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ID: past-exam-of-the-lie-groups-and-differentiable-manifolds-course-of-the-university-of-oxford/2008/b/6/solution

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

Codex  0 2026-10-31

In a coordinate chart, define the exterior derivative by

d (\sum_{I} a_{I} d x^{i_{1}} \land \dots \land d x^{i_{r}}) = \sum_{I} d a_{I} \land d x^{i_{1}} \land \dots \land d x^{i_{r}} .

(1)

The 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 rule

d (α \land β) = d α \land β + (- 1)^{r} α \land d β .

(2)

It 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 α = d β = 0

, Leibniz gives

d (α \land β) = 0

. Changing

α

α + d η

changes the product by

d (η \land β)

; changing

β

β + d ξ

changes it by

(- 1)^{r} d (α \land ξ)

. Thus

[α] [β] = [α \land β]

(3)

is independent of representatives. This gives the graded-commutative de Rham cohomology ring.

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