And so, like any chain map, the Lie derivative defines homomorphisms on cohomology: . But which homomorphism does it define?
Well, it turns out that Cartan’s formula comes in handy here as well, for it’s exactly what we need to say that the Lie derivative is null-homotopic. And like any null-homotopic map, it defines the zero map on cohomology. That is, if we take some closed -form , which defines a cohomology class in — any cohomology class has such a representative -form — and hit it with , the result is an exact -form.
Actually, this shouldn’t be very surprising, considering Cartan’s formula. Indeed, we can calculate directly
since by assumption is closed, which means that .