For a compact Riemannian manifold $M$, we know that the Hodge map $\ast$ and Laplacian $\Delta$ commute. From Hodge decomposition and its implied isomorphism between harmonic forms and cohomology classes we now that an indued on the cohomology ring $H(M)$, which is usually denoted again by $\ast$.

Now one could also naively define a map $$ \ast:H(M) \to H(M), ~~~~~~~~~~~~~ [\omega] \mapsto [\ast(\omega)], $$ and hope that they coincide. The thing is, as far I can tell, this map doesn't even seem to be well-defined. Can someone please prove whether or not it is well-defined.

(In the case that it is not well-defined, any philosophical motivation for why the naive approach doesn't work would be appreciated.)

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