TITLE: Inner product of De Rham cohomology classes
QUESTION [6 upvotes]: 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?
Thanks

REPLY [6 votes]: Assuming you're working on a compact Riemannian manifold, each cohomology class is represented by a unique harmonic form, so use the harmonic forms.