Suppose $X$ is an infinite countable CW complex which satisfies the following property: for all $k$-cells $e$, the number of $(k+1)$-cells incident to $e$ is at most $c_k$, where the latter is some number that depends on $k$. Let $X_k$ be the set of $k$-cells.

Let $\ell^2_k(X)$ be the set of functions $a_k : X_k \to \Bbb R$, such that the series $$ \sum_{e \in X_k} a_k(e)^2 $$ converges. Then the incidence bound assumption of $X$ implies that coboundary operator $$ \delta: \ell^2_k(X) \to \ell^2_{k+1}(X) $$ is defined (this uses the same formula that arises when defining the cellular cochain complex of $X$).

When $\dim X =1$ this construction was introduced by Dodziuk and Kendall in

Dodziuk, J.(1-CUNYG); Kendall, W. S.(4-STRA) Combinatorial Laplacians and isoperimetric inequality. From local times to global geometry, control and physics (Coventry, 1984/85), 68–74, Pitman Res. Notes Math. Ser., 150, Longman Sci. Tech., Harlow, 1986.

Questions

1. Has this construction been investigated in the generality described above?

2. How is the cohomology of this complex related to the usual cellular cohomology of $X$?

3. Under what conditions is this cohomology finite dimensional?

4. How does the above relate to other notions of $L^2$-cohomology?