Is there a well-defined inner product between cohomology classes? In particular, is it possible to extend the Hodge inner product? If I try, I obtain this:
$$\int *(\omega + d\lambda)\wedge (\sigma + d\mu) = \int (*\omega\wedge\sigma + *d\lambda\wedge\sigma + *\omega\wedge d\mu + *d\lambda\wedge d\mu)$$
Is it possible to discard the last terms as an exact form, or to define anyway a product that does something similar?