Motivation
The question arises because in the paper arXiv:1005.2405 the authors pull back the Hodge decomposition on a simplicial complex from $C^1$ to $(C^0)^N$ for some integer $N$ via some homomorphism $D:(C^0)^N \rightarrow C^1$ that does not depend on the inner product, and they claim that the decomposition obtained in $(C^0)^N$ is canonical. I haven't checked the details yet, but this seems strange, since the Hodge decomposition on $C^1$ seems to be inner-product dependent - which is my question.
Some notation on simplicial homology and combinatorial Hodge theory:
Let $K$ be a finite dimensional simplicial complex, and denote by $O_k$ the finite set of oriented $k$-simplices in $K$. For any $\sigma \in O_k$, $\sigma'$ is the oriented simplex with same vertices and opposite orientation as $\sigma$.
Let $C_k$ be the vector space of $k$-chains, that is $$ C_k = \{ f: O_k \rightarrow \mathbb{R}: f(\sigma') = -f(\sigma) \, \forall \sigma \in O_k \} $$ with the obvious vector sum and scalar multiplication.
Let $C^k = \text{Hom}(C_k, \mathbb{R})$ be the dual space of $C_k$ and let $\delta_k: C^k \rightarrow C^{k+1}$ be the coboundary homomorphism.
Endow each $C^k$ space with an inner product, and let $\delta_k^{\dagger}: C^{k+1} \rightarrow C^k$ be the adjoint of the $k$-th coboundary homomorphism with respect to such inner product.
Finally, define the Laplacian homomorphism as $$ \Delta_k = \delta_{k-1}\circ\delta_{k-1}^{\dagger} + \delta_{k}^{\dagger} \circ \delta_{k}: C^k \rightarrow C^k $$
- call the $k$-cochains in $\text{im }\delta_{k-1} \subset C^k$ exact
- call the $k$-cochains in $\text{ker }\delta_{k} \subset C^k$ closed
- since $\delta_k \circ \delta_{k-1} = 0$, exact $\Rightarrow$ closed
- $\text{im }\delta_k^{\dagger} = \left(\text{ker }\delta_k \right)^{\perp}$
- $\text{ker }\delta_{k-1}^{\dagger} = \left(\text{im } \delta_{k-1} \right)^{\perp}$
The Hodge decomposition theorem for simplicial complexes says that
$$ C^k = \text{ker }\Delta_k \oplus \text{im }\delta_{k-1} \oplus \text{im }\delta_k^{\dagger} $$
furthermore, $$\text{harmonic }k\text{-cochains} := \text{ker }\Delta_k = \text{ker }\delta_k \, \cap \, \text{ker }\delta_{k-1}^{\dagger} = \text{ker }\delta_k \, \cap \, \left(\text{im } \delta_{k-1} \right)^{\perp}$$
Question
Is this decomposition canonical, or does it depend on the chosen inner product?
At a first glance, $\text{im }\delta_{k-1}$ is canonical, hence the exact component is canonical, whereas $\text{ker }\Delta_k$ and $\text{im }\delta_k^{\dagger}$ depend on the chosen inner product (via the adjoint of the coboundary homomorphism), so the corresponding components seem inner-product dependent. Is this correct?
Edited to include corrections from comments.