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This is the kind of a natural question which can come to mind after completing the standard course in differential geometry and homology theory: lety us start with a smooth manifold $M$. One can construct the differential $d$ on differential forms which satisfies $d^2=0$. With the help of this differnetial one can consider de Rham (cochain) complex and one naturally arrives to the de Rham cohomology. However, it is also possible to consider the codifferential $\delta$ defined as $\pm \star d \star$ where $\star$ is the Hodge star operator. However, as far as I understood, the Hodge star depends on the choice of metric (Riemannian) structure on $M$. This codifferential satisfies $\delta^2=0$ so also gives rise to some (chain) complex and therefore to some homology theory.

Question: what is the connection of homology corresponding to this codifferential with (for instance) singular homology? In patricular, is it true that this homology does not depend from the choice of metric structure?

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The Hodge star is an isomorphism and you've transported the differential along that isomorphism, so you just get de Rham cohomology with the indices backwards.

A more fun thing to do is to study both differentials at the same time, which leads to Hodge theory.

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