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?