In Petersen's Riemannian geometry text, he defines the Hodge operator $*: \Omega^k(M) \to \Omega^{n-k} (M)$ in the standard way.
He then proves (Lemma 26, Chap 7) that $*^2: \Omega^k(M) \to \Omega^k(M)$ is multiplication by $(-1)^{k(n-k)}.$
So far no problems.
However, he seems to argue that this lemma implies that the Hodge star gives an isomorphism $H^k(M) \to H^{n-k}(M),$ where we are considering the de Rham cohomology groups.
It is clear by the lemma that we have an isomorphism from $\Omega^k \to \Omega^{n-k}$ given by the Hodge star. But why must this descend to an isomorphism on cohomology?
I guess one would need to show that the Hodge star maps closed forms to closed forms, and exact forms to exact forms? Is this clear from the lemma, or am I to conclude that Pedersen is foreshadowing a theorem to come?
Best Answer
No, what is obvious is that $\star$ gives an isomorphism from the space of harmonic $k$-forms ($\mathscr H^k(M)$) to the space of harmonic $(n-k)$-forms. The Hodge Theorem gives an isomorphism $H^k(M) \overset{\cong}{\to} \mathscr H^k(M)$. (I don't have Petersen's book here with me, so I can't check to see his context.)